ºjºſº. |sſ- *Fºr lettº º ..º.º.º. . . . . . . . . . . . . . . . . ... • * * * * * * vºte tº ºffs ºf , sſ ºf eſs sº “s ºſ-ºs is 1-4 as - 'stºº ºvels ***.*.*.*, *. st º:::::::::::::::::::::::: º !. sºlº ºr ºf tº !: ºf rººf: - ; ºn- ºr “tº rººmſ ſºlººlººl *f - ºf ºſsºwſ: -i-. “ºhº;; etºn - tº is leſſ * * * * * * * * * * * * * * * * * * * ºf 1:1:1: ...---ºf-ºf-ſºw-ºn-tº-ºººººººº-ºº::istºr º C. tº ſet st º "F -º-º: 1, . . . . . ,, . . .tºſ sneſs tº be ſº tº . . . .” 14 ſervº, i-ºf-tºl-i eſºsºl, sº . . . * > * > y - 1:1: : -, -ºº ºff. ...-ſºº is •fººts ºustºsºsºls - ºr . . . . Rºſſ º iſ s | tºº-ºººººº... . . . . . ; ; ; ; ; ; ; ; -**** - Is tº -1-a ºf "fººtstºrs. lººſ--- ºf-ºf-isºlºist-ſº-ſº-sºlº ºſsil- sº-ſºlet-ºfeſsissists ºf º ſolº iºnºtº'ſ tº sº *ſº sº tºº lºſiſ: 1 tº 1: ". . . ; * *ſ-ſº sº, ſº slºwer & 1. - . . .” --- ºr “s is st-sº-ſº tº t-ſeº. f*** *...*.*.*.*.*.*.*.*.*.*.*.*.*.*.*, aſsists ºftº rat is lººr, alsº tºº lºst lºst ºf fººtºººººººººººººº; . . ... --- ºf-ºn-tº-lºv ſºlº º ... . . . [...] *:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: tº "ºº" tººls sº tºok sºn- ..ſººn-lºſsºſºft Wºl º:::::::::::::::::::::::::::: ºſ- | º º ºlº::::::::::::::::::::::::::::::::::Fººl'ſ * * : * ~ *** - - , , , , . . . . . sºlºi...ºtsiºlºmºnºtºr ºteſ ºf snº *:::: !. . . . . . . . ºº::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: * . . . . . . . . . . ; : tºº. . . . . . . . . tºº lººſely lºſs *::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: ; Pººl.º º: tº ºn. lºt ºf . , tº lººrs tº ºt-fººt * * *- :: ** -- . . . . . ºt-fººt, tº. * -- shººtººſ::ſ-fºr-º-; nº-1-stºſ-fººt sºlº latºº |- is ſº |ºt. tºº.;sſºlº is lºw 1-1. ******** -- " -- - - - - - - - - - , , , , , , ºf eſsets 's sºlºſsºns: 1...ſºlºisſºlºsſº ºn-ſet-lºſs sºlº eſcºrer. ºfeſsiveſ ºne ºf ſº * 1: . . . . . . . . . . . . . . . . . . . . . . . . . is -º ºr - ºr " ºrhººf'." ºne -ſ ºr st-tº-º-º- †::::::::::::::::::::::::::::::::::::::::::::: • aſ ... . . . . . . . . . . . . . . cºis le. ". . ... lºsiºſ-ºſ-ºl;º: - a . . . . . . . . . . . . ºl, tº it is ſºlºsſ; . . . . . . . . . . . . . .t-nºseve lºſsrººts ºt... ºº isºtº-jºº'ſ d † : º!!ºſsº-ººrººt-lès-tººl::::::::::::::::::::::: ºf tººl-lººt-ºf-ºr-liººnººſ---ſºlºisſºtsiºſ::ſºlºist ºf::::::::::::::::::::::A; eſ jºi ſº a ſºlº ſº. . . is tº is tº jº • * >|--|- sºlº sº sº sº. 3 ºs tº • * * * * * * * * * * * * * *- : * * * * * * • * *** * * * * * *f; ſº tº lººr - sº “tº seſsſsſsſs tº tº 1- is is º: | s ºf-ºw tº sº tº ºn tº . . . . . . ºf: * ºf $1 is • * : * > . . . . . . . . . . . . . . . . . * * ~1. ºf "ſº lºſsºſ- ... ſºrt; ºſsºſ; ºf •ºtºſ.stºrsfººt tº:: - º - * * * *...* - “ , ” 1.1 tº slº -us e-º-º-º-º-º-º-º-º: * tº e? . . . . . . . . . . . . . º. ºil. * , , ſ: • * * * * * * * * * * * * * * * * ºf stººleſalººlººlººlººlººlººlºº Cººl, Kºtº ºne ºf sº ºweſ aſ eraſºl ºf... . . ſ: Weſ. --- * : *-m-tºº--ºn, tºº-- ºw... • ******* - *** ****** * * * is is stººſ. ººººººººººº; :*::::A;|xists; cºlºſ:: ſºlº Rººt ºf ; ; ; ºleº º # º, ſººn st-ſºusſº is tº 1, 1: ... . ." ºr " º - . - a a sº a a ºf º * . . . . ." • * *** * * * * ºliº'ſ ºilº |- st ºn.' ...ſº * . . . . ºvº-º-º: , ſº ºlºt ******* *** - “ - 15 - * * * * * * * * - ... • , ...: ; , ; ; ; ; ; ; ; ; ; “rºº ********* -1; ; ; ; , . . . . .” -- ºf eſseſsiºns, sº sº. wºrst is ſº isſeſ ºf: - t |- ſº &ºts -1.11eel.": ".. |* is tº whº is tº • * * * * * * * * * * * * > . . . . . ." ºn-ºn-ºn.... . . sºlºſ: 1; *****{ ºisºtrºls sººtºs.stºl-ists sinſ ...!!!... º.º. ººr-teºrºiº-ºº:::::::::::::::::::: ºf ºne-hººl ºr ºf 'ºt -ºº: º;ººl; lºº; refºis . . . . . . . . . . . . . . . * * * : * : " . . . . . . Sº tº 1-1-1;"Riº" ---tº-shººtººſ; ſº-ºº::1: ; ** * * * * * * * * * * *ive tº a 1- ºr sº easis shººtstºwstºt-fºr--ºf -ſeſ- - - - , ºf ºf sºlºisºn |-|- tº: tºº, ſº-ſets fººt |ºl, ...&iºſ;ſsiºn-crºsſ-ºf-lºsſrººtsº-ºº::st-ſºus-tº-tºº-ºº: * * * * * * * * º jºst---ºf-lºt ºf ſºft-tº-viºus fºſetºſe's º, ſº ſet-tº-tº .ſet tº sº. t; ſº tº * * * : ***. -1. As ºf ºst-1--, -º- i. 1-, -i-. * "X" ºr . . . . tºº “º is is is sº sººn-ſ-chaetetsu: ; , , --> ºr isſ-ſ-ſ- - -tsºº-ººººººººººº-ºº: "ºss rºls ºf rºtº-stºrists ºf ... . . . . ;-) ..tº sists sº ſº tº - tº-ºn-º-º-º-º-º: sists sºlºſ:::::::::::::::::::::::::::::::::::::::::::::: ... -sſeſsiºn: 1,121.4% ºsts ºn Alsºstºſºlstetsºº-tsºº-tsºº-ts st-ſe º ... ſºlºi. 1 & ºwe a & D º * : * * - I e “ ( - e i < *** **** ****ſ-, -i- is ºpers -ºſ.º.º.º.º.º.º.º.º.º-ſ: 1... tº º tº º wºº. late fºreſ sº tº ºne sº-º: º *:::::::::::::::::::::::::::::::::::::::::::::::::::::A; *****'. * * * *º-stºrskº stafsſºs s sº ſº he , - ... we ... a sº send º is a ſet is ºs ſet e i s sºlº - - - - flºº - G ºf-ſiºlºsſ.ſºſ-i: ‘i:::::::::::::::::::::::::::::::::::: ºteſ ºffe . . . . . tº -º-º: stººls statests - alsº st-sº-ſºlºſ---i-º-º-º-ºººººººººººº. * | * ~ * sºlºisºtºtºkºtºtººrººtsfºtº sºlºsſesſe sº rt ºff-ºf-cº-ºwº is; ſº tº ..ºus cielº isº-ºxºſ-tºº-tsºº-ºº-ºº:::: • * * * * * * * * * * * * * * * * * *** ºteſ. 1...fººt-º-º-º-tººlººlºtsterºſºfsteſs #; :*ſ, *f; |--|--tº lºº |s lºsiº 2 " . . . ºf-tºº-ººººººººº-ºº-ºº-ºº::::::::::::::...r.º.º.º.º.ºtºtºººººººººººººººº-ſº º::::::::::::::::::::::::::::: * * * * * * St. ºº . . . .” --> * * * * : * ..." º-s, . . . * = . . . . . . . . . . . a . . . . . . ºn tº * Hº * : * I ºf ºr * * * * * º's º * * * * * * * * * * * * * * * *** - I - ..., . . . . . . . . . . ; Aſººts sºlº ſº tº “...t: “tet - 1st sº ſets * * *º-ſº ºf ºtºs? sº [.. • ºt-, -is is r-, “”, “ . . . . ." |* I & ſº “ . . . * * * * ºº:::::::::::::::::::::::::::::::::::::::::: tº W. ºt. ..º.º.º.º.º.º.º.º.º.º.º. ºlºlºlºº . --tº ºr ºil. is tº r is tº twº cº-º-tets ºf ºfeſsº ſetwº- - - - - - -ºº ºn. -ſº ºtelsºn fººt.º.º.º.º.º.º.º.º.º. º ºn-i-tº-letºstºſ-fºr-tº-lºst ſet ºf wºrſe ºt-sº-strº- C . . . . .,, ºr sº . . . . . . * *ſ.” * - ". . . . . . . . . . . . . . . . . . . . . . º:::::::::::::::::::::: exists. ºº:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::A; lººr, ſº ºf i. i. 21-rººt-, -, ºn-sistºſ-i: - . . . . . . . . ºlºtºſ.ſ., ſº sts tº sº.---wºº. --rººm--- sºlºsſeſsisterºissºtskº º - |s ſº tº tº sº." ſwººds tº ...t-ºf-twiste -- estate ºweſs tº c is evº ºf elºnsºleſ. ... º “ſ tº “ºlºrs ſº ſº tºº ... ....feſsºnsists is she ſesſºr ºne-nyeº sists:: *****º-ſeafºſsić ete Iºna - ...º.º.º.º.º.º. 3 cº-ºº:::: *-ºſ--tº-sists -ºst-fººt-tº-is-ºf-t-. ºº: ºst-i-º-º-º-º-º-º-º-º-º-º-tetºs sehºlsº stºneſesiºſºs wºrst sººt stºlsº ſºlºsſºſºfºrs-ſº sº ſº. ---ºf selefeleiºts ... ." -- sº I -1; sº shººtº - . . st-ºf-tºº-ººººººººººººººº-º-º-º-º-º: --- 4:1. . . . . . . . . . ......nº.ºtss :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: - a - I -, - . . . . . . . . . . . * * * * * ***.*.*.* tº stºls is aſ a s. ‘. . . . . ; ; ; ; ;ſºrsº -i-º-º: sº.-, ºr ºi... - rººm. ---wºº; ... ºf ºtºſ-fºr tº wºn tº ºº::::::ſºnsists Jºjº tº tº sº a ſet ºf tºtsºlatºvº-ºwn. , , sº is sº sºrºrººz -t, -tº, ºsſ--tº cº-ºr-linkſ...ºniº & º º lºſºtºrºiºſºſºfºr: sºlº ºlº"; * | • * * * ºfºil; º; eſsive *::::::::::::::::::::::::::::: º::::::::::::::::::: º º G Tº ºt. He ºt. Tº F is is ºr " * * | * : * * * * * * * * * * * * * * * * * * * * * * sºle is ſº ºwe & sº sº º (ºººººººº &ſº eſ wºº Hººſºº"ºº ſºlºſ º ºſ eſſes tº º • ‘ſs sist" tº sº; ...; tº let whº ſº, tº | |< || * * * * * * ſºlels ºf sºles ste §sº º: ſºleſ ºf lºſs ". * * * is . . * * * * : * : sts ºf , º, ºf “1”: "ſº " " - tº 'ºlº is ºf "ſº ºf sea. - ºf tº is tº . . . . . . . . . . . . . . ſº - . . . . . . . . . . . . . | • *- : * : * * * * * * * * * * * * ~ * * * f º: |*|| stºls - eſs ºf wºº! ... tºtal -i-. . . . .rººt ºf 4 is aſ ºhº Hººlººl ºf stºo!" * * * * * * *pº e. el ºff ºf ºsſ w • * * * * * * * * * * * * * s tº 1-fººt ºf ºf , t , a wºmateſseſ ºf esºtateſ an a . . . .tºs ºsſeſs eſs'ſ ºw r º º * * * º . . . . . a fºr tº: "ºsher: . . . . -- º º!!!...º.º.º.º.º.º. '''''''': ºf-ſºº'ſ wºfºl; lººtentiºſefsſºn ºil-fººtºººººº...ſººn; ::::::::::::::::::::::::::::::::::::::::::::::::i-iºlºr:ºrºſ;ſº ºf: fººt-i-tº-ſºlºisſºlºisſºjºſºfºrº. tº is 1: ... tº ſº; y . . . . . . * Iºſ" ºf tº :::::::::::::::::-1: . . . . . . . . . . . . . . . . . . St. . . . est-lººt ºr ºf: lºss ºf ess; sºme ºf sºlets tº sº ºf . , , ; 13 tº tºº. ** 14-3 lºſsºist ºf ºº::::::::::::::::::::::::::::::::::::::::::::::::: Rººſ a º “Wºº ºº::::::::::::::::::::::::::::::::::::::::::: 1. º. º.º. tº ſº º Aºtº ºsº is ſº rejº ºt ºf Hººtº: Nixºnºr ºf isſºl:l-leſ-tº-nºnet.* r * : * * * * ºlºisiºn.ºrs stºriºſ::::::::::::::::::::::: efs ºf ºil mºtº ºſtſ ºf ºt-º-º-º:::::::::: "fººl; ****** * ****** *-ºſ-i: ºt.ſ.º.º.º. i-ſºlº-tººl: *-ºn-ºn-ºr-tº-1 tº ºſºtºyºs ºjºſºfºrsſsſ iſ . . . . . . . [- - - - - - - tº ſeſs tº ſet #4. ". . . . . . At "A" • is ſº ºf . . . . tº º 'º 3. '' . . . . . . .ºns tº a ſº is tº ºf ºst ſºil; º ſets is sº |-|3| 1:1: Wºłºsſº"; * . . . . . ; , i.e. '" . º sº, sº tºº:::::::::::::::::::::::::::::::::::::Isºſºſºlºiskiºsſºſ fºllº ºf- ºf ºſsºs.ſºſsºsſeſs ºfesſel ºf; * -i- ºr tº . . | 1:1. ". |- A ºfºº ºº:::::::::::::::::::::::::::::::::::::::: ºr n a ºº:::::::::::::::::::::::::::::::::::::::::::::::: * ºlº;; tºlalººſe Ai-º-º-º- . . . º, rººf::: . . . . . . 12 tº gº tº ºff. : :"… . . . . . . . . . . . . . ; ; ;">“y ºf ºſs! º::::::::::::::::::::::::::::::::::::::::::::::::: ºft-sºusſºtºpºlºtºfºtºstºlerºtºtº ſºlº tº 1:1:1:1::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::tºiºnºiºiºſºlºſ::::::::::::::::::::::::::::::::::::::::::ſº ºf-f-14 lºſs ----|--|- “ . . . . .” -- - - - - - - - - , , ºr " - ****** **** 1: ºff-tºº-º-º-º- *4-1 tº . . . . . . . . . - e º – 5 - ? - . . . .eſ- ſee tº • vºte ºf a lºt stºº - sº sººfs eſseſ ºf wººets st-lºsteºs slºt esſeſ ... . . . . . . . . . . . . . . . ºrst tº "... tºº ſº tºº lººk 1. ºf... . . . . . . . . . . . . . . ); tº jº'º - ºr stºps is . . . . tº sts sº tº ſº. . .tºſ.ſ.º.º. 1 . . . .º.º. ºººº; ſº sº; tº ºſºtsiºs ºf "sºlºist ºbseſs ºtº ºt: ºf-ºº-ºº: !...] ſº...º.º.º.º.º. ºº is lºw-tº ºf ſºlºlººlºº. ::::: -, -, -, -º-º-º-tº ºt ſºrts ºf...tºº..., -º-º-º-º-º:::::ists: shººtsiºsºft-tº-ºf-sºº rºſy-tº-º-º: eſ ºisºners cº-ºws tº cººle ºsſeº ºfºſs's rººts lººſ-ºſºft fººtsºlsº ºn-tºſ. ſºlºists stºlººlººtstºfºssº - * ... • * * * * * ~ ; ; , . . . . . . . .” *i-,-,-]º ºne --- st-sº sºlºr: “ - ºr “ºfº” ******* * * * * * *** *** **** *********** - - - - - - - - - - - - wºmestºſºfsists ºf ºt-rººt-tºsºl fººt ºr st-ſºlºist ºf ººls fººt ºn asſº eleſ: “ lºſs tº H.R. ſkſ is ſes tº leſs-ºs is " *1-3 -)-º ºus team. -º-º-º-º-º-º-º: tº leſ a e º is ... • rººf-ºf lºst ºtºſ.stºreſ, sºlºsſy-tºº-º-º-º-º: ºsſºſºlºtºſ.ſº ſ: twº ºr st-sº-ººººººº-º-º-º-º-º-º-tºſº ſ ºtºstºlºſsºſºlsº . . . . ºf ºsi- C • * 1: . . . . . . . . . . º * * * : * : * * * º |- sº- ºf ºwlsº ºw- ... . . . . . . ." sign toº ( - teſ: eſs - ºt. ºf sºciº-º-º: tº ºn. Sººn-tº-ºº-ºº::::::::::::::::::::::::::----...ſ.l.i.jirº.º.º.º.º.º.º.º.º.º. |- st-tº lºst; tº st--ſº-tºº-ºº! ºr lººk ſet-lesſºr-elet-tº-ºººººº-º-º-º-ººººººººster-sºn::sºlº. • sists - twº sºmeteºs slºwſ:stºl-isºtºſs. ºf rººfs ; : ****** vºw: * ~ * º |*|| || • * * *-i- - - - -º-º-º-º- " - sº-ſafer. [.. isºt - º-º-º: - **** - tº . ~4-fº ºsº. 4 ſº - - - - - * : *i; ; }; - ºf-14ſ-91. - - - - - - - lºſsºis-ſº *** *********ºſºſºl º ſº lººrſ = 1 Aſeſºta.iº.ſv fºr ºf . , set. . ... aſsº. ſever, tºº is tº a .e. a sist Hººlºº ºisſolºisielsſeſ ºf **I* -º-º-º-º-tº-1-fººt-ºf" sº tººſ º fºr tºº. º.º.º.º.º. ºr ſarcº º * -- - - º: - - - -t-t-pºt: ... .º.º.º.º.º.º.º.º.º.º. nºt…t-tsºº-tsºº-ºººººº sists ſº sººn-tºi-ſº lº. --!> †. ºf tºter-tº-tº-stºrſ ºf sºlº tº 1-tº-ºf-----4--- ºl: ºctºr-tºº... º: º º #################### lºſsº.-->'', º' -- * * * . . . . . ºlºrsk, "tº cº-w, slºwl sister tº: **** ***, ºr ſº tº . . . ; ; ; ; ; ; ; ; ; ; ; ; ; lºts . . . . . . . . . sººt-ºf-teſ--isſ-law. -sſ- -lºsiºſºſºſtºſºlshºtsks ºfºſs ºf ill-ºf-tºº-º-º-º::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: |< *s tº 4- **.*.*.*.*** |-º-º-º-º: * * * ºf-ſeºs tº tel- ºf-stoſºſº ---is lºſsºlºtºw-tº-wººt-ºu- º lºſºls|-ºf- tº ... *:::::: ºtºſłºńskº-º-º-tººls ºf sºlºist ºf: …tº ſeaſe º,... ſºlºisiºn ºrº-rºtºtººl: •ºts j-, -, -, -ſs- Sºº - ºf: ; ; ; *ſ-, -º-; . . . . . ºf tººl: ºſsº:: rºl-lºſs ºfºl;1-1-1; lºſºſºlºſs alsº aſ “ſeſ ºf ºf lº cºsts: "hººls ºr , ºet. -- |- * * * be tº 1-tº-:4sſ-fººt-ſº *** *****-isºsts ºf:1: *** tºº lººt *s is lºſs |º iſ . . . . . . i-ſºſ ſºlºist ºf: isiºſ::ſºlºiºſºſ: st; sºft • * * * *******hºlºt: • * . . . . . . . . . . . . . . tº * - 1: ... --, i.e. 14-, ********- : * :-}... ºr sº Mºjº: sº ºf $1 ºf 'ſ-ſ-ſ: tº ********{s-lººp: ºf lºſºl. • Pºlº **::Infºſº º; * | * º “lº isºfº is ºl. ºn. ºf st.---nºt-fºr-tººt:::::::1:1"ſºº"; *i; ; ; ; ; ºr "yºtº ſ ſº ºlºſ:::::::::::::::::::: ( ºfsſºns:ſ:4::1 ºts ºf . lºſeſ, tºº ſºlºis set-tº-nºt lººt; tººl:ººt. ...sºſ.....ſº lººt-tº-lºſºvº -º- is sº-ſºº. º. ºjºſº. ºilºilºlºſ: |sºlº º ºtºlyſºſ: -º-º-º-º-º-º-º-ºººººººººº-º-º: sistºiºſ-ºs-ºſ-i:::::::::::::::::::sºlºisſºisºtºiºſºl- s *-ºſ-ºſ-ºs--lºſs ..º.º.º iſºtº-tº-ºn-ºr-tººtºººººº. ºf;ºlºr:; ºr:tº:-1:1:1:fºliº isſºlºſºlº ºf aſ ºſ-ºs- º-rºsſ-ºf-tººt::- •ºſs ºffs ºś ºº:::::::::::::::::::::::::: E. º: • nº ºf ::::::::::::::::::::::::::::::::::::::::::::::::: lºſſ ſ & §§§ ſ º Tºrºstate tº º , , .jsp.tº is is is lºº...!, - tº ºf “ -- - - - - - - - - ºr ºr Wºº “ºlº" ſº ºffers - e º stºº is ſº ºf: - - - is • a ſeſs & tºº. Hellº sºlºe's sºleſs-tºe ºf sº ſº ºf ºsts ſº ſeis. eiºſºſºkºtº lººſe.s Jºſé ºlº # º fºllº!!! | ºil. | º: ############### *I's - tº . . . . . sººn ºf tº tº 1-fººt tº sº. ſº # * | * sets tº swis is is ºwns a set-tºſe.tvetchºj, ter---fell is a tºp-twº ſºi - - - - - - - - : * ~ * º st sº sº. -º-º-º-º-º-º-º-º-º-º-º-º: ºº::::::i-fºr-tººls -- •ºleſ ºf , ºf - * ~ * : * ~ * ...tº ſafe-fººt-i-º-º-º-º-º: ,-i, 2 riºtstet-leſ: - L. !. º º º º - • *- : * * * * * * -º- ºr ſoft-shºt . º: . . . º' -j-. is lºſt aſkſ jº-º-º-º-º-º-º- elºſs is tº aiatºfºrº - fºliº lºſſ tº ºf º - º * I tº w = nºt sº º W º º * * * & Lº me I ºf & º 'º º - - , is vi. º.º.e.'ſ tº º::::::::::: at . stºl | ** Helens ºf sºlº ... sºcial ºt sº-º-º-, -, - * * * * * º º ºº:::::$tºn-tº-tº-º-º-º: - ; ºf ºrſyt º: ºftº:#; - . . º § - lºw-ſel ºf - * - - sºlsh*tºlºist ºf: - º º tº ſet ºf sºlºſs ºf ºf --|--|-- º:::::::::::::::::::::::::vºstºl--lºsſ-ºf-tºº-ºº::::::::::::::::::::::::::::::::::::::::::: ºfºſsietºſseſ ºf ºr a sic |-tº-ºn...i. **** ****** fºr tº: ºº:::::::::::::::::::::::::::::::::::::::...] ***** **** 1: ſ tº jºſs' sº lºſsºsºsſ ºtºs ºf slººſe C ºn certº sººn. , i.e. - º: º; º! ºf-tº lºst; ; “tººt, tººl; lººººº; ; ; ; ; ;ºsis, tº ":: º º, 1-ºxºlºtºtºººººººº...". ºfessiºſ-ºſ--lºſsº sº. -- 1:: * ***** -isºſ. --Sº fºsſ-slººs ºſsists ºf stºn-tººt-sº -- - jº-il, sº tºº::::::::::::::::: ºººººººº; ; ; ; ſºlºſ ºf |-tººl: tºº lºsſeſsiºns ºf * * ...tºs ºſsºt, tººls ºf ºn- º ºt-ºf-ºf-k-ºr-tºº-tsºº-ººrººlºſsºſºlºiºtsiºlºr: - r 1. ** - ". . . . * **** ****s ºf , is is 1-1-1; sº “… . . . . . . | º * - - , ºe is - vist. . . . . . . . . . . . . . U - d ºf sº t ſºlº rººt ºfen's tº a ſecºnºmºus islatºrs ºf...ſºſs Is Lººſe twº ºf st enºſ lysis. else ſ |- •ººrººººººººººººººººº- ** * * * * * * * º C | s >{ *- : *a a meſ ºr sº - aſ a gºe ºr a t- - sº sº... . . (stºl ºve a sº tº ſº 1. * * * * * * -- a.m. - ºr ºf 'ſ-leſ-ºs -º-º-º: Đ |-tº-ºº: |aſs vºte Fºlº • • * * * * * * * *ſ tº a tº as a ſell. t rets tº in a t-ſº- a s - I - I - I - , ſº , , § e º ſº sittºſ; sº e i s is †i. stºsts ºff-tº-º-º-º-º-º-º-º-º-te-v-f-teſ Alºist. Histºresterstatestºsº teleſs; sº stateſ º: • * * * * *** * * * *- - * ~ * * * * * ºn tº Rºß et at n | sº wºr- ºt taſkſ at ſº ſeleſetwº º ºfºliº Cºº º º ſ t Satº, ºr sº as *sº stºreſ-º-º-Hºls/stºſ-reſº = º sº at •Ü• * * * * º • º O | º: º ! •ſ; sº tº ºlººlºº |- - , , , , ºf resºl-- " " - " - sº aſ ºlºſſ - tººt-sºlºſsºs. ººººººººº ſºlºſ-ºſ.º.º.º.º.º. ºf::::::A; Cºncºcktºn-tºtals cº-º. tº tº custºne ºf zººl tº ſ ºleistſ tº tº Heidiº º; º º º sº ºf ºl- ist ºf: sº º Fº º: º: lº * * a ſist ºlºlºlº. §: |-ºffs: ºrsetstºſºlº aſºjº • aſ a ºf -ºſs . . . . . . . . . . . . . . . . . I-ſ - " - • . . . . . . * * * * *, *, * * ~ * • * * * * * * * * * * * * * * * * C ſº sº E- ºś |-ººººººº, ºlºt: ºt-fººt-lºsſ tº: º, ſº sºlºistess ºf ºfeſs ſº º º elesſalººſeſ º: º ºº::::::::::::::::::::stºsºlº C - º r ** ſw ºitiºn: *|| sº set ºf * { * * * * ºº::::::ºººººººººººººººººººººººººº; - ſº I e : ſ - - ºffeºus hºt-ſº-ſ-ſ-ſ-ſ-s-s-ſ-s :::::::::::::::::::::::::::::::::::::::: ºf-ſets sºlºſstººls sºlºisºfºrºsis, ºtºlºlºſsº lºſsºſ º Jºſefºre s - •elºw: ºfºſs-tººl: ***** *** * * - ſ ſºlºrºfºl; tº . ~~ * - - ſ - a “w” tº be eſseſ eſºteºsºtiºn ºffs tº sºmeº iſ sºlº sists; ºr . . . . . . ;--- ºr -t- sºlºs. 1.-1.-1.-1.-14 , sº e º ºr e º - tº tº jº suicy cºv d C - C f Suetººl, sº tºº sº-tº-, ... •ſ. … •ººtº: -ºn- ºr-tºrytelºt-ſºlº is stºlºgº ºº:::::::::::::: º #; sts.º-isºrs tººl: ºſsº º ºfeſsºlºist ºf eſºl tººt-i-º-º-º-; 4. . . . . . . . . . . . . .tºst-sººt *fººls ºº:::::::::::::::::::::::::: ºtestsehºlsº | | ºf sº , tº | *i-sººtºsis ºf sºlesiºlºgisterºſºlºiºsºlºis-i-lººkſ: ºº::::::::::::::::::flºº! ºf ºff-i-º- ºr- G ſe Sºlºsſ ºtºl-lº - "tºſsºs i-º-º-º: Rººt ºn ſº *** * * * * * º, º . " . * } - lºſer . . . tº a ºf lºſºlºſºstºlºſsºſºlº lºſºlºſºlstºtºiºſºfºrºººººººººº tºº º a - tº | ºve ºn tº - - | eſ ºº::::::::::: º ºlº sisſists left- l; †† ſ ºf ºf ºf ſ ºf . I sº º sº & E. * T is * * * º - º º tº 1-fºº!. ºf ºt sº º: * . . . . . ºf s *** * : * ~ * sºlº º ſ lºſs is ſºlºsſe º - & [. ºlºr. º º ºn ºi nº tº ºr a º * whº is - t - - ************, ºr • * |-1. º.º.º.º.º.º. tº ºt, lººſ. º is a lººſ-ºs-ºſ) º ſº lºſſ. | ! º: ºº:::::::::::::::::::::::::::::::::::::::::::::::::::::::: ºfºº ſºlº • *-d-: ****** * * * * * ºf:::::ſºlº ºf: e i & II º lºº. ſ º º [. r Mºtº ºne ſº ſchººls ºnat- ºr- ºr-.' * * *º ſº. st a tº |alºſ * Weſ. f sts. . . .esºt; ºf ºustºſ---i-rºs - i.e. ºº:::::::::::::::::::::::::::::::::::::::::::::::::::: sts ºwl ºf esſes ſºlº ſºlº t-ſ-ſ-ſ-ſ-ſ-ſ-s-ſ-ſ-ſ-rººf: - - |- st ºf whºlesk st s sº. --, -ºº: st steeſ, sº sº e º sº a . , t , t , ~" ºr " !" ºr ºf::::: º … ºº::::::::::::::::::::::::::::::::::::::::::1: • * : * I sº a & e ſ eſſ * a . . . . . . . . . . . . st-nº releºsºls tº yºkº an e-º-º-, * * *- : * 'wº ºn- * Heſses -i-, * sº. 8 a ſºlºsseſ statest stºº-wºrsſºns sº [. ſºlº-leſ-leſ-isºlºisie stººſe *** * * *** **********ºlºſslatººs - !º...!!! º etºlº ºr-º-º-º-º-ºº-ºº-ºº:::::::::::::::::::::::::::::::sºlstººlººlerºtºtºººººº-ººººººººººººººsleless sºlºisſºlºiºkºlºlºtºsſ * **!... : * : *-* ~1-y-º-º-º: , “ lººrºººººººººººººººº-ºº: º ºſºt - ". . . . . ~~~ * “ººt ºt " " - . . . size º a as ſº * sº dº ael's ſºlº. º ºl tºus wºr-ºw .…. . . cº-teſ-º-º-º-º-º-º-º: - ºº:::::::::::Rººt-ºutsiºſalskºlºlºs::::::: tºº-º-º-º-º-ººººººººº-º- • *te, sº vesſee- eſs tº * : ****** * * * ºº:: *** * *ſ-, -, -}. •ºveſe ſºlºſs ºf r ºlºrs - sº * * * * * * * * * | *ſº ºº::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: lºſsºlºi-ſºlºistſ: st:-1-1-1-tº-ºº: - - G aſ a Q- - - - - - • * * * * * * * * * * * * * * * º::::::::::A; • * * * * º ºf - (-i-ſºtº-jºº'ſ set ºf ºil . . . . . . . . . . . . . . . . . . . . . . . . ºr ºf - º º!...º.º.º. r ºf: - -, -nºſsº -i-. - ſº tºº -º-fººt ºn tº ºlº ſº ºn-sº-sºº'ſ fºſsiétºt-º-º-tº-ſº sº ſeſ-tºſ, ſº ºn; ſº ºf: º ººijº-ºº: -º-º-ººrºº::::::::::::::::::::::::::::::::::::::::: º: º - *******k-tº-isºlº lººk. ºlºsſºters...sºsºlate ºil--sººt-ſº-statewºfstekeleſslºhe ºfºlº - ºft ef *1: . . . . tº 1- tº ºr ºf ºt. --> ſº "... Nº • * > 1 . . . . . . . . . . . . . . ºº::::::::::::::::::::::::: |*| y : | i. eſſeſ ºf e ºf ºf stºpºſº | ºilºilº ºfs less tº . . . . . . . . . . . . . ... . . kº-1-1 seſ: "sº erºtºsſ-ºr-1s is ºf . tº 3 tº ºx!'"; ºr ºf atº's sº ºn. - . . . . . . . is is sº ºf . ºf Rºº. ºf::::::::::::: ſº ºfºtºſ.stºfºrºſºft tºº! *tºlº sºlº, 1st G ºfºº ºf ºne ºf * * * * * * * * * * ºf sºlºix hºlºſsº ſ Hººlººlºſ sºlºſsºſºkºlºlºlºlº- ºissºl::::::::::::::::::::::::::::::::::::::Hºtel, ſºlºisºtºſs ºf *** * * * * * * * * - ºf e s = }• * - * * * * * * ****** * * * * * Hºsſels -º-º-º-º: a s c ºf . , x - sº - sº- | • **ts" ºf . • * * * * * * * * * sts ºf $1&ſº ºf ...º.º.º.A: ; ; ,-- ºr ºn-º-º: sº-ºººº- state ºf ºivº is mºst-ſets ºn-ſ-s - ...t-º-º-º-º-º-º-º-º-º-º-º-º-º-i-, -º-º-º-º-ººººººººººººººººººººººº-º-º-º-º-º-º-º-º-º- ºº::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: sº en.d.º.º-º-teſ: º | Sºrºrºº::::::-ºf-jºº. yºsis ºſºtsiºſ lºſsºſ tºº.º.º.º.º. late ſet ºf º ºº::::::::::::::::::::::::::::::::::::::::::::::::: * | afonºſº's sº." - ſº . . . . . * * * * * * * *s-ſº sº sººts ſets. -- ~ *r- ſº sº. ---ſº ºf-tºº-ººººººººººººººººº-ºº: º::::pººfs: aſºn-lºsſºt ºfeſsis ºff-net ºf ºt-º ºst-º-º-º-º-º-ºººººººº- *:::::::::::::::::::::::::::::::: *tºlº; | st lºſiº it wººfeºff- - º Histºff-ſe ºfeſsºrs tº sº; ſºft-ºf-sºº lºſsºt - tº st-ºffs * , "tºº isº-tº tº “1” - ºf . , ------- *-* * * * * * ******* **** ºº::::::::::::::::::::::::::::::::::::::::::::::::::::::: ºteſ - º º tº. ºf: -ºº:::::::::::::::43 sº-, ºtºstelºtºfºrº, i-º-º::::::: rººf-ºf-jºº -t-t-ſº-ſº | ºwſ i-1 st soººººººººººººººººººººººººººº tº * * : * ~ * sºlºist ſºlºſsºt -ºº-ºº::1:ffººt- ºr ºf jºsº wººts sººt . . . . . . . . . . Yººtºſ.ſyſºlº tºº sº. - ºne ºrºfssº ºneſºteºsº.º.º.º.º. º-rºtº-º-tºe ºr sº tºº ſºlsº - - waſ ºr - rs :*::::::::::::ºtºkºtºsſ ºf ſºlº ºf-ſºº rººs ºf 'Aºst ºf sº tº ºf lºst I lº eft-isſa fººt-tº-º-º-º: ºº:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: lºſſ ſ *** * º::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: º | ſe ſºlºſ cºlºne ºrſº ºf . . . . . sºlº º eſſel seſsels) -ºſ e º ſeſs sº tº teleſs ºf slººſeſ sºleſetºſseſseſ ºf sºmeº ſet ºf º º! º: sº sº sº... is ... • * : *t. | -ºs i-ty! • * * * * * * * * ********* * * * Hºº: tº ºr “ſ ºf sºlºsſeſ eºsºvºsºlºists sºlºist. ---|- (2 - ºf . , 14ſ-ºſsſsſsſs ºsſ º º ºf-1-fººtsºeteºrº §§ºlºſſ # | ºft º ºº:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: # - - tº ºn.... tº . . . . . ;- º;;lºſſ; •ºi...º.º.º.º.º.º.º.º. ººlºſs ºf::$tº 1:1:1:1:1:ffs º * - sºlºſ::ſ: ot º isſºciºlºrs ºftºn-ºf-tº st- tº tºº. - ºf rºtº- lºſſ º ºiºſ ºf fººt-ſº *::::::::::::::#fºliºſ::sºlºiºiºiºſº # ºlºſ Tºſº. |ºtºriº ºf- ºr ºf:- ºver-eſ-, 1... -- " - tº ºu lººſe!" fººt ºt-lºſsºſºlºſ...tº ::::: ºlº ºf:-ſº ***** *{-} • *-ºſt. jºian sºlºisieſ-ſºlº ºt::::::::::::::::::::::::::::::::::::::::: ºlºr-ſºº jºiºiºiºi: CIes [. tº - is ſets ºf: -, * * *** -º-º-º: ſºlº ºt;" ºt: , Wºº-jºº "stººſ, ºt-ſº-ſº º ºº::::::::::::::::::: lººkstºwfººt-ºf- :::::::::::1: ... º *isſºlseſ-lºſsºsts ** { ºn-lºſºft-tº-º-º-º-º-º-º-º-º-º: ºº::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: iº. sistºieſel- |- ºf ºi!!ºr. *::::::::::::tºre º “...ſº ºº::::::::::::::::: tºsſ-tººt-lºts, ººlººlºlet-isºtºt: :*::1:1:1:1: ...tº cº-ſ: ºſ- sº-ſºſyº-tsºn: ºfe ºf: ºf • . . . . . ºf ſºlº ºf sºi:1:1:...nºſ. . . . . . . . ºf 1: . . . ºſ-tº-ſº * : * *isleſ ºf ſº ºº:::::::::::: ºlºlºſſ; cº-ſºlºkº" its isſºſºſºft tº is ºf-tººlººlºº." º, was ſººtºſ. tº it. 'wºsſºn-slººſ-- • ſºlº - ºr ºt-fººts tº ſºlº vºwlākwººl ºtetºr ºf:1:1:1:1:1:1:-1:};lºſº-ſºlºi-sºlºis . . . tº ºn- ºr ºt:4::Fºº ºlººlºº. ºf ºxº~ : ***** *- ºf:::::::::::ſºrºiº ºfºº rººf iſſ Fºl-ſºº tº stateſ sialºlºlºr ºt. ſºlº vºtes ºf ºf-tºrs tº else saleſ, ſets. ºº::::::::::::::::::::::::::::::::Hººt: ºilºlº. ºlºt. -- ſº tº ºf-fººt-tººls sºn-tº-tºnri º ºr...…tº sº. #1s ºr ºf tº 1.-1.-jºiºsº ºn tº lºſsº Rººſ--ºft'ſ -º-º-º-º-º-º-º: ºiºſ:::::::::::::::::::::: ºlºist; lºt ºf , #4:1:1:1:1:1:1:1:1:1: Here ºr-tºº-rººt ºr-tº-ſº"; Is ºlºr, sº wºn'twº-isºtºyº ſºlº ºſºtiºn stºriºsºlº º ºriºiºiºſº C tº eſºt - tensſee . . .º.º. ºº::::::::::::::::::::::::::::: ºfºl; ºº:::::::::::::::::::::::::::::: ºº:: ; ::::::::::::::::::::::::::::::::::::1 lºſs-ſºus-leſ- 'hººl ºf |- sºi -1 tº tºº lºsſºſºl...sº ſºlº ºff-ſº ºt-ºº::::Hºº-ºº-ººrººt jºrrºſºlº ſeſsiºſºls fºr “ſºlº i.i. Lºº. is •ſakſsºve; aſ at , , ºr $4. tº ºustº ºustºn-ºf-14s-ſ-ſ-ſºekºlºr:;1; ; ; ; ºr ºf 're- :::::::::::::::::::::::ſºlº ºlºlºſ; -lºſsºtsiºſ : *** * * * * * * - 1.1-rººs º ºx! .º.º.º is ſº is is ºne sº-ººrºº . . . . . . . . . . . .i-ºº-ºº: . . . . ; ; ; ; ; iſ º lººººººººº. ºisºfºteſ ºf::::::::::::::::::::::::::Sfºr Isºtº ºtswººt. ... .º.º.º. ººººººººººº-ºº-ºº: -º-, -º-º: sº. "...is ºw- ºr . . . ºff-ſel 34 º: Welsh*te, ---, -4. , ſ: , ;": ºfºsſºſºlºisiºn. º *Rºleleſs-tº-a-teſ-lººk cºvkºkºlºsivist tººkººl-sºº ſºlº ſºlº cºelº ºº:::::::::::::::::ſºſºsºſºſºlºiºſº lºſsº • * : * is 1- ºr . . . . . . º * * *, *****ºtºtº tº ... - . . . . . evºr ºn-tºº-i ºx; fººt ºf . * ºff- ::::::::::::::::::::::::::::: > †: *:::::::::::::::::::::::::::::::: CIA ºn . -- tº a “ . . " --fººtºººººº-ºº: ſº tº jºjº. ſ jºisºtº ºf: º * - 1:1 cºº ſº ~ſeſ ſºlºſsºſ lººſen: ſº - al • * * *ſatº aſsº ºweſsºrs." º;;"ºilº tº ºr ºf wº ºf ºr ºf lººſ-tº-º-º-º-º-º: ºals; ºtsiºſtºl; lyſºlºiºsºſºfsº-ſcisſºſºvºrºv sts is tº ſº ſet-ºf- iseſe whº'ſ nº *** ****-tººl sº tºe ºff-ſet-ºf-1- sº ºn......" ºf sºlºſs ºff-sººtsºvº :*:1; beft-fººts ºf ºtºkºek, feels leſsesſeſsºrsºvºsts stºlerºciºenºl- lººſcºw's ---|- ºf-ºfºl; tº º - º ºlºr:l-tºlet-ºf iſºlº ºtsiºſ:::::::::::::::::::::::::::::::::::i-ijºlºfsºlºiºſ ºlºtºfºtº º *** ***** ** fººt-sºº •ºst; tº tºº-ºº-º-º-º: *** -º-º-º-º: -et-tº-ºneºus, erºſeſ ºustºs- lººſetºstºl-º-º-º-ºº:::::::::::::::::::::::1:1:1:1: sº ºtsºlstºwns - ºt-mºnº ſºlº ºr . ...ºu. ...... i-tºº- ſºlºiſ; ºººººººººº. sh twº-ſº-º-º-º:º ºfºrſ::: #: lºsſeſsiºn; Fººts lººtiºisºtºrºiºiºſº. lºsiºsºſºlst-ſº-ſºlºr:lºſº; ;-ſºlºist tººls: | ºº::::::::::::::::::::::::::::::: sº lººſ & # tº sº tºº sº º • * : * is ºf affei º ºst- -i. ſºlº, Rºſewººtstvº. --º-º-º-º-º-º-º-º-º-º-º: - ::1:1: … Jºſefººt-, *i-ſºº tºº ºf: &nt tº º:::::::::::::::::::::::: ºfºrºi...º.º.º.º. ſ::::::::::::::::: º *** * * * * *, *te ºf . . . . . ; ºr vºiºſº | Iskº tº ſet sºlº Hº - †† ºilº.º.º.º.º.º.º.º.º.º. º |-|x-ſº-sº six tºpºl- tº telºtºſ-4-4 tiºn:---ſºº isºlº •ººl ºf wireſ. ºfººtº-º-º-º-º-º-º: ºf sºjº-j-r-ţ-ºº::::::: visiºſ:- ºśl. sºfthºſeſ [e ſe lºsis º .*.*.* tº ºt. ſeſs P Rºſs ºf sº; tº ſº “ſº tº 15 tº ſºrº: W*** * l,'º', ; ºt-º-º: sº ºwner-º-º-º-º-º-º-º-º-º-º: at-isſeſ sºlº f º - * He º ºſºvº, aſsºsºst stºº. §; º sºft'ſ # * + i = a- | *Hººſe is ſº twº stateſ--ºriº - . Weien. ºt. s wºº I.irrº - |-ºriº -, ºr ſeſs ºfºrºlolºº ºf rº stººt-lºsſºſ- ºyº “ſºmeºwu-º-º:*tºlº; sº e s = < * * * * * * * * * * * * * * * * ſ º sincideie º: º eieſc eaſe sº aſºsts º - º: ºf ºst-ſº | ete leſ. º skish-ſº * : * * * . . . . . . ; • * * * * * * º -º - º § - * * * * * * * * * * * º: | " " ' ". . . -- ſ . . . . . . . . . . . . . . . . . . ." “ ( V .. ). tº = , . . . . . . . . ~ : º | " . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .", * * * * * * * * ...'...}, ... v.) ºr . ." . . . . . . . . . . * * * * * * * * * * : * ~ * . . . . . . . . . . . . . . . . . º ...'...!!!...'"; . . . . . . . . . . . . - - º * * * * * * * * * * * .. º † - ſº . . . . . . . . . . . . . . . . . . . . . ~ * * * : * ~ : * ~ * . . . . . . * * * * * * * * * - - * * * * * * * * * * * * * * * * * * * * * º * * * * > º' * * * * C- -- ºº::::::.." . . . . . . . . . . . . . . . . * * * * * * * . . . . . . . . . . . . . . * * * • . . . . . * * * * * * * * * * U : I v . * * * * * - . . . . . . . . . . . . . . . . . . . . . . . . -- - s = e = { z c s = i = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . " * > . . . . g . . . . . . * * * • * : * ~ * * * * * * * . . . . . . . . . . . . . . . * * * * , . . . . . . . " ' - - . . . . . . . . " : " " . . . . . .'; … ." . .'; ' ' ' ' ' ' ' ) i. * *, *, *, *, *, * * : ' ' ' " :: * * * - || > | * . . . . . . . . . . . . * * * * * 5 d * * * * * * : * * - . . . . . . . . . . . . * - t < * > 3 i - . . * * * * tº-mºm . . . . . . . . . . . . . . " ' ' ' ' ' - - - ( * * * * --- a 4 - , - . . . . . . . . . A. : , , , w w Economic Control of Quality - of ~ * Manufactured Product BOOKS FROM BELL TELEPHONE LABORATORIES, INC. THEORY OF WIBRATING SYSTEMS AND SOUND. By IRVING B. CRANDALL, Late Member of the Technical Staff, Bell Telephone Laboratories. Illustrated, 282 pages, 6% X9}. Full Buckram. Second Printing. $5.00 CONTEMPORARY PHYSICS. By KARL K. DARROW, Member of the Technical Staff, Bell Telephone Laboratories. Illustrated, 512 pages, 6% X9%. Full Buckram. Second Printing. $6.00 SPEECH AND HEARIN G. 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Sec- ond Printing. $6.50 ECONOMIC CONTROL OF QUALITY OF MANUFACTURED PRODUCT. By W. A. SHEwBART, Member of Technical Staff, Bell Telephome Laboratories. Illustrated. 491 pages, 6; X9%. Full Buckram. PUBLISHED BY D. VAN NOSTRAND COMPANY, INC. ******* ************-* of Manufactured Product By w: A SHEWHART, Ph.D. Mºr of the Technical Staff BELL TELEPHONE LABORATORIES, INC. NEW YORK D. VAN NOSTRAND COMPANY, INo. 25o FourTH Avenue I93 I MATHEMATICŞ TS ISS , S55 292/2 2 By D. WAN NOSTRAND COMPANY, INC. All Rights Reserved This book or any parts thereof may not be reproduced in any form without written permission from the publishers. : i : © : i © Printed in U. S. A. PRESS OF E RAU NVVO RTH & Co., I NC. B O O K NMAN UFACTU FERS E ROO KLYN, N EVV YORK “When numbers are large, chance is the best warrant for certainty.” A. S. EDDINGTON, The Nature of the Physical World “A situation like this merely means that those details which determine the future in terms of the past may be so deep in the structure that at present we have no immediate experimental knowledge of them and we may for the present be compelled to give a treat- ment from a statistical point of view based on con- siderations of probability.” P. W. BRIDGMAN, The Logic of Modern Physics PREFACE Broadly speaking, the object of industry is to set up economic ways and means of satisfying human wants and in so doing to reduce everything possible to routines requiring a minimum amount of human effort. Through the use of the scientific method, extended to take account of modern statis- tical concepts, it has been found possible to set up limits within which the results of routine efforts must lie if they are to be economical. Deviations in the results of a routine process outside such limits indicate that the routine has broken down and will no longer be economical until the cause of trouble is removed. This book is the natural outgrowth of an investigation started some six years ago to develop a scientific basis for attaining economic control of quality of manufactured product through the establishment of control limits to indicate at every stage in the production process from raw materials to finished product when the quality of product is varying more than is economically desirable. As such, this book constitutes a record of progress and an indication of the direction in which future developments may be expected to take place. To get as quickly as possible a picture of the way control works, the reader may find it desirable, after going through Part I, to consider next the various practical illustrations given in Parts VI and VII and in Appendix I. The material in this text was originally organized for presentation in one of the Out-of-Hour Courses in Bell Tele- phone Laboratories. Since then it has undergone revision for use in a course of lectures presented at the request of Stevens Institute of Technology in its Department of Eco- nomics of Engineering. Much of the work recorded herein is the result of the cooperative effort of many individuals. To a V11 viii - PREFACE considerable extent the experimental data are such as could have been accumulated only in a large industry. On the theoretical side the author wishes to acknowledge the very helpful and suggestive criticisms of his colleague Dr. T. C. Fry and of Mr. E. C. Molina of the American Tele- phone and Telegraph Company. On the practical side he owes a great debt to another colleague, Mr. H. F. Dodge. The task of accumulating and analyzing the large amount of data and of putting the manuscript in final form was borne by Miss Marion B. Cater and Miss Miriam S. Harold, assisted by Miss Fina E. Giraldi. Mr. F. W. Winters contrib- uted to the development of the theory. The Bureau of Publication of the Laboratories cooperated in preparing the manuscript for publication. To each of these the author is deeply indebted. The author is particularly indebted to R. L. Jones, Director of Apparatus Development, and to G. D. Edwards, Inspection Engineer, under whose helpful guidance the present basis for economic control of quality of manufactured product has been developed. W. A. SHEw HART. BELL TELEPHONE LABoratoRIES, INC. New York, N. Y. April, 1931. TABLE OF CONTENTS PART I INTRODUCTION CHAPTER I Characteristics of a Controlled Quality. . . . . . . . . . . . . . . . . . . . . . What is the Problem of Control?—Nature of Control—Definition of Control. CHAPTER II Scientific Basis for Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three Important Postulates—When do Fluctuations Indicate Trouble?—Evidence that Criteria Exist for Detecting Assignable Causes—Róle Played by Statistical Theory—Conclusion. CHAPTER III Advantages Secured through Control. . . . . . . . . . . . . . . . . . . . . . . . Reduction in the Cost of Inspection—Reduction in the Cost of Rejections—Attainment of Maximum Benefits from Quantity Production—Attainment of Uniform Quality Even though Inspec- tion Test is Destructive—Reduction in Tolerance Limits—Con- clusion. PART II WAYS OF ExPRESSING QUALITY OF PRODUCT CHAPTER IV Definition of Quality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introductory Note—Popular Conception of Quality—Conception of the Quality of a Thing as a Set of Characteristics—Conception of the Quality of a Thing as an Attribute—Quality of a Number of the Same Kind of Things—Quality of Product: Distribution of Quality Characteristics, Quality Statistics, Quality Rate— Quality as a Relationship—How Shall Quality be Defined? PAGE 3 8 1X X CONTENTS CHAPTER V The Problem of Presentation of Data . . . . . . . . . . . . . . . . . . . . . . . Why We Take Data—The Problem of Presentation—Essential Information Defined—Statement of the General Problem—True versus Observed Quality. CHAPTER VI Presentation of Data by Tables and Graphs Presentation of Ungrouped Data—Presentation of Grouped Data —Choice of Cell Boundaries—Conclusion. CHAPTER VII Presentation of Data by Means of Simple Functions or Statistics. Simple Statistics to Be Used—Fraction p Defective or Non-Conform- ing—Arithmetic Mean X as a Measure of Central Tendency— The Standard Deviation a as a Measure of Dispersion—Skewness k—Flatness 32—Calculation of Statistics: Errors of Grouping, Number of Figures to be Retained—Measures of Relationship— Other Statistics. CHAPTER VIII Basis for Determining How to Present Data . . . . . . . . . . . . . . . . . The Problem—Statistics to be Used When Quality is Controlled —Why the Average X and Standard Deviation a Are Always Useful Statistics—Tchebycheff's Theorem—Importance of Skewness k and Flatness 82–Conclusion. CHAPTER IX Presentation of Data to Indicate Relationship . . . . . . . . . . . . . . . Two Kinds of Relationship: Functional and Statistical–Observed Relationship—Information Given by the Correlation Coefficient— Relationship between Several Qualities—Measure of Relationship, Correlation Ratio—Measure of Relationship, General Comments. PART III BASIS FOR SPECIFICATION OF QUALITY CONTROL CHAPTER X Laws Basic to Control Control—Exact Law—Law of Large Numbers — Point Binomial in Relation to Control—Evidence of the Existence of the Law of Large Numbers: Tossing a Coin or Throwing Dice, Sampling Experiment, Distribution of Number of Alpha Particles, Macro- scopic Properties of Matter—Controlled or Constant System of Chance Causes—Meaning of Cause—Variable System of Chance Causes—Statistical Laws—Exact and Statistical Laws, A Com- parison–Summary. • * * * * * * * * * * * * * * * * * * * * * * * * * * e s e º e º e 63 71 85 99 CONTENTS xi Statistical Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maximum Control Sampling Fluctuations CHAPTER XI Conditions for Control—Necessary and Sufficient Conditions for Statistical Control. CHAPTER XII Maximum Control Defined—Characteristics of Maximum Con- trol—Molecular Phenomena—Necessary Conditions for Maxi- mum Control—Simple Cause System—Necessary Conditions, Some Criticisms—Some Practical Conclusions. PART IV SAMPLING FLUCTUATIONS IN QUALITY CHAPTER XIII Sample—Sampling Fluctuations—Simple Illustration of Sampling Fluctuations—Sampling Fluctuations in Simple Statistics—Simple Problem in Prediction of Sampling Fluctuation, Problem of Dis- tribution—Relation of Sample to Universe—The Problem of Determining the Allowable Variability in Quality from a Statis- tical Viewpoint. CHAPTER XIV tº º e º & e º $ tº € 8 s tº e º ºs e e º ſº tº º ſº º ſº e º g º º e º ºs º e º ºs e º & & º e s e e s e º e º e s e º e s tº e s & º is e º e º e e e g tº e f Sampling Fluctuations in Simple Statistics under Statistical Control Sampling Fluctuations in Simple Statistics, Correlation Coeffi- Clent Method of Attack—Fraction Defective—Average, Normal Universe —Average, Non-Normal Universe—Standard Deviation, Normal Universe—Standard Deviation, Non-Normal Universe—Variance— Ratio z = , Normal Universe—Distribution of Average and Standard Deviation—A Word of Caution—Skewness k and Flatness 69–Other Measures of Central Tendency—Other Measures of Dispersion—Chi Square—Summary. CHAPTER XV • * is e º s e e s a e º e º is e e s is e a e º e º e º is e s e s is a e e s e is e e s ∈ e s a e g a s Correlation Coefficient—Distribution Function of Correlation Coefficient—Standard Deviation or of Correlation Coefficient— Modal and Expected Values of Correlation Coefficient—Trans- formed Distribution of Correlation Coefficient—Conditions under Which Distribution of r Has Significance—Commonness of Causa- tion Measured by r–Simple Example Showing How Correlation Coefficient Measures Commonness of Causation—Interpretation of r in General. PAGE xii CONTENTS CHAPTER XVI PAGE Sampling Fluctuations in Simple Statistics, General Remarks . . 230 Two Phases of Distribution Theory: Mathematical Distribution, Objective Distribution—Importance of Distribution Theory—Math- ematical Distribution Theory, Method of Attack: Some Numerical Results, Comparison of the Two Methods—Mathematical Distri- bution Theory, Important Results—Mathematical Distribution Theory, Present Status—Importance of Distribution Theory— Further Comments. PART V. STATISTICAL BASIS FOR SPECIFICATION OF STANDARD QUALITY CHAPTER XVII Design Limits on Variability. . . . . . . . . . . . . . . . . . . … 249 Tolerances—Tolerances Where Ioo Per Cent Inspection Cannot be Made—Importance of Control in Setting Economic Tolerance— Tolerances Where Ioo Per Cent Inspection Cannot be Made, Importance of Control—Tolerance for Quality of Finished Product in Terms of Tolerances of Piece-Parts—The General Problem of Setting Tolerances on Controlled Product—Design for Minimum Variability. CHAPTER XVIII Specification of Standard Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 Standard Quality—Types of Specification—Importance of Specifying the Function f—Specification, Further Discussion—Conclusion. PART VI ALLOWABLE VARIABILITY IN QUALITY CHAPTER XIX Detection of Lack of Control in Respect to Standard Quality 275 The Problem—The Basis for Establishing Control Limits—Choice of Statistic to Detect Change in Average Quality—Choice of Statistic to Detect Change in Standard Deviation—Additional Reason for Choosing the Average X and Standard Deviation a-Choice of Sta- tistic to Detect Change Ar in the Correlation Coefficient r—Choice of Method of Using Statistics—Choice of Method of Using Statistics, Simple Example—Choice of Method of Using Statistics, Con- tinued—Choice of Statistic for Detecting Change in Universe of Effects—Detection of Failure to Maintain Standard Quality. CONTENTS CHAPTER XX Detection of Lack of Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Problem—Choice of Method of Estimating X and or—Choice of Test Criterion for Detecting Lack of Control—Criterion I, General–Criterion I, Attributes—Criterion I, Variables, Large Samples—Criterion I, Variables, Small Samples—Use of Criterion I, Some Comments—Criterion II. CHAPTER XXI Detection of Lack of Control, Continued . . . . . . . . . . . . . . . Introductory Statement—Criterion III—Criterion IV—Criterion V —Criticism of Criterion V–Rôle of Judgment in Choice of Criteria— Sampling Inspection in Relation to Control, Attributes: A Priori Method, A Posteriori Method. PART VII QUALITY CONTROL IN PRACTICE CHAPTER XXII Summary of Fundamental Principles. . . . . . . . . . . . . . . . . . . . . . . Introductory Statement—Object of Control—Physical Properties —Physical Laws—Causal Explanation—Measurement of Average X and Standard Deviation or: A Posteriori Probability Method, Maximum Likelihood Method, Empirical Method—Measurement of Average X and Standard Deviation or, Practical Example. CHAPTER XXIII Sampling, Measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Place of Measurement in Control—All Measurement a Sampling Process—Good Data—Correction of Data for Constant Errors— Errors Introduced by Constant Systems of Chance Causes—Correc- tion for Constant Chance Errors of Measurement—Analysis of Bad Data—Analysis of Good Data—Minimizing Cost of Measurement, Simple Example—How Many Measurements?—Law of Propagation of Error, Practical Significance—Measurement through Statistical Relationship: Calibration, Effect of Error of Measurement, Conclu- sion, Example. CHAPTER XXIV Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental Considerations—Random Sample—Sampling for Protection—Representative Sample—Size of Sample—Sampling in Relation to Specification. xiv CONTENTS CHAPTER XXV The Control Program Résumé—Control in Research—Control in Design—Control in Development—Control in Commercial Production—Control in the Purchase of Raw Material—Quality Control Report. APPENDIx I Resultant Effects of Constant Cause Systems Introductory Remarks — Practical Significance of Results—Ana- lytical Results—Economic Significance of Control from a Design Viewpoint. APPENDIX II Experimental Results Presentation of Original Experimental Results Useful in Obtaining an Understanding of the Fundamental Principles Underlying the Theory of Quality Control. APPENDIx III Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References for Parts I and III: Exact and Statistical Laws, Empirical Laws, Frequency Distribution Functions, Probability, Quality Con- trol—References for Part II: Economics, Texts on Statistics, Curve Fitting—References for Part IV—References for Parts VI and VII: Estimation, Detecting Lack of Control—Other References: Errors of Measurement, Tables, Magazines, Mathematics, Graphical Methods. * u e = * * * * * * * * * * * * * * * * * * * * * e e º e e º 'º e º e & * * * * * * * * * * * * g º º * * * * * * e º & º e º e º is e º s e e º ºs e º e e s tº e º e º e º 'º e PART I Introduction Fundamental Concepts of Statistical Con- trol and an Outline of Five Economic Advantages Obtainable through Statistical Control of Quality of Manufactured Product CHAPTER I CHARACTERISTICS OF A ContROLLED QUALITY I. What is the Problem of Control? What is the problem of control of quality of manufactured product? To answer this question, let us put ourselves in the position of a manufacturer turning out millions of the same kind of thing every year. Whether it be lead pencils, chewing gum, bars of soap, telephones, or automobiles, the problem is much the same. He sets up a standard for the quality of a given kind of product. He then tries to make all pieces of product conform with this standard. Here his troubles begin. For him standard quality is a bull's-eye, but like a marksman shooting at a bull's-eye, he often misses. As is the case in everything we do, unknown or chance causes exert their influence. The problem then is: how much may the quality of a product vary and yet be controlled? In other words, how much variation should we leave to chance? To make a thing the way we want to make it is one popular conception of control. We have been trying to do this for a good many years and we see the fruition of this effort in the marvelous industrial development around us. We are sold on the idea of applying scientific principles. However, a change is coming about in the principles themselves and this change gives us a new concept of control. A few years ago we were inclined to look forward to the time when a manufacturer would be able to do just what he wanted to do. We shared the enthusiasm of Pope when he said “All chance is but direction thou canst not see”, and we looked forward to the time when we would see that direction. In other words, emphasis was laid on the exactness of physical * J 4. ECONOMIC CONTROL OF QUALITY laws. Today, however, the emphasis is placed elsewhere as is indicated by the following quotation from a recent issue, July, 1927, of the journal Engineering: Today the mathematical physicist seems more and more inclined to the opinion that each of the so-called laws of nature is essentially statistical, and that all our equations and theories can do, is to provide us with a series of orbits of varying probabilities. The breakdown of the orthodox scientific theory which formed the basis of applied science in the past necessitates the introduction of certain new concepts into industrial development. Along with this change must come a revision in our ideas of such things as a controlled product, an econ- omic standard of quality, and the method of detecting lack of control or those variations which should not be left to chance. Realizing, then, the statistical nature of modern science, it is but logical for the manufacturer to turn his attention to the consideration of available ways and means of handling statistical problems. The necessity for doing this is pointed out in the recent book 1 on the application of statistics in mass production, by Becker, Plaut, and Runge. They say: It is therefore important to every technician who is dealing with problems of manufacturing control to know the laws of statistics and to be able to apply them correctly to his problems. Another German writer, K. H. Daeves, in writing on somewhat the same subject says: - Statistical research is a logical method for the control of opera- tions, for the research engineer, the plant superintendent, and the production executive.” The problem of control viewed from this angle is a compar- atively new one. In fact, very little has been written on the subject. Progress in modifying our concept of control has been and will be comparatively slow. In the first place, 1 Anwendungen der Mathematischen Statistik auf Probleme der Massenfabrikation, Julius Springer, Berlin, 1927. * “The Utilization of Statistics,” Testing, March, 1924. CHARACTERISTICS OF A CONTROLLED QUALITY * 5 “y it requires the application of certain modern physical concepts; and in the second place, it requires the application of statistical methods which up to the present time have been for the most part left undisturbed in the journals in which they appeared. This situation is admirably summed up in the January, 1926 issue of Nature as follows: A large amount of work has been done in developing statistical methods on the scientific side, and it is natural for anyone interested in science to hope that all this work may be utilized in commerce and industry. There are signs that such a movement has started, and it would be unfortunate indeed if those responsible in practical affairs fail to take advantage of the improved statistical machinery now available. 2. Nature of Control Let us consider a very simple example of our inability to do exactly what we want to do and thereby illustrate two characteristics of a controlled product. Write the letter a on a piece of paper. Now make another a just like the first one; then another and another until you have a series of a's, a, a, a, a, . . . . You try to make all the a's alike but you don’t; you can’t. You are willing to accept this as an empirically established fact. But what of it? Let us see just what this means in respect to control. Why can we not do a simple thing like making all the a's just alike? Your answer leads to a generalization which all of us are perhaps willing to accept. It is that there are many causes of variability among the a's: the paper was not smooth, the lead in the pencil was not uniform, and the unavoidable vari- ability in your external surroundings reacted upon you to introduce variations in the a's. But are these the only causes of variability in the a's? Probably not. We accept our human limitations and say that likely there are many other factors. If we could but name all the reasons why we cannot make the a's alike, we would most assuredly have a better understanding of a certain part of nature than we now have. Of course, this conception of what it means to be able to do what we want to do is not new; it 6 ECONOMIC CONTROL OF QUALITY does not belong exclusively to any one field of human thought; it is commonly accepted. The point to be made in this simple illustration is that we are limited in doing what we want to do; that to do what we set out to do, even in so simple a thing as making a's that are alike, requires almost infinite knowledge compared with that which we now possess. It follows, therefore, since we are thus willing to accept as axiomatic that we cannot do what we want to do and cannot hope to understand why we cannot, that we must also accept as axiomatic that a controlled quality will not be a constant quality. Instead, a controlled quality must be a variable quality. This is the first characteristic. But let us go back to the results of the experiment on the a's and we shall find out something more about control. Your a's are different from my a’s; there is something about your a’s that makes them yours and something about my a’s that makes them mine. True, not all of your a’s are alike. Neither are all of my a's alike. Each group of a’s varies within a certain range and yet each group is distinguishable from the others. This distinguishable and, as it were, constant variability within limits is the second characteristic of control. 3. Definition of Control For our present purpose a phenomenon will be said to be controlled when, through the use of past experience, we can predict, at least within limits, how the phenomenon may be expected to vary in the future. Here it is understood that prediction within limits means that we can state, at least approximately, the prob- ability that the observed phenomenon will fall within the given limits. In this sense the time of the eclipse of the sun is a predictable . phenomenon. So also is the distance covered in successive intervals of time by a freely falling body. In fact, the prediction in such cases is extremely precise. It is an entirely different matter, however, to predict the expected length of life of an individual at a given age; the velocity of a molecule at a given instant of time; the breaking strength of a steel wire of known CHARACTERISTICS OF A CONTROLLED QUALITY 7 cross section; or numerous other phenomena of like character. In fact, a prediction of the type illustrated by forecasting the time of an eclipse of the sun is almost the exception rather than the rule in scientific and industrial work. In all forms of prediction an element of chance enters. The specific problem which concerns us at the present moment is the formulation of a scientific basis for prediction, taking into account the element of chance, where, for the purpose of our discussion, any unknown cause of a phenomenon will be termed a chance cause. CHAPTER II SCIENTIFIC BASIS FOR ContRoL I. Three Important Postulates What can we say about the future behavior of a phenomenon acting under the influence of unknown or chance causes? I doubt that, in general, we can say anything. For example, let me ask: “What will be the price of your favorite stock thirty years from today?” Are you willing to gamble much on your powers of prediction in such a case? Probably not. However, if I ask: “Suppose you were to toss a penny one hundred times, thirty years from today, what proportion of heads would you expect to find?”, your willingness to gamble on your powers of prediction would be of an entirely different order than in the previous case. The recognized difference between these two situations leads us to make the following simple postulate: Postulate 1–All chance systems of causes are not alike in the sense that they enable us to predict the future in terms of the past. Hence, if we are to be able to predict the quality of product even within limits, we must find some criterion to apply to observed variability in quality to determine whether or not the cause system producing it is such as to make future pre- dictions possible. Perhaps the natural course to follow is to glean what we can about the workings of unknown chance causes which are generally acknowledged to be controlled in the sense that they permit of prediction within limits. Perhaps no better examples could be considered than length of human life and molecular 8 SCIENTIFIC BASIS FOR CONTROL 9 motion. It might appear that nothing is more uncertain than life itself, unless perhaps it be molecular motion. Yet there is something certain about these uncertainties. In the laws of mortality and distribution of molecular displacement, we find some of the essential characteristics of control within limits. A. Law of Mortality The date of death always has seemed to be fixed by chance even though great human effort has been expended in trying to rob chance of this prerogative. We come into this world and from that very instant on are surrounded by causes of | 45 50 AGE | N YEARS FIG. I.-LAw of MoRTALITY-LAw of Fluctuations ContRolled witHIN LIMITs. death seeking our life. Who knows whether or not death will overtake us within the next year? If it does, what will be the cause? These questions we cannot answer. Some of us are to fall at one time from one cause, others at another time from another cause. In this fight for life we see then the element of uncertainty and the interplay of numerous unknown or chance causes. However, when we study the effect of these chance causes in producing deaths in large groups of individuals, we find some indication of a controlled condition. We find that this hidden host of causes produce deaths at an average rate which does IO ECONOMIC CONTROL OF QUALITY not differ much over long periods of time. From such obser- vations we are led to believe that, as we approach the condition of homogeneity of population and surroundings, we approach what is customarily termed a “Law of Mortality” such as indicated schematically in Fig. I. In other words, we believe that in the limiting case of homogeneity the causes of death function so as to make the probability of dying within given age limits, such as forty-five to fifty, constant. That is, we believe these causes are controlled. In other words, we assume the existence of a kind of statistical equilibrium among the effects of an unknown system of chance causes expressible in the assumption that the probability of dying within a given age limit, under the assumed conditions, is an objective and constant reality. B. Molecular Motion Just about a century ago, in 1827 to be exact, an English botanist, Brown, saw something through his microscope that caught his interest. It was motion going on among the sus- pended particles almost as though they were alive. In a way it resembled the dance of dust particles in sunlight, so familiar to us, but this dance differed from that of the dust particles in important respects, for example, adjacent particles seen under the microscope did not necessarily move in even approx- imately the same direction, as do adjacent dust particles sus- pended in the air. Watch such motion for several minutes. So long as the temperature remains constant, there is no change. Watch it for hours, the motion remains characteristically the same. Watch it for days, we see no difference. Even particles sus- pended in liquids enclosed in quartz crystals for thousands of years show exactly the same kind of motion. Therefore, to the best of our knowledge there is remarkable permanence to this motion. Its characteristics remain constant. Here we certainly find a remarkable degree of constancy exhibited by a chance system of causes. $ Suppose we follow the motion of one particle to get a better SCIENTIFIC BASIS FOR CONTROL II picture of this constancy. This has been done for us by several investigators, notably Perrin. In such an experiment he noted the position of a particle at the end of equal intervals of time, Fig. 2. He found that the direction of this motion observed in one interval differed in general from that in the next succeeding interval; that the direction of the motion y- Y /D § Ó V N sº NZN k º ^ſº N Sº FIG. 2.-A Close-UP of Molecu LAR Motion APPEARING ABsolutELY IRREGULAR, YET CONTROLLED witHIN LIMIts. presents what we instinctively call absolute irregularity. Let us ask ourselves certain questions about this motion. Suppose we fix our attention on the particle at the point A. What made it move to B in the next interval of time? Of course we answer by saying that a particle moves at a given instant in a given direction, say AB, because the resultant force of the molecules hitting it in a plane perpendicular to I 2 ECONOMIC CONTROL OF QUALITY this direction from the side away from B is greater than that on the side toward B; but at any given instant of time there is no way of telling what molecules are engaged in giving it such motion. We do not even know how many molecules are taking part. Do what we will, so long as the temperature is kept constant, we cannot change this motion in a given system. It cannot be said, for example, when the particle is at the point B that during the next interval of time it will move to C. We can do nothing to control the motion in the matter of dis- placement or in the matter of the direction of this displacement. Let us consider either the x or y components of the segments of the paths. Within recent years we find abundant evidence indicating that these displacements appear to be distributed about zero in accord with what is called the normal law." Such evidence as that provided by the law of mortality and the law of distribution of molecular displacements leads us to assume that there exist in nature phenomena controlled by systems of chance causes such that the probability dy of the magnitude X of a characteristic of some such phenomenon falling within the interval X to X + d x is expressible as a function f of the quantity X and certain parameters represented symbolically in the equation dy = f(X, \1, N2, . . . , \m).dx, (2) where the X's denote the parameters. Such a system of causes we shall term constant because the probability dy is independent of time. We shall take as our second postulate: Postulate 2–Constant systems of chance causes do exist in nature. To say that such systems of causes exist in nature, however, is one thing; to say that such systems of causes exist in a 1 That is to say, if x represents the deviation from the mean displacement, zero in this case, the probability dy of x lying within the range x to x + dx is given by 22 T 273 dx, (1) I dy = =6 Or 2 Tr where a is the root mean square deviation. SCIENTIFIC BASIS FOR CONTROL I3 O.O4 c. 0.03 O O [1] > H. O Ull Ul- # 0.02 O 2 9 H. § —º -*— § 0.0 O Ul- O O O O O O O O —l I | | | | | | – l | | J. F M A M J J A S O N D MONTH S (a) APPARATUS TYPE A O.O4 r- O Cl- O.O3 – u! O > H. O Ul Ul- uſ O,O2H- C 2 o H. O O O § 0.01– O Ul- O O O e O ° e O | | l | I | | | l | | i h J. F M A M J J A O N D MONTHS (b) APPARATUS TYPE B FIG. 3.-Should THESE WARIATIONS BE LEFT to CHANCE: I4. ECONOMIC CONTROL OF QUALITY production process is quite another thing. Today we have abundant evidence of the existence of such systems of causes in the production of telephone equipment. The practical situation, however, is that in the majority of cases there are unknown causes of variability in the quality of a product which do not belong to a constant system. This fact was discovered very early in the development of control methods, and these causes were called assignable. The question naturally arose as to whether it was possible, in general, to find and eliminate such causes. Less than ten years ago it seemed reasonable to assume that this could be done. Today we have abundant evidence to justify this assumption. We shall, therefore, adopt as our third postulate: Postulate 3–4ssignable causes of variation may be found and eliminated. Hence, to secure control, the manufacturer must seek to find and eliminate assignable causes. In practice, however, he has the difficulty of judging from an observed set of data whether or not assignable causes are present. A simple illus- tration will make this point clear. 2. When do Fluctuations Indicate Trouble? In many instances the quality of the product is measured by the fraction non-conforming to engineering specifications or, as we say, the fraction defective. Table I gives for a period of twelve months the observed fluctuations in this fraction for two kinds of product designated here as Type A and Type B. For each month we have the sample size m, gº ſo 721 the number defective n1 and the fraction p = *. We can 7? better visualize the extent of these fluctuations in fraction defective by plotting the data as in Fig. 3-a and Fig. 3-6. What we need is some yardstick to detect in such variations any evidence of the presence of assignable causes. Can we find such a yardstick? Experience of the kind soon to be con- sidered indicates that we can. It leads us to conclude that SCIENTIFIC BASIS FOR CONTROL I 5 it is feasible to establish criteria useful in detecting the presence of assignable causes of variation or, in other words, criteria which when applied to a set of observed values will indicate whether or not it is reasonable to believe that the causes of variability should be left to chance. Such criteria are basic to any method of securing control within limits. Let us, there- fore, consider them critically. It is too much to expect that the criteria will be infallible. We are amply rewarded if they appear to work in the majority of cases. Generally speaking, the criteria are of the nature of limits derived from past experience showing within what range the fluctuations in quality should remain, if they are to be left to chance. For example, when such limits are placed on the fluctuations in the qualities shown in Fig. 3, we find, as shown in Fig. 4, that in one case two points fall outside the limits and in the other case no point falls outside the limits. TABLE I.—FLUCTUATIONS IN QUALITY of Two MANUFACTURED PRODUCTs Apparatus Type A Apparatus Type B Fraction Fraction Number | Number | Defective Number | Number | Defective Month |Inspected| Defective 711 Month |Inspected| Defective 711 77 711 = —- 77 711 p = - 77 77 Jan. . . . . 527 4. o.OO76 || Jan. . . . . I69 I o.OO59 Feb. . . . . 6Io 5 o. oo82 || Feb. . 99 3 O. O3O3 March 428 5 o, or 17 || March 2O8 I o.OO48 April 4OO 2 o.Ooso || April I96 I O.OO 5 I May 498 I 5 o, ogo1 || May I 32 I o. oo76 June 5OO 3 o. ooéo || June 89 I O. O I I 2 July 395 3 o. oo76 || July 167 I o.oobo Aug. 393 TX o.OOSI || Aug. 2OO I o.OOSo Sept 62 3 o.OO48 || Sept. 171 2. o, or 17 Oct. . . . . 465 I 3 o.O28o || Oct. . . . . I 22 I o. oo82 Nov 446 5 o. of 12 || Nov Io'ſ 3 o. oº 8o Dec. . . . . 5 IO 3 o.OO59 || Dec. . . . . I 32 I o. oo76 Average 483. O& 5.25 o. olog | Average I49. 33 I. 42 | O. OO95 I6 ECONOMIC CONTROL OF QUALITY Upon the basis of the use of such limits, we look for trouble in the form of assignable causes in one case but not in the other. O.O4 O. O3 O.O2 O. Ol O.O4 Cl- O. O 3 O.O2 O.O | (b) APPARATUs TYPE B a—“No.” b—“YES.” pºse O O ſº ſº O O O O O O O O 1–1 | | ! | I | 1 l —1– -1– I—l I J F M A M J J A S O N D M O N T H S (a) APPARATUS TYPE A H. H–– — — — — — — — — — — — — — — - O O C O J O O O O © O O --------—— — — — — — — — — — — J F M A M J J A S O N D M O N T H S FIG. 4.—SHOULD THESE WARIATIONS BE LEFT TO CHANCE: However, the question remains: Should we expect to be able to find and eliminate causes of variability only when deviations SCIENTIFIC BASIS FOR CONTROL 17 fall outside the limits? First, let us see what statistical theory has to say in answer to this question. Upon the basis of Postulate 3, it follows that we can find and remove causes of variability until the remaining system of causes is constant or until we reach that state where the probability that the deviations in quality remain within any two fixed limits (Fig. 5) is constant. However, this assumption alone does not tell us that there are certain limits within which all observed values of quality should remain provided the causes cannot be found and eliminated. In fact, as long as O O O O O : AS TIME GOES ON FIG. 5.—JUDGMENT PLUs MoDERN STATISTICAL MACHINERY MAKEs PossIBLE THE ESTABLISHMENT of SUCH LIMIts the limits are set so that the probability of falling within the limits is less than unity, we may always expect a certain percentage of observations to fall outside the limits even though the system of causes be constant. In other words, the accept- ance of this assumption gives us a right to believe that there is an objective state of control within limits but in itself it does not furnish a practical criterion for determining when variations in quality, such as those indicated in Fig. 3, should be left to chance. Furthermore, we may say that mathematical statistics as such does not give us the desired criterion. What does this situation mean in plain everyday engineering English? Simply I8 ECONOMIC CONTROL OF QUALITY this: such criteria, if they exist, cannot be shown to exist by any theorizing alone, no matter how well equipped the theorist is in respect to probability or statistical theory. We see in this situation the long recognized dividing line between theory and practice. The available statistical machinery referred to by the magazine Nature is, as we might expect, not an end in itself but merely a means to an end. In other words, the fact that the criterion which we happen to use has a fine ancestry of highbrow statistical theorems does not justify its use. Such justification must come from empirical evidence that it works. As the practical engineer might say, the proof of the pudding is in the eating. Let us therefore look for the proof. 3. Evidence that Criteria Exist for Detecting Assignable Causes A. Fig. 6 shows the results of one of the first large scale experiments to determine whether or not indications given by 4.O T- 3.5 H- uj 3.OH- - > F PER CENT PER CENT PER CENT § 2.5H CONTROLLED CONTROLLED CONTROLLED º C 2.O!— H. 2 § 1.5F AVERAGE Or. UL, Ol. ºf AVERA GE O.5 H. O I I | 1923 - 1924 | 925 1926 (9 MonTHS) YEAR FIG. 6.-Evidence of IMPROVEMENT IN QUALITY witH APPROACH to CoNTROL. such a criterion applied to quality measured in terms of fraction defective were justified by experience. About thirty typical items used in the telephone plant and produced in lots running into the millions per year were made the basis for this study. As shown in this figure, during 1923–24 these items showed SCIENTIFIC BASIS FOR CONTROL I9 68 per cent control about a relatively low average of 1.4 per cent defective." However, as the assignable causes, indicated by deviations in the observed monthly fraction defective falling outside of control limits, were found and eliminated, the quality of product approached the state of control as indicated by an increase of from 68 per cent to 84 per cent control by the latter part of 1926. At the same time the quality improved; in 1923–24 the average per cent defective was 1.4 per cent, whereas by 1926 this had been reduced to O.8 per cent. Here we get some typical evidence that, in general, as the assignable causes are removed, the variations tend to fall more nearly within the limits as indicated by an increase from 68 per cent to 84 per cent. Such evidence is, of course, one sided. It shows that when points fall outside the limits, experience indicates that we can find assignable causes, but it does not indicate that when points fall within such limits, we cannot find causes of variability. However, this kind of evidence is provided by the following two typical illustrations. B. In the production of a certain kind of equipment, considerable cost was involved in securing the necessary electrical insulation by means of materials previously used for that purpose. A research program was started to secure a cheaper material. After a long series of preliminary exper- iments, a tentative substitute was chosen and an extensive series of tests of insulation resistance were made on this material, care being taken to eliminate all known causes of variability. Table 2 gives the results of 204 observations of resistance in megohms taken on as many samples of the proposed substitute material. Reading from top to bottom beginning at the left column and continuing throughout the table gives the order in which the observations were made. The question is: “Should such variations be left to chance?” No a priori reason existed for believing that the measure- ments forming one portion of this series should be different from those in any other portion. In other words, there was * Jones, R. L., “Quality of Telephone Materials,” Bell Telephone Quarterly, June, I927. 2O ECONOMIC CONTROL OF QUALITY no rational basis for dividing the total set of data into groups of a given number of observations except that it was reasonable to believe that the system of causes might have changed from day to day as a result of changes in such things as atmospheric conditions, observers, and materials. In general, if such changes are to take place, we may readily detect their effect if we divide the total number of observations into compar atively small subgroups. In this particular instance, the size of the subgroup was taken as four and the black dots in Fig. 7-a show the successive averages of four observations in the order, in which they were taken. The dotted lines are the TABLE 2.-ELECTRICAL RESISTANCE OF INSULATION IN MEGoHMs— SHOULD SUCH VARIATIONS BE LEFT to CHANCE 2 5,045 || 4,635 | 4,700 || 4,650 |4,640 || 3,940 4,570 4,560 || 4,450 | 4,500 || 5,075 |4,500 4,35o 5, Ioo 4,600 4,170 || 4,335 | 3,700 || 4,570 3,075 || 4,450 || 4,770 || 4,925 |4,850 4,350 | 5,450 |4,IIo 4,255 5,ooo 3,650 || 4,855 2,965 || 4,850 | 5,150 | 5,075 |4,930 3,975 || 4,635 | 4,4Io 4,170 || 4,615 || 4,445 || 4,16o 4,080 || 4,450 | 4,850 || 4,925 |4,700 4,290 || 4,720 | 4,180 || 4,375 || 4,215 4,000 || 4,325 || 4,080 3,635 | 4,700 5,250 |4,890 4,430 || 4,810 || 4,799 || 4,175 || 4,275 || 4,845 || 4,125 || 4,425 | 3,635 | 5,000 || 4,915 |4,625 4,485 4,565 4,790 || 4,550 || 4,275 5,000 |4, Ioo 4,300 | 3,635 | 5,ooo 5,600 |4,425 4,285 || 4,4Io |4,34o 4,459 || 5,Coo 4,560 |4,34o 4,430 3,900 || 5,000 || 5,075 |4,135 3,98o 4,065 4,895 || 2,855 4,615 || 4,700 || 4,575 || 4,840 || 4,34o 4,700 4,450 |4, ISO 3,925 4,565 5,750 | 2,920 | 4,735 | 4,310 || 3,875 || 4,840 || 4,34o 4,500 || 4,215 |4,080 3,645 5,190 4,740 || 4,375 4,215 4,310 || 4,050 |4,310 | 3,665 |4,840 || 4,325 |3,690 3,769 || 4,725 | 5,ooo 4,375 4,700 5,Ooo 4,050 || 4,185 3,775 | 5,075 || 4,665 5,050 3,300 || 4,640 || 4,895 || 4,355 |4,700 || 4,575 || 4,685 4,570 || 5,000 || 5,000 || 4,615 |4,625 3,685 4,640 || 4,255 || 4,090 || 4,700 || 4,700 || 4,685 4,700 || 4,850 || 4,770 || 4,615 |5,150 3,463 4,895 4, I 7O || 5,OOO 4,7CO || 4,430 || 4,430 4,440 || 4,775 4,570 || 4,5CO || 5,250 5,2Ool 4,790 3,850 | 4,335 | 4,095 || 4,850 | 4,300 || 4,850 | 4,500 || 4,925 || 4,765 | 5,ooo 5, Iool 4,845 || 4,445 || 5,ooo 4,095 || 4,850 || 4,690 || 4,125 || 4,770 || 4,775 4,500 || 5,000 limits within which experience has shown that these observa- tions should fall, taking into account the size of the sample, provided the variability should be left to chance. Several of the observed values lie outside these limits. This was taken as an indication of the existence of causes of variability which could be found and eliminated. Further research was instituted at this point to find these SCIENTIFIC BASIS FOR CONTROL 2. I causes of variability. Several were found, and after these had been eliminated another series of observed values gave the results indicated in Fig. 7-b. Here we see that all of the points lie within the limits. We assumed, therefore, upon the basis of this test, that it was not feasible for research to go much further in eliminating causes of variability. Because of 5 2 ſ O 5 2 - game sº tº-ºº--> sºme *mº dºmº *E==º gº *º -O— * -º-º- * 4 8 hºmº O O O 4 B - 4 6 ºmº O O O O 4 6 }* 44Hºs-a-e-ºr- 4 4. - OO O O O 4 2 º O 4 2 m ºmº gººmsºmº ºmºmº ºmºmº º Ok— 3- *º-º-º-º-º: –0 O- &E =º tºº-º-º-º: *E=- —* tº- * = *mº sºmºmº 3 8 hºmºs O O O i 38 }* O O 3. 6 ºmº 3 6 lº 3 4. ſ ſ f 1 3 4. - I | | | O |O 2O 3O 40 5 O O |O 2O SA M P L E SA M P L E ANSWER "NO" ANSWER "YES" FIG. 7.-SHOULD THESE WARIATIONS BE LEFT to CHANCE: the importance of this particular experiment, however, con- siderably more work was done, but it failed to reveal causes of variability. Here then is a typical case where the criterion indicates when variability should be left to chance. C. Suppose now that we take another illustration where it is reasonable to believe that almost everything humanly possible has been done to remove the assignable causes of variation in a set of data. Perhaps the outstanding series of observations of this type is that given by Millikan in his famous measurement of the charge on an electron. Treating his data in a manner similar to that indicated above, we get the results shown in Fig. 8. All of the points are within the dotted limits. Hence the indication of the test is consistent with the accepted conclusion that those factors which need not 22. ECONOMIC CONTROL OF QUALITY be left to chance had been eliminated before this particular set of data were taken. 4. Rôle Played by Statistical Theory It may appear thus far that mathematical statistics plays a relatively minor rôle in laying a basis for economic control of quality. Such, however, is not the case. In fact, a central concept in engineering work today is that almost every physical property is a statistical distribution. In other words, an observed 48.o.B— - - - - - - - - - - - – – — — — O Q O o X 4 7, 9 H. O (l) O § O O O U- 47.8 O w w < 47.6 F- - - - - - - – – — — — — — — — — 4 7.5 | l | | ſ I I J O 2 4. 6 8 |O | 2 |4 | 6 SA M P L E FIG. 8.-VARIATIONS THAT SHOULD BE LEFT To CHANCE—DoES THE CRITERION WoRK ? “YES.” set of data constitutes a sample of the effects of unknown chance causes. It is at once apparent, therefore, that sampling theory should prove a valuable tool in testing engineering hypotheses. Here it is that much of the most recent math- ematical theory becomes of value, particularly in analysis involving the use of comparatively small numbers of observa- tions. Let us consider, for example, some property such as the tensile strength of a material. If our previous assumptions SCIENTIFIC BASIS FOR CONTROL 23 are justified, it follows that, after we have done everything we can to eliminate assignable causes of variation, there will still remain a certain amount of variability exhibiting the state of control. Let us consider an extensive series of data recently published by a member of the Forest Products Lab- oratories, Fig. 9. Here we have the results of tests for modulus of rupture on 1,304 small test specimens of Sitka spruce, the kind of material used extensively in aeroplane propellers “ſ 55 H. 50 3.44 5 3 O ol—º I ſ ſ | º 1–1 | f | | | I —! 3O 4O 5 O 6O 7O 80 90 IOO MODULUS OF RUPTURE IN IOO POUNDS PER SOUARE INCH FIG. 9.-VARIABILITY IN MODULUS OF RUPTURE OF Clear SpecIMENs of GREEN SITKA SPRUCE TYPICAL OF THE STATISTICAL NATURE OF PHYSICAL PROPERTIEs. during the War. The wide variability is certainly striking. The curve is an approximation to the distribution function for this particular property representing what is at least approxi- mately a state of control. The importance of going from the sample to the smooth distribution is at once apparent and in this case a comparatively small amount of refinement in statistical machinery is required. * Newlin, J. A., Proceedings of the American Society of Civil Engineers, September 1926, pp. 1436–1443. 24 ECONOMIC CONTROL OF QUALITY Suppose, however, that instead of more than a thousand measurements we had only a very small number, as is so often the case in engineering work. Our estimation of the variability of the distribution function representing the state of control upon the basis of the information given by the sample would necessarily be quite different from that ordinarily used by engineers, see Fig. Io. This is true even though to begin with we make the same kind of assumption as engineers have been 1.8 r 1 7 |N E W .2.3.4.5.6 -mº|Fºs - OLD .O º | I 1 1—1—1–1 | | ſ f ſ ſ ſ 1–1–1–1 3. 5 |O 2O 50 |OO SA M P L E SIZE ſh O. 9 º FIG. Io.—CoRRECTION FActors MADE PossIBLE BY MoDERN STATISTICAL THEORY ARE OFTEN LARGE–TYPICAL ILLUSTRATION. accustomed to make in the past. This we may take as a typical example of the fact that the production engineer finds it to his advantage to keep abreast of the developments in statistical theory. Here we use new in the sense that much of the modern statistical theory is new to most engineers. 5. Conclusion Based upon evidence such as already presented, it appears feasible to set up criteria by which to determine when assignable SCIENTIFIC BASIS FOR CONTROL 25 causes of variation in quality have been eliminated so that the product may then be considered to be controlled within limits. This state of control appears to be, in general, a kind of limit to which we may expect to go economically in finding and removing causes of variability without changing a major portion of the manufacturing process as, for example, would be involved in the substitution of new materials or designs. CHAPTER III ADvantAGEs SECURED THROUGH ContROL I. Reduction in the Cost of Inspection If we can be assured that something we use is produced under controlled conditions, we do not feel the need for inspecting it as much as we would if we did not have this assurance. For example, we do not waste our money on doctors' 2Or- O O O o- 16H- U > 5 12 #. O LL O O O 8H O É O O O O lil O O Ö 0. O O O O O O O O O ol————-i-i-t-t-T-I-T-I-T-F J A S O N D J F M A M J J A S O N D J F M A M J |927 | 928 | 92.9 MO N T H S O.6 rº O –––––––––– O O ſ>< O.4– —a––2–3–––?— — — U O ; - O 8 O.2 O > () § OH- O O < CC Q O O tal > O 3 -O.2H- O ° e -O.4 l ſ | | I t | f | | | | | ſ | 1–1 | ſ ſ ſ | f ſt | J A S O N D J F M A M J J A S O N D J F M A M J |927 | 928 | 92.9 MONTHS FIG. I. I.-APPROACH to STABLE EQUILIBRIUM or ContRol. As AssIGNABLE CAUSEs ARE WEEDED OUT, THUs REDUCING THE NEED FOR INSPECTION. 26 ADVANTAGES SECURED THROUGH CONTROL 27 bills so long as we are willing to attribute the variability in our health to the effects of what in our present terminology corresponds to a constant system of chance causes. & In the early stages of production there are usually causes of variability which must be weeded out through the process of inspection. As we proceed to eliminate assignable causes, the quality of product usually approaches a state of stable equilibrium somewhat after the manner of the two specific illustrations presented in Fig. II. In both instances, the record goes back for more than two years and the process of elimination in each case covers a period of more than a year. It is evident that as the quality approaches what appears to be a comparatively stable state, the need for inspection is reduced. 2. Reduction in the Cost of Rejections That we may better visualize the economic significance of control, we shall now view the production process as a whole. We take as a specific illustration the manufacture of telephone equipment. Picture, if you will, the twenty or more raw materials such as gold, platinum, silver, copper, tin, lead, wool, 'rubber, silk, and so forth, literally collected from the four corners of the earth and poured into the manufacturing process. The telephone instrument as it emerges at the end of the production process is not so simple as it looks. In it there are 2OI parts, and in the line and equipment making possible the connection of one telephone to another, there are approximately I IO,OOO more parts. The annual production of most of these parts runs into the millions so that the total annual production of parts runs into the billions. How shall the production process for such a complicated mechanism be engineered so as to secure the economies of quantity production and at the same time a finished product with quality characteristics lying within specified tolerances? One such scheme is illustrated in Fig. 12. Here the manu- facturing process is indicated schematically as a funnel, at the small end of which we have the Ioo per cent inspection screen 28 ECONOMIC CONTROL OF QUALITY to protect the consumer by assuring that the quality of the finished product is satisfactory. Obviously, however, it is often more economical to throw out defective material at some of the initial stages in production rather than to let it pass on to the final stage where it would likely cause the rejection of a DEFECTIVETMATERNAL PART S - -- WTTTTTTTTTTTTTTTT |NSPECTION TO REDUCE COST • * * * * * * * * * * * * OF PRODUCTION &::... . . .'; ; , ºº, PARTS DEFECTIVE PARTS * *-* *- :-º ºms - “º ºmº ºm º ºsmº -ºº ºm mº º DEFECTIVE PARTS FINISHED PRODUCT DEFECTIVE ASSEMBLIES GOOD UNITS ~s DEFECTIVE UNITS | OO 9/o INSPECTION TO -º- PROTECT CONSUMER FIG. I.2.-AN EconoMIC PRODUCTION SCHEME. finished unit of product. For example, we see to the right of the funnel, piles of defectives, which must be junked or reclaimed at considerable cost. It may be shown theoretically that, by eliminating assignable causes of variability, we arrive at a limit to which it is feasible to go in reducing the fraction defective. It must ADVANTAGES SECURED THROUGH CONTROL 29 suffice here to call attention to the kind of evidence indicating that this limiting situation is actually approached in practice as we remove the assignable causes of variability. Let us refer again to Fig. 6 which is particularly significant because it represents the results of a large scale experiment carried on under commercial conditions. As the black sectors in the pie charts decrease in size, indicating progress in the removal of assignable causes, we find simultaneously a decrease in the average per cent defective from 1.4 to O.8. Here we see how control works to reduce the amount of defective material. However, this is such an important point that it is perhaps interesting to consider an illustration from outside the telephone field. Recent work of the Food Research Institute of Stanford University shows that the loss from stale bread constitutes an important item of cost for a great number of wholesale as well as some retail bakeries. It is estimated that this factor alone costs the people of the United States millions of dollars per year. The sales manager of every baking corporation is interested, therefore, in detecting and finding assignable causes of variation in the returns of stale bread if by so doing he can reduce this loss to a minimum. Some time ago it became possible to secure the weekly record of return of stale bread for ten different bakeries oper- ating in a certain metropolitan district. These observed results are shown graphically in Fig. I 3. At once we see that there is a definite lack of control on the part of each bakery. The important thing to note, however, is that the bakery having the lowest percentage return, 1.99 per cent, also shows better control than the other bakeries as judged by the number of points falling outside the control limits in the 36-week period. 3. Attainment of Maximum Benefits from Quantity Production The quality of the finished product depends upon the qualities of raw materials, piece-parts, and the assembling process. It follows from theory that so long as such quality 3O ECONOMIC CONTROL OF QUALITY characteristics are controlled, the quality of the finished unit will be controlled, and will therefore exhibit minimum vari- ability. Other advantages also result. For example, by 6.63 6. 6 3. 8 | 4. 5 2 9 9 6. 2 O | | .7O 3.78 FIG. O O ====e ======s=*-e-–2=========="—-----— BAKERY | == *——FFFFF * = ** = ′ = - F-C-T-T-F-C-e- e O O O • * © O O H O O O O →7--------s ---------- ============= O O =============s*-*===== a--- *—— ==e=–-s---º-º-º- — BAKERY 3 Hs-s-s---------- ========Frº--------- O C BAKERY 4 BAKERY 5 BAKERY 6 BAKERY 7 BAKERY 8 BAKERY 9 © * * * * ================ BAKERY Io !-mºm- \º-O-ºſ- * - *-* *-* *-m-m-m * sº- •------- O F**s * ---------- O == ER |--|--|--|-1–1–1–1–1–1–1–1–1–1–1–1–1–1–1–1–1–1–1–1–1–1–1–1–1–1–1–1–1–1–1 || | | | O 5 |O | 5 2O 25 3O 35 4 O W E E K S 13–RESULTs SHowING How ContRol EFFEcts A REDUCTION IN THE Cost of REJEcTIONs. gaining control, it is possible, as we have already seen, to establish standard statistical distributions for the many quality characteristics involved in design. Very briefly, let us see ADVANTAGES SECURED THROUGH CONTROL 31 just how these statistical distributions representing states of control become useful in securing an economic design and pro- duction scheme. Suppose we consider a simple problem in which we assume that the quality characteristic Y in the finished product is a function f of m different quality characteristics, X1, X2, . Xm, representable symbolically by Y = f(X1, X2, . . . , Xm). (3) For example, one of the X's might be a modulus of rupture, another a diameter of cross section, and Y a breaking load. Engineering requirements generally place certain tolerances on the variability in the resultant quality characteristic Y, which variability is in turn a function of the variabilities in each of the m different quality characteristics. It follows theoretically that the quality characteristic Y will be controlled if the m independent characteristics are controlled. Knowing the distribution functions for each of the m different independent variables, it is possible to approx- imate very closely the per cent of the finished product which may be expected to have a quality characteristic Y within the specified tolerances. If, for example, it is desirable to minimize the variability in the resultant quality Y by proper choice of materials, and if standard distribution functions for the given quality characteristics are available for each of several materials, it is possible to choose that particular material which will minimize the variability of the resultant quality at a minimum of cost. & © 5 4. Attainment of Uniform Quality even though Inspection Test is Destructive So often the quality of a material of the greatest importance to the individual is one which cannot be measured directly without destroying the material itself. So it is with the fuse that protects your home; with the steering rod on your car; with the rails that hold the locomotive in its course; with the propeller of an aeroplane, and so on indefinitely. How are 32 ECONOMIC CONTROL OF QUALITY we to know that a product which cannot be tested in respect to a given quality is satisfactory in respect to this same quality? How are we to know that the fuse will blow at a given current; that the steering rod of your car will not break under maximum load placed upon it? To answer such questions, we must rely upon previous experience. In such a case, causes of variation in quality are unknown and yet we are concerned in assuring ourselves that the quality is satisfactory. Enough has been said to show that here is one of the very important applications of the theory of control. By weeding out assignable causes of variability, the manufacturer goes to the feasible limit in assuring uniform quality. 5. Reduction in Tolerance Limits By securing control and by making use of modern statistical tools, the manufacturer not only is able to assure quality, even though it cannot be measured directly, but is also often able to reduce the tolerance limits in that quality as one very simple illustration will serve to indicate. Let us again consider tensile strength of material. Here the measure of either hardness or density is often used to indicate tensile strength. In such cases, it is customary practice to use calibration curves based upon the concept of functional relationship between such characteristics. If instead of basing our use of these tests upon the concept of functional relationship, we base it upon the concept of statistical rela- tionship, we can make use of planes and surfaces of regression as a means of calibration. In general, this procedure makes possible a reduction in the error of measurement of the tensile strength and hence the establishment of closer tolerances. This is true because, when quality can be measured directly and accurately, we can separate those samples of a material for which the quality lies within given tolerance limits from all others. Now, when the method of measurement is indirect and also subject to error, this separation can only be carried on in the probability sense assuming the errors of measure- ment are controlled by a constant system of chance causes. ADVANTAGES SECURED THROUGH CONTROL 33 It is obvious that, corresponding to a given probability, the tolerance limits may be reduced as we reduce the error of measurement. 40,000 20,000 30,000H ſt I R I I ſt 6O 8O RocKw'ELL HARDNESS (E) (a) x - DENSiTY IN GRAMS PER CENTIMETER CUBE 2.4 2.6 2.8 3.O DENSITY 'th GRAMS, PER CENTIMETER CUBE (b) z = TENSILE STRENGTH IN LBS/SQ.. IN. P Y=Rockwel L HARDNESS (E) FIG. I.4.—How ContRol MAKEs PossIBLE IMPROVED QUALITY THROUGH REDUCTION IN ToLERANCE LIMITs. Fig. 14 gives a simple illustration. Here the comparative magnitudes of the standard deviations of tensile strength about 34 ECONOMIC CONTROL OF QUALITY two lines of regression and the plane of regression are shown schematically by the lines in Fig. I4-d. The lengths of these are proportional to the allowable tolerance limits corresponding to a given probability. It is customary practice to use the line of regression between tensile strength and hardness. Note the improvement effected by using the plane of regression. By using the hardness and density together as a measure of tensile strength, the tolerance range on tensile strength cor- responding to a given probability can be made less than it would be if either of these measures were used alone. 6. Conclusion It seems reasonable to believe that there is an objective state of control, making possible the prediction of quality within limits even though the causes of variability are unknown. Evidence has been given to indicate that through the use of statistical machinery in the hands of an engineer artful in making the right kind of hypotheses, it appears possible to establish criteria which indicate when the state of control has been reached. It has been pointed out that by securing this state of control, we can secure the following advantages: I. Reduction in the cost of inspection. 2. Reduction in the cost of rejection. 3. Attainment of maximum benefits from quantity pro- duction. 4. Attainment of uniform quality even though the inspection test is destructive. 5. Reduction in tolerance limits where quality measure- ment is indirect. PART II Ways of Expressing Quality of Product A Review of the Methods for Reducing Large Numbers of Observations of Quality to a Few Simple Functions of These Data Which Contain the Essential Information CHAPTER IV DEFINITION OF QUALITY I. Introductory Note When we analyze our conception of quality, we find that the term is used in several different ways. Hence, it is essential that we decide, first of all, whether the discussion is to be limited to a particular concept of quality, or to be so framed as to include the essential element in each of the numerous conceptions. One purpose in considering the various definitions of quality is merely to show that in any case the measure of quality is a quantity which may take on different numerical values. In other words, the measure of quality, no matter what the definition of quality may be, is a variable. We shall usually represent this variable by the symbol X. In future chapters when we are discussing quality control, we shall treat of the control of the measurable part of quality as defined in any one of the different ways indicated below. The more important purpose in considering the various definitions of quality is, however, to examine the basic require- ments of effective specifications of quality. 2. Popular Conception of Quality Dating at least from the time of Aristotle, there has been some tendency to conceive of quality as indicating the goodness of an object. The majority of advertisers appeal to the public upon the basis of the quality of product. In so doing, they implicitly assume that there is a measure of goodness which can be applied to all kinds of product whether it be vacuum tubes, sewing machines, automobiles, Grape Nuts, books, cypress flooring, Indiana limestone, or correspondence school 37 38 ECONOMIC CONTROL OF QUALITY courses. Such a concept, is, however, too indefinite for practical purposes. 3. Conception of the Quality of a Thing as a Set of Characteristics Quality, in Latin qualitas, comes from qualis, meaning “how constituted” and signifies such as the thing really is. Suppose we consider a simple thing like water. What is it that makes water what it is? One might answer that it is the chemical combination of hydrogen and oxygen represented by the symbol H2O. To do so is to evade the question, how- ever, for to begin with we must know what we mean by the symbol H2O. If we turn to a textbook on chemistry, we find that the quality of water is expressed in terms of its chemical and physical properties. For example, it is colorless in thin layers and blue in thick layers. It is odorless and tasteless, has a density of unity at 4 deg. C., a heat of vaporization of 54o calories at IOO deg. C., and remains a liquid within a certain temperature range. It dissociates at 1,000 deg. C. in accord with the formula H2 + Ozº H2O, I.8% 98.2% and is an active catalyst. Even this description, however, is only an incomplete specification of water in terms of that which makes it what it is. In general, the quality of a thing is that which is inherent in it so that we cannot alter the quality without altering the thing. It is that from which anything can be said to be such and such and may, for example, be a characteristic explainable by an adjective admitting degrees of comparison. Going a little deeper we see that possibly without exception every conceptual “something” is really a group of conceptions more elementary in form. The minimum number of con- ceptions required to define an object may be called the qualities thereof. For example, Jevons says: “The mind learns to regard each object as an aggregate of qualities and acquires DEFINITION OF QUALITY 39 the power of dwelling at will upon one or other of those qualities to the exclusion of the rest.” The same conception underlies the definition of quality of manufactured product as given by a prominent author on this subject. Thus he says: “The term ‘quality’, as applied to the products turned out by industry, means the character- istic or group or combination of characteristics which dis- tinguishes one article from another, or the goods of one manufacturer from those of his competitors, or one grade of product from a certain factory from another grade turned out by the same factory.” ” In this sense a thing has qualities and not a quality. For example, a piece of material has weight density, dimensions, and so on indefinitely. For our purpose we shall assume that, had we but the ability to see, we would find a very large number m' of different characteristics required to define what even the simplest thing really is. A thing is therefore formally defined in this sense, if the specific magnitudes of the m' characteristics are known. Admittedly we do not know a single one of these—not even the number of possible ones in any given case. Those that we take as elementary we believe to be but a combination of several truly elementary ones, so that the nearest we can approach to the description of any physical thing is to say that it has a finite number of measurable characteristics, X1, X2, ..., Xn, where of course, m' is presumably greater than m. Thus we might take the characteristics of capacity, induct- ance, and resistance as defining the quality of a relay. Geo- metrically speaking, the quality of a relay in this sense can be thought of as a point (P = X11, X21, X31) in three dimen- Sional space with coordinate axes X1, X2, and X3, see Fig. I 5. Of course, to define the quality of the relay in terms of those characteristics which make it what it is would require a space of m/ dimensions, where m' is the unknown number of inde- * The Principles of Science, 2nd Edition, page 25. * Radford, G. S., The Control of Quality in Manufacturing, published by Ronald Press Company, 1922, page 4. 4O ECONOMIC CONTROL OF QUALITY pendent characteristics required to define a relay. For example, to characterize a monatomic gas molecule we need a space of six dimensions, since one dimension is required for each of three space coordinates and for each of three velocity components. Quality then as we shall use it may be a quantity having known physical dimensions such as length, velocity, resistance; a quantity representing the magnitude of any entity in-units of the same kind; or merely a number such as a rate, number defective, and so on. X X3 3. P = X11, X21, X31 | | | * P = x1,x2 x3) | | | | | | | ! l X2 | X2 | | ! /– — — — — — J | | | sºme ame = * * * * * * * * -] X| X] FIG. I 5.-QUALITY As A Point IN FIG. I6.—QUALITY ConForMs IF witHIN SPACE. Volume. 4. Conception of the Quality of a Thing as an Attribute Customary engineering practice specifies the limits or tol- erances within which the different quality characteristics are supposed to lie provided the single piece of apparatus or thing under study is to be considered as satisfactory or conforming to specifications. Geometrically this can be represented for the previous example involving three quality characteristics by Fig. 16. A piece of apparatus or thing having a quality falling within the rectangular element of volume is said to possess the positive attribute of conformance to specified standards. Obvi- ously this element of volume may be large because often only DEFINITION OF QUALITY 4. I a single lower or upper bound is given to some one or more of the quality characteristics. If the quality falls outside this volume, the piece of apparatus or thing is said to possess the negative attribute of non-conformance. The property of positive attribute is variously characterized as good, satis- factory, conforming, standard, and that of negative attribute is characterized as unsatisfactory, non-conforming, and so on. 5. Quality of a Number of the Same Kind of Things To begin with, let us consider the information presented in Table 3 giving the measurements of tensile strength, hard- ness, and density on sixty specimens of a certain aluminum die-casting. This table gives three quality characteristics for each specimen." To picture the quality of the group of sixty specimens, it is therefore necessary to consider the one hundred and eighty measures of the different quality charac- teristics given in this table. Now our graphical representation of quality becomes a real aid because we must have some method of visualizing the significance of a set of data such as that in Table 3. First let us think only of the sixty values of tensile strength. How shall we arrive at a simple way of expressing the quality Aº. Aſ º mºl- A *—— a aſſº. AºA • A A.--a Aſh. vºy wºr vºy --—w vºy wºr- wºr wº wº-vºz vºy T ENS LE STRENGTH X FIG. 17-QUALITY IN RESPECT to TENSILE STRENGTH. of the sixty specimens in respect to this characteristic? The answer is simple if we think of the sixty values of tensile strength plotted along a line such as indicated in Fig. 17. Here, of course, we have plotted only a few of the sixty points. This graphical presentation at once suggests that we seek some distribution function to represent the density of the points along the line. If we can find such a function and if this function can be integrated, it is obvious that the integral within 1 The abbreviation psi is used here and elsewhere for pounds per square inch. All hardness measurements are given throughout this book in Rockwell’s “E” even though the “E” may sometimes be omitted. 42 ECONOMIC CONTROL OF QUALITY specified limits gives us the number of specimens having a value of tensile strength within these limits. TABLE 3.−QUALITY ExPRESSED IN TABULAR ForM Speci Tensile Hardness in Density Speci Tensile | Hardness in Density * | Strength Rockwells in *| Strength Rockwells in In en - e. « g T 7 y - men - º « g Tºº y 3 1n psi E gm/cm3 1n psi E. gm/cm3 I 29,314 53. O 2.666 3I 29,250 7.I.3 2.648 Q 34,860 7o. 2 2.708 32 27,992 52.7 2.4OO 3 36,818 84.3 2.86.5 33 31,852 76.5 2.692 4. 3O, I2O 55. 3 2.627 34 27,646 63.7 2.669 5 34,O2O 78.5 2. 581 35 31,698 69.2 2.628 6 30,824 63.5 2.633 36 30,844 69.2 2.696 7 35,396 7I.4 2.671 37 31,988 61.4 2.648 8 31,260 53.4 2.650 38 || 36,640 83.7 2. 775 9 || 32, I84 82.5 2.717 39 4I,578 94.7 2.874 IO 33,424 || 67.3 2.614 4o 30,496 7o. 2 2. 700 II 37,694 69.5 2. 524 4 I 29,668 8o. 4 2.583 I 2 34,876 73. O 2.74 I 42 32,622 76.7 2.668 I3 24,660 55.7 2.619 43 32,822 82.9 2.679 I4 34,760 85.8 2.755 || 44 || 30,38o 55. O 2.609 I 5 38,02O 95.4 2.846 45 38,580 83.2 2.721 I6 25,680 5.I . I 2. 575 46 28,202 62.6 2.678 17 25,810 74.4 2. 561 47 29, 190 78.o 2.6 Io I 8 26,460 54. I 2. 593 48 35,636 84.6 2.728 I9 28,070 77.8 2.639 49 || 34,332 64.o 2. 709 2O 24,640 52.4 2.61 I 5o 34,750 75.3 2.88o 2. I 25,770 69. I 2.696 5 I 40,578 84.8 2.949 22. 23,690 53.5 2.606 52 28,900 49.4 2.669 23 28,650 64.3 2.616 53 34,648 74. 2 2.624 2 32,38o 82.7 2.748 54 3I 3244 59.8 2. 705 25 28,2 Io 55.7 2. 518 55 33,802 75.2 2.736 26 34,OO2 7O. 5 2.726 56 34,850 57.7 2.701 27 | 34,470 87.5 2.875 57 || 36,690 79.3 2.776 28 29,248 5o. 7 2.585 58 32,344 67.6 2. 754 2 28,710 2. 3 2. 547 59 34,440 77.o 2.660 3O 29,830 59.5 2.606 6o 34,650 74.8 2.8.19 In a similar way we may represent the sixty observed values of tensile strength and one other property, such as hardness, by sixty points in a plane. Again the graphical representation suggests the need for some distribution function DEFINITION OF QUALITY 43 which will give us the density of the points in this plane. In just the same way, the graphical representation of the values of tensile strength, hardness, and density in three dimensional space suggests the need for a distribution function indicating the density in space. The graphical representation of the sixty points in a plane and in space was given in Fig. I4. In the inspection of product manufactured in quantities running into the thousands or even millions of pieces per year, it would be a very laborious task to measure and record as a olaſ O |2 O O º O O 8 }* O O.O6 – O.O4 H. O.O2 H. O L –1 | | ſ | ſ f L– L | 1– J F M A. M J J A S O N. D M ON THS. FIG. I 8.—REcoRD OF QUALITY IN TERMs of FRACTION DEFECTIVE. variable the quality characteristic for each piece of apparatus or piece-part. Instead, the practice is usually followed of recording only the fraction non-conforming or defective in each lot of size N. In the course of a year, then, we have a record such as shown graphically in Fig. 18 representing the quality of a given kind of apparatus measured in terms of fraction defective. In the general case each piece of apparatus is supposed to possess several quality characteristics and the results of an inspection of a lot of size N on the basis of, say m, quality characteristics, X1, X2, . . . , Xm, can be reported either as the fractions, pi, p2, . . . , pm, within limits for the respective 44 ECONOMIC CONTROL OF QUALITY characteristics or the fraction p within all the limits. Obviously, this fraction p does not give as much information about the product as the set of m fractions. 6. Quality of Product Thus far we have considered the meaning of the quality of a number of the same kind of things such as the set of sixty specimens of a given kind of die-casting. Now we come to the problem of expressing the quality of a product for a given period of time where this product is composed of M different kinds of things, such as condensers, relays, vacuum tubes, telephone poles, and so on. g We must define quality of product in such a way that the numerical measure of this quality serves the following two purposes: I. To make it possible for one to see whether or not the quality of product for a given period differs from that for some other period taken as a basis of comparison. 2. To make possible the comparison of qualities of product for two or more periods to determine whether or not the dif- ferences are greater than should be left to chance. A. Distribution of Quality Characteristics Let us assume that there are N1 things of one kind such as condensers, N2 things of another kind such as relays, and finally NM things of the Mth kind. Let mi, m2, . . . , my represent the number of quality characteristics on the M different kinds of things. From what we have already seen, it is obvious that our picture of quality must be derived in some way from the mi + m2 + . . . -- my observed frequency distributions of the quality characteristics. The quality of product for two different periods consists of two such sets of frequency distributions. For example, Fig. 19 shows 12 observed frequency distributions for a single quality char- acteristic, efficiency, for a given kind of product over a period of twelve months. Since there were five quality characteristics DEFINITION OF QUALITY 45 for this particular kind of apparatus, the complete record of quality requires five sets of frequency distributions similar to “I JULY Fº AUGUST p- SEPTEMBER 3OOH- F- |- 200- H. pºe | OOH- }- He O ſl ſº l f i i —a il A In 1 A A 1 ſ l 1 ſ | 1 ſº “I OCTOBER Ps NOVEMBER Fº DECEMBER 3OOH- yº. }* (ſ) 2 2OO }* }* P- u! > }= H. 5 looH CC * O l 1 - 1 1–1 — I | 1 – 1 – 1 1 — I 1 - 1 1 2 ta- O 4OOF- JAN, UARY }* FEBRUARY pºs MARCH Cº. tal on 300 }* }* >: ID pe 2 2OO | OO yºs }= O f i ſl R Il | ſi I I l I I 400 r APRIL. P- MAY }* JUNE 3OOH- H. is 2OOH- }* * |OOH- E F F 1 C E N C Y X FIG. 19.-QUALITY REcoRD IN TERMs of OBSERVED FREQUENCY DISTRIBUTIONs. those shown in Fig. I9. As already said, the corresponding picture of the quality of product consisting of M different kinds of apparatus or things would require as many sets of 46 ECONOMIC CONTROL OF QUALITY such distribution functions as there are quality characteristics. Such a picture contains the whole of the available information. B. Quality Statistics The information presented in the form of frequency dis- tributions does not permit readily of quantitative comparison. – i.2 r- O |X O O O § - 1:4 º < — (ſ) O.8 O O 5.O r- O (\! O Q, O o 4.O H. (ſ) Lil • O Z O H 3.O - i ( ) O <[ ; * O - —! O O u- } O 2.0 L | I ſ | | * 1– | | 1 –1 J A S O N D J F M A M J WM O N T H S FIG. 20.-QUALITY REcoRD IN TERMS OF STATISTIcs. To get around this difficulty, we may use instead of the fre- quency distribution itself some characteristic or statistic of this distribution, such as the fraction within a given range, the average, the dispersion, or the skewness. For example, the information given in Fig. 19 is presented in terms of certain . DEFINITION OF QUALITY 47 of these statistics in Fig. 20. Whereas we have only one frequency distribution for each characteristic, we have one or more statistics for each distribution. These statistics, however, give us a quantitative picture of the variation in the given quality characteristic. C. Quality Rate The two measures of quality just considered are based upon the conception of quality as that which makes a thing what it is and, therefore, involve the use of as many quality characteristics as are required to define the product. In this sense, the quality of one thing cannot be added to that of another; for example, the quality of a condenser in. terms of capacity, leakage, and so forth, cannot be added to the quality of a telephone pole in terms of its modulus of rupture and other physical properties. - If, however, we can find some measure of the goodness of a thing, no matter what it is, we can then get a single quanti- tative measure of quality of product. One way of doing this is to weight each quality characteristic. As an example, let us assume that for some one quality characteristic Xi of the product, we have the observed relative frequency distribution Xii, Xiz, . . . , Xij, . . . , Xin, (4) Pil, Piz, . . . , Pij. . . . ; pin, 4. where the X's represent the n, different observed values of the variable Xi, pij N. is the number of times that the characteristic Å; was observed, and N, is the total number of things having the quality characteristic Xi. By choosing a weighting factor wi(X) where w, is a functional relationship different, in general, for each characteristic, we get a transformed frequency dis- tribution wi(Xil), wi(Xiz), . . . , wi(Xij), . . . , wi(Xin) |. pil; Pi3, . . . ; Pij. . . . ; pin, It is assumed usually that the weights are additive so that the total weight //; for the quality characteristic X, on the N. pieces of product having this characteristic is W = Wiſpiiw:(Xa) + pºw,(Xià) + ... + pºw;(X;) +...+ pin, w;(Xin)]. (6) (5) 48 ECONOMIC CONTROL OF QUALITY The corresponding total weight W for the whole product then becomes W = W1 + W., + tº e tº + W. + . . . + 7 m + m, + . . .--mºr, (7) where as above there are supposed to be m1 + m2 + . . . -i- m ºf quality characteristics. It is obvious that the total weight from month to month for any given product will vary because of the effects of un- known or chance causes which, as we have already seen, produce variations in the observed distributions of the re- spective quality characteristics. We also see that to be able to interpret the significance of variations in respect to this weight, we must be in a position to consider the significance of variations in the observed frequency functions from which this weight is calculated, assuming that for a given kind of product the number of pieces produced each month is approx- imately the same. In general, an attempt is made to obtain a weighting factor which represents approximately the economic value of a quality characteristic having a given magnitude. Obviously, however, it is very difficult to attain such an ideal, and con- sequently the weights usually represent empirical factors." By dividing the weight W of product for a given period by the weight Ws of the same product over some previous period taken as a base, we get the customary form of index // I = −. (8) Ws It should be noted that the statement that the index of quality is such and such does not give any indication of what the quality is unless we take into account the details of the method underlying the formation of the index. In fact a high or low index does not necessarily mean that the quality is good or bad in a given case unless it is known that for the particular index with which * One very simple form of rate used extensively in the Bell System is described by Mr. H. F. Dodge in an article “A Method of Rating Manufactured Product,” Bell System Technical journal, Volume VII, pp. 350–368, April, 1928. DEFINITION OF QUALITY 49 we are dealing, a high index means good and a low index means bad quality from the accepted viewpoint. 7. 9tuality as a Relationship Often the quality of a thing, such as the quality of a manu- facturing process, is of the nature of a relationship. As an example, we may consider the process of creosoting telephone poles. In general, the depth of penetration of the creosote appears to depend upon several factors, one of which is the depth of sapwood, as is evidenced by the data given in Table 4, showing the depth of sapwood and the corresponding depth of penetration for 1,370 telephone poles. In this case the relationship between these two factors is an important char- acteristic of the quality of the process. To compare the quality of the creosoting process of one plant with that of each of several others, we must try to inter- pret the significance of observed differences in the results obtained by different plants, such as the seven records shown in Fig. 21. To facilitate comparisons of this character, we need to have available quantitative measures of the correla- tion or relationship between the quality characteristics corre- sponding to a given process. The importance of the concept of relationship in specifying quality is more deeply seated than might be indicated by this simple problem. In trying to define the quality of a thing in terms of those characteristics which make it what it is, we called attention to the fact that we make use of what are perhaps secondary characteristics. For example, in expressing the quality of a thing in respect to strength we make use of meas- ures of ductility, brittleness, and hardness—characteristics which are likely dependent to a certain degree upon some com- mon factor more elemental in nature. Hence it follows that not only the magnitudes of the characteristics but also their interrelationships are significant in characterizing a thing. The representation of quality in m space as outlined in a previous paragraph lends itself to a quantitative expression of quality relationship. ‘3 TABLE 4.—QUALITY As A RELATIONSHIP DY Y X Y X Y A Y X Y X V A y X Y X y A P M y M y X Y Ä y 4.60 | 1.90 || 3.oo | 1.20 || 3.20 | 1.5o || 3.70 || 2.oo 2.20 2.15 || 3.oo | 1.8o || 3.25 2.20 | 2.60 | 1.5o || 2.20 | I. Io | I-35 | o.70 || 4.4o | I.6o || 4.5o || 2 4o || 3.90 1.40 || 3.1o 2.15 2.15 | 1.3o 2.90 | 1.8o 2.3o | I’75 2.70 | o.8o | I-90 | 1.5o | I.6o | I.oo || 3.20 | I.85 2. Io I.20 2.20 | I.20 || 3.30 | I.o.5 || 5.4o 2.4o 2.8o | 1.30 || 2.70 | 1.5o 4.1o 3.46 3.70 | 1.60 || 3.55 I.5o || 4. Io 2.oo || 4.40 || I.90 || 4. Io 2.90 || 4.20 | I.o.5 2.75 2.20 || 3.5o 2. I5 || 2.70 || I. Io || 3. Io | I.40 || 2.05 || I.6o || 2. Io | I.70 || 2.05 || I.oo | 1.30 o.8o 2.70 || 2.20 || 3.30 || 2.oo || 3.5o | I.45 || 2.5o | I.20 | 2.8o 2.45 || 2.90 || 2:30 || 3.5o | I.6o 3.60 | I.70 || 3. I5 |o.8o || 3.8o | I.20 | 2.90 I. Io | I. I5 |o.go || 3.30 I.oo || 4.oo | 1.8o 3.20 | 1.5o 3.40 | 1.85 | 1.60 |o.go || 3.60 2.05 || 2. I5 | I.90 || 3.oo o.95 || 2.35 | I. Io || 3.60 | I.45 || 2.25 o.90 || 2 7o o 8o || 3.30 | 2.20 || 3.3o I.6o 2.8o | 1.60 || 2.40 | 1.20 2.40 | 1.70 || 3.8o 2.4o 3.70 | 1.90 || 3.70 | 1.8o | I.go | I-3o || 2.85 |o.85 || 2.5o | I.20 || 3.20 |o.85 || 2.4o I. Io | I.55 | I.oo 2.8o | I.20 | 1.75 o.95 || I. 15 o.8o 2.60 | 1.35 2.15 | 1.15 | 1.70 | 1.oo | 1.75 | I.Io | 1.5o o.85 | 1.8o | I. I5 | 1.95 o.8o 2.70 || 2.5o 2.90 2. Io | I.6o | I.45 2.3o | I.oo || 3.60 | 2.3o || 2. Io | I.5o | 1.85 | 1.4o 2.60 | I.Io I go | I 5o || 3. Io | 1.8o 2.5o | I.5o 2.60 | 1.60 || 3.60 2.20 | 2.90 | I.4o 2.oo | I.o.5 2.35 | I.oo || 3-oS I.4o 2.60 | I.go || 3.4o | I.70 || 3.20 2.oo 2.8o | 1.5o || 2.55 | 1.40 3.5o | 1.5o || 2.5o | 1.90 || 2.3o 1.55 | 2.oo | I.20 | I 75 |o.90 | 2.60 | I. Io 2.45 I-35 | 3.3o | I. I5 || 3.65 2.oo || 3.35 | I.85 | I.65 | 1.15 || 3.70 | I.oo 2.oo | 1.35 | 2.1o | 1.oo 3.5o 2.70 || 2.95 | 1.35 | 3.40 | 1.5o || 3.60 | 2. I5 || 3.90 2.55 || I-90 | I.5o || 3-oS | I.25 || 3.20 | 2.65 | I-95 | I.8o 2.65 | I 20 || 2 6o 2.05 || 2.25 | I.65 | I.95 I. 15 || 3.55 2.05 2. Io | I.oS | 3.oo 2.oo 2.85 | 1.25 | 1.90 | I.20 | I.90 | I.6o 2.85 I.25 || 3-oo 2.3o || 2.35 | I-80 || 4 4o o 9o 2.65 | I. Io 2.45 I.I.5 3.20 | 1.45 || 3.70 | 1.80 || 3.40 | 1.35 2.45 | 1.60 | 2.75 I.20 | 2.60 | 1.5o || 3.65 2.oo 2.25 | I.6o 2.8o | I.4o 2.95 I.5o || 3.70 | 1.5o || 2.4o | I-40 || 3.05 |o.70 || 3.55 | 1.65 || 4.oo 3.60 | 1.45 | I.Io 3.5o | 1.5o 3.30 | 1.45 || 2.65 | 1.65 || 3.5o | 1.40 || 3.oo I. Io 2.30 o.90 || 2.20 |o.90 || 2.60 | I.oo 2.60 I.I.5 2.5o o.75 | I.5o |o.8o | I.4o o.75 | 1.90 | 1.3o 2.70 || 2.4o 3.4o | 1.1o 2.90 | 1.65 2.25 | 1.35 | 3.70 | 1.35 | 3.05 || I.5o 2. Io |o.95 || 3. Io I.5o | I.90 |o.90 || I.6o | I.o.5 || 3.4o I.20 | I.6o | I.45 || 3.05 | I.20 || 3.60 | 1.65 3.70 | 1.8o || 3.oo 2.20 3. Io 2. Io || 3.90 | 3.3o 3.90 | 1.3o 4.oo 2.85 2.3o | I.75 | 2.60 o.8o || 4.20 2. Io 3.85 | 1.85 || 3.75 | I-95 || 2. Io | I.I.5 || 3.8o 2.75 || 3.oo | I.os | 4.oo 2.65 |3.3o 2.5o 3.5o | 1.60 | 1.70 | 1.55 2.05 | 1.60 | 2.70 || 2.oo || 3.35 | 2.85 || 3.70 || I. Io || 3.25 | I.95 || I.95 || I.5o 2.65 I.oo 3.25 || 2.75 | I.85 | I.o5 || I.75 I. Io 3.40 | 1.60 || 2.90 | 1.4o 1.5o o.85 2.70 | 1.25 3.15 | 1.75 || 3.Io | I.5o || 3.25 | 2.3o. 2.90 o.90 || 3.25 | I.25 || 2.45 I-55 || 2.20 | I-40 || 2.85 | I.6o | I-55 o.85 | 2.3o | I.7o 2.70 | 1.40 2.45 | 1.25 2.95 | 1.55 || 3. Io | 1.35 | 3.35 | 1.90 || 3.20 | 1.75 3-3o | I.65 2.8o 2.55 || 2.8o |o.90 2.45 | I.oo 2.90 | 1.45 || 3.20 2.oo || 3.5o 2.4o 2.75 I-30 || 3.25 | 2.20 | 2.35 | 1.35 2.90 o.go 4.ro | I.Io || 4.oS | 1.3o 2.4o I.5o 2. I5 I.20 || 3.oS I-90 || 2.60 |o.90 || 2.25 | I.óo 2.70 || I.oo 2.70 || 2.4o 4.3o I.3o || 4.20 | I.oS 3.30 | 1.70 || 2.5o | 1.5o I.70 | o.85 | 2.55 | 1.4o | 2. Io | 1.40 || 2.8o | I.40 || 2.3o | I.5o 2.4o | I.45 || 3.40 |o.95 || I-95 I-35 | 2.20 | I. Io || 3.40 | I-90 || 2.85 I.90 || 2.8o | I.90 || 2.70 | 1.35 | 2.85 2.oo I. Io I.oo 2.4o | 1.3o 3.05 || I.OS 3.4o 2. Io | I.65 o.95 || 3.20 | I.70 || 3-oo 2.5o 2.5o I.8o || 2.5o I.oo || 2.90 | I-35 | 2.70 || I.20 | 2.60 | I.70 || 2.5o o.75 2.70 | 1.60 2.70 | 1.20 || 2.8o | I.óo 2.20 o.go | I.65 o.75 3.o.5 | 1.3o | I-85 | I.6o || 2.5o | I-80 || 2.90 I.4o | I-95 || I.20 || 3.oo 2.oo 3.70 || 2.Io 2. Io | I.20 || 3.oo o.65 4.4o | 1.55 3. Io o.85 || 4.70 | 1.5o || 2.40 o.go 4.5o I.90 3.90 |o.70 || 5-oo 2.60 || 4.90 | I-70 || 4.oo 2.5o 3.Io o.85 || 3.4o 2.90 || 2.70 || 2. Io || 3.05 || I.95 || 3.15 | 1.60 || 3. Io | 1.56 3.30 | 1.8o |4. I5 || 3.o.o 4.25 2.65 3.4o | I.40 || 3.60 | I.95 || 2.70 || I-55 || 3.95 || I-35 || 4.20 || 3. Io 2.25 | I. Io || 3.oo o.70 || 2.40 o.4o 4.oo | I.6o 4.3o 2.55 || 3.5o | 1.30 2.8o I.I.5 || 2.75 o.82 || 3'20 2. Io | I.3o o.6o 2.85 o.90 || 4.oo 2.4o 2.85 | 1.3o 3.60 | I.20 || 3.20 2.20 2.3o | I.65 || 2.25 o.85 || 3.5o 2.3o 2.55 | 1.55 || 3.oo | 1.15 3.20 | 2.55 || 2.85 | 1.90 || 2.65 I.I.5 2. Io I. Io | 2.05 | I.5o 2.15 | 1.65 || 2.65 I.20 | 3.65 2.4o 2.95 || I.25 2.90 2.60 || 2.70 | 1.8o || 3.Io o.8o | 1.90 | 1.25 || 5.20 || 4.76 3.5o | 1.8o || 3.70 | 1.70 || 4.8o 2.70 || 4.8o 2.90 || 4.oo | I-80 || 3.70 | I.70 || 2.70 || 2.95 || 3.20 2.05 || 2.60 o.95 || 2.70 || I-75 | I.I.5 |o.55 | 1.75 I.6o | 1.40 | I.os | 3.60 | 2.65 1.70 | I.oo | I.90 | 1.5o | I.95 I.oo || 3.75 3.35 | 2.85 2.3o 3.20 | I.25 || 3.oo | I.6o || 2.70 || 2.99 || I.90 I.oo || 3-oo 2.85 2.05 | 1.3o 2.4o o.90 || 2.70 || 2.60 || 2.55 | 1.46 2.5o o.8o 3.45 2.90 2.15 | 1.5o 2.05 I. Io 2.20 | I.4o 2.40 | 1.40 || 3.45| 2.20 | I.20 o.8o 2.oo I.95 || 3.45|o.85 | I.6o o.8o || 3.1o 2.15 4.oo 2.70 || 4.40 | 1.5o 4. Io 2.60 || 3. Io | 1.5o 3.3o o.8o || 3.05 || 2.oo || 4.55 || 3.15 || 4.oS 2.5o || 3. Io I.oo 3.75 2.5o || 3.3o o.8o 2.35 | I.o5 || 2.oo I.6o | I.25 o.75 || 2.40 | I.Io || 4.15 | 1.75 2.70 | I.oS | 3.90 | 1.25 || 4.oo I.25 || 3.90 3.20 | 3.70 | 1.8o || 3.40 || 2:30 || 3.40 | 2.40 || 2.5o | I. Io 2.30 I.4o | 2.90 | I.70 || I-3o | I.oo 2.oo | I.70 || 2.5o | I.Io || 3.96 || 3:30 2.70 || I.20 I.6o o.90 || 3.60 o.90 || 2.5o | I.oo | I.óo o.90 2.70 || I.5o 3. Io I.20 | I-90 o.70 || 3-3o I.4o 2.oo | I.20 || I.8o | I.oo || 2.30 || I.3o 3.3o | I.Io | 1.70 | 1.4o 2.60 | I.go || 4.70 | 1.20 || 3.5o | I.70 || I.6o o.90 || 2.8o 2. Io || 2. Io o.70 || 2. Io o.90 2.80 I.oo I.90 o.8o 4.oo | I.20 || 4-3o 2.Io 4.20 | 1.5o 2.60 | 1.40 || 3.90 | 1.8o 3.4o I.8o || 2.60 | 1.5o 4. Io 2.70 || 3. Io I.5o 2.40 o.70 || I-95 o.70 || I.90 | I.25 || 2.20 | I.25 || 2.5o | I-30 || 2.35 | I.25 || 2.oo o.75 3.20 I.8o || 2.15 I.Io 4.15 2.8o 4.5o || 3.55 || 5.oo 4.15 || 3.35 | I.oS | 2.20 | 1.25 || 2.5o o.6o 2.55 I.oo | I.6o |o.6o 2.40 | I.20 | I.70 | I.I.5 || 3.15 I.4o 2.60 | I.Io || 4.1o 2.60 || 3.8o | I.oo || 3.85 2.60 2.95 | 1.20 | 2.95 || 2.40 || 2.4o | 1.5o || 3.40 | 1.40 || 3:30 | 1.55 || 3.8o | I-90 || 3.8o 2.30 || 3.4o o-go || 4-oo 2.65 || 3.40 | 1.45 || 3.8o | I.os | 4.20 | 3.5o 2.05 | 1.35 | 2.56 | 2.35 3.15 I.25 || 4.20 | 2. Io |4.20 2.20 || 3.40 | 1.25 2.15 o.7o 2.95 || 2. Io 3.oo 2. Io || 3.5o | I-80 || 3.25 I.I.5 || 4.30 2.oo 2.3o o.95 || 3.60 | 2.Io 3.3o I.20 || 4.60 2.75 2.30 o.8o 2.30 | 1.8o 3.5o | I-35 | 3.3o I.25 | 2.3o | I.20 | I.6o I.o.5 || 3.5o I-80 || 2.90 | I-90 || 2.85 I-55 || 3.60 2.60 I.65 | 1.3o | 1.25 o.go || 2.95 | 1.4o 3.1o | 1.56 I.8o I.20 2.35 | 1.35 | 1.65 |o.75 | 1.40 o.8o | I-55 o.75 || 3. I5 2.90 || 3.25 | I-35 | 2.70 | I. Io I.65 I.o.5 3.25 | I. Io 2.85 I.5o 2.35 | 1.15 || 2.15 | 1.25 || 2.65 o.go I.8o o.95 || 3.oo | 1.8o | 1.70 || o.70 || 3.25 || 2:30 || 2.8o | I.95 || 3.45|I.óo || 2.oo | I.oo 2.75 2.20 | I.75 I.25 | 2.85 | I.oo || 2.25 | I.oo 2.15 | 1.30 || 2.25 | 1.60 || 3.oo | 1.1o I.70 || I.o.5 | 2.8o 2.90 || 3.5o | I.70 || 2.90 I.5o || 3.70 || 2.90 3.60 | I.oo || 3.60 | I.6o || 2.45 I.5o || 3.60 | 2.20 || 3.95 || 2.15 2.90 | 1.5o | I.95 | 1.60 || 3.25 | 1.35 | 2.90 | 1.35 I-55 | 1.40 || 3.5o I.85 2.5o I.45 || 4. Io 2.15 2.8o | I.85 || 3.5o | I-30 || 3.20 | I-3o 2.85 2. Io | 2.75 2.25 || 3. Io | I.20 || 4.oo 2.40 || 3.60 | I. Io || 3.8o | 1.60 || 3.70 || 2.66 2.95 |o.8o 3.40 I.6o 2.55 |o.90 2.55 I.6o 2.90 || I.85 3.8o 2.8o | I-75 I-35 | 2. Io I.oo 2.35 | I.os | 3.65 | 2.25 || 3.95 |o.85 2.05 | 1.20 | 2.50 | 1.25 2.40 | 1.5o 2.8o | I.I.5 | 2.35 | 1.70 || 3. Io | I.4o 4.oo 2. Io 2.Io I-3o 2.20 | I-3o |4. Io 2.30 || 2.oo o.90 4.4o I.20 3.20 | 1.60 || 3.20 o.8o || 3.75 3.40 2.40 | 1.95 || 4.1o | 1.5o 4.20 2.65 || 3.5o 2.90 || 3.5o | I.5o 2.25 o.90 || 3.3o I.65 || 2.5o | I.70 || 3.60 | 1.75 || 2.55 2.20 || 3.8o | I-30 || 3.40 2.8o || 3.14 || 2.95 || 2.70 | 1.3o 2.30 | 1.70 || 2.60 | 1.70 2.oS o.85 || 2.5o 2.05 || 2 go I. Io 3.05 || I.55 || 3.20 | I-95 3.20 I.5o 2.oS I-3o | I.75 I. Io 2.85 I.4o 2.oo o.go || 1.8o | I.oS 2.65 || 2.35 | 3.70 | 1.5o 2.5o o.8o 3.20 | 1.20 ! I.70 jo.6o | 1.70 | o.5ol 3. Io | I.6o 2. Io o.8o 3.70 | I.65 || 3. Io 2.60 l 3.60 2.ool 2.60 | I.oS 2.oo | I.I.5 || 3.3o 2.30 || 3.70 || 2.1o 2.oo O.75 || 2.65 | 1.90 DEFINITION OF QUALITY 5 I o e co co - or to *4 ºr ºf H & H ex, * 9, 2 3 & 2.33 9 gº º lf) K-4 *s-, * }-i O H oz ºz | S9， º į oºz || oőrz | O & I || o6ºz || oŞ’I Į oºz || oț” I || OI’z | OI ºz. ſ o9º& ! o 4° I || o/ºz | OS: I | C3: z I o Iºz | OS-9 || o6. 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I | Sgrz || oº！ I || očiº | og I || oÍ-9 || OS: I | oſº | oor I || o޺z | 99” I || oŐº | OS: I || S.Șº || Sº Ioºº z o6-9 || osºr | oőt | otrz | oğ-9 || of: I | og 9 || oź·o | Scºz I oz. z | osºg | os: I || oğ·9 | oz. I || oort | ož· I l ooºº | o9. I l ooºº | orº I | oſº | Soºz | olº l o Iº I l osºz I OS™I I oÔºz 52 ECONOMIC CONTROL OF QUALITY C O M PA NY 5 C O M P A NY C O M PA NY 6 C O M PANY 2 q � 4.0 3.4 5- 2.8 ^} tae S 3 H © D N | Q N | COMA PANY 7 COM PANY 3 «so o O (IO) 31, Xj2, • * - 5 3i, . . . , Xin g Xml, Xm2, . . . , Xmi, . . . , Xmn THE PROBLEM OF PRESENTATION OF DATA 59 Naturally, we always have a certain purpose in accu- mulating such a series of data, and the object of tabular and graphical presentation is to assist in the interpretation of the raw data in terms of the object for which they were taken. As already noted, the distributions of values of depth of sapwood and depth of penetration as given in Table 4 illustrate the first form (9) in which raw data may occur. Similarly, the two distributions taken together illustrate the second form (IO). Later we shall have occasion to make use of several simple geometrical conceptions in our study of the ways and means of presenting data. It will be helpful, therefore, for us to keep in mind some of the problems involved in the analysis of data, both from the viewpoint of presentation of facts and from that of causal interpretation stated in terms of these geometrical conceptions. For example, the problem of presenting a series, such as (IO), of m qualities on each of n things may be looked upon as that of locating a set of n points in a space of m dimensions in reference to certain lines, planes, or hypersurfaces. A simple illustration is that previously given in Fig. I4 where we may think of the points as being located in respect to the coordinate axes in one case and in respect to either the lines or planes of regression in the other case. There are many ways in which we may set up this problem. For instance, in the case of two variables X and Y, we may seek some function f(X, Y) such that f(X, Y)dXd Y tells us approximately how many of the observed values lie within the element of area X to X + d X and Y to Y + d'Y. Such a function would give us approximately the density of the observed points in the plane. Sometimes, however, it is more convenient to have some measure of the clustering of the points about a curve Y = f(X). It may be sufficient to know that approximately a certain per cent of the points lie within some band f(X) + e as shown in Fig. 22-6. It may be of interest to note how some of the problems of causal interpretation mentioned at the beginning of this 6o ECONOMIC CONTROL OF QUALITY chapter can be expressed in terms of certain geometrical representations of the data. Thus, if we represent a series of m measurements of some quality characteristic by points along a straight line, we are often interested in knowing whether or ~1.5I O O O O 2.1 5 9. O Q O / >{ __ - O O © Y + d Y H - – --— — — — — — — O / O O O YH - — — — — — — *— — —- O !. O Y O O O | " | | | • e • | | e e " | O O O | | / | | | | mº-ºº: X X+ dx X (a) (b) FIG. 22.-Two Methods of REPResenting DATA. not the particular spacing of the points indicates that the causes of variation between the observed values are such as should be left to chance, Fig. 23-a. Assuming that we have decided that the causes of variation should be left to chance, *-O-O-O-O-O-O-O- OUALITY CHARACTERISTIC X QUALITY CHARACTERIST I C X SAMPLE UNIVERSE (a) (b) FIG. 23.−ScHEMATIC RELATION BETween SAMPLE AND UNIVERSE. we are usually interested in discovering the distribution of the variable to be expected if these same causes are allowed to operate for an indefinite period of time. In other words, we seek the universe of effects for a given cause system, Fig. 23-6. THE PROBLEM OF PRESENTATION OF DATA 61 It is obvious that the other problems of causal interpretation may also be given a geometric significance. 5. True Versus Observed Quality Thus far we have purposely avoided the problem of trying to distinguish between true quality and the observed mag- nitudes of the quality characteristics. Obviously, it is necessary to try to do this since all measurements are subject to error. Hence, to obtain the essential information in respect to the distribution of true quality from a set of observed data such as either (9) or (IO), we must have some means of correcting for errors of measurement existing in the original data. To get a picture of what we mean by true quality, let us consider first a very simple illustration. What is the true length of the line ABF Strictly speaking, it does not have a A. - B true length in the sense of an unchangeable value which is a constant of nature. On the contrary, we believe that the molecules at the ends of the line are jumping around in random fashion so that in the last analysis the line does not have a length except in the sense of some distribution of length or in the sense of some characteristic of a distribution function, such as an average. Whereas, in the case of the length of the line (in fact the magnitudes of most physical quantities) the objective or true quality is a frequency distribution function, there are instances where we believe that the true quality is perhaps a fixed constant of nature. As an illustration, it appears that most physicists regard the charge on an electron as such an objective constant. Even the most precise measurements of such a quantity, however, are subject to chance causes of variation or, as we say, errors of measurement. As evidence that there always remains a nucleus of chance causes of variation in even the best physical measurements, we may take the series of observed values of 62 ECONOMIC CONTROL OF QUALITY the charge on an electron originally given by Millikan, Table 5. The problem of presenting the essential information contained in such a set of measurements of some quantity assumed to be a constant is that of finding the best estimate of this con- Stant. TABLE 5.—MILLIKAN’s OBSERVATIONs of CHARGE on AN ELECTRoN e X Iol C 4. 78.I 4.764 4. 777 4. 809 4. 761 4.769 4.795 4.776 4.765 4. 790 4. 792 4.806 4.769 4.77I 4. 785 4. 779 4. 758 4. 779 4. 792 4.789 4.8oš 4.788 4.764 4. 785 4. 779 4.772 4.768 4.772 4.8IO 4. 790 4. 775 4.789 4.8ol 4. 79 I 4. 799 4. 777 4.772 4.764 4. 785 4.788 4. 779 4. 749 4. 79 I 4. 774 4.783 4.783 4. 797 4. 78.I 4.782 4.778 4.808 4. 74O 4. 790 4.767 4.79 I 4.77I 4. 775 4. 747 Now let us consider the meaning of true quality where we have one or more series of measurements (9) or (IO) on a number of different things. It is obvious from what has been said that the true quality in such a case is a frequency distri- bution function. It is, however, not the objective frequency distribution function of the observed values, for these contain errors of measurement. It is rather this frequency distribution function corrected for errors of measurement. Since, in com- mercial work, the error of measurement is often large, it follows that the distribution of observed values may differ significantly from our best estimate of the true distribution function. Hence, in our discussion of the ways and means of presenting data, we must lay the basis for correcting, insofar as possible, the original data for errors of measurement. * These data are those given in the first edition of Millikan's book The Electrom, published by the University of Chicago Press. For our purpose, we shall neglect in all further discussions of these data the fact that certain corrections should be made as outlined by Millikan if we are concerned with the problem of giving the best estimate of the charge on an electron. To do this, it would also be necessary to weight the values as he has done. For the latest discussion of the use of these data in estimating the most probable value of the charge on an electron, see “Most Probable 1930 Values of the Electron and Related Constants,” R. A. Millikan, published in the Physical Review, May 15, 1930, pp. 123.I-1237. CHAPTER VI PRESENTATION OF DATA BY TABLES AND GRAPHS I. Presentation of Ungrouped Data Perhaps the most useful way of presenting an ungrouped distribution of raw data in tabular form is that in which the values of the variable are arranged or permuted in ascending order of magnitude. Such a permutation is termed a frequency distribution. Let us consider this form of presentation for the fifty-eight observed values of the charge on an electron given in Table 5. TABLE 6.-TABULAR PRESENTATION of PERMUTED SERIES OF DATA 4.740, 4:747, 4:749, 4,758, 4.761, 4,764, 4,764, 4.764, 4.765, 4,767, 4.768, 4.769, 4.769, 4.771, 4.771, 4.772, 4,772, 4.772, 4.774, 4.775, 4.775, 4.776, 4.777, 4.777, 4.778, 4.779, 4.779, 4.779, 4.779, 4.78 I, 4.78 I, 4.782, 4,783, 4,783, 4.785, 4.785, 4.785, 4.788, 4.788, 4.789, 4.789, 4.799, 4,790, 4.799, 4,791, 4,791, 4,791, 4,792, 4,792, 4,795, 4.797, 4.799, 4.8or, 4.805, 4.806, 4.8o8, 4.809, 4.810. With this tabular arrangement we can easily obtain such characteristics of the observed distribution as range, mode or most frequently occurring value, and median or middlemost value, of the permuted variable. Naturally we can present such a permuted series of mag- nitudes graphically in numerous ways, only one of which is given by way of illustration in Fig. 24. In a similar way a set of observations representing measure- ments of several characteristics on each of several things may be arranged in tabular form by permuting one of the series of observations in ascending order of magnitude and then tabulating the corresponding values of the associated char- 63 64 ECONOMIC CONTROL OF QUALITY acteristics. Table 7 shows two such tabulations, there being in each case two quality characteristics. Table 7-a gives the observed current I in amperes through a certain kind of carbon contact as the voltage E is changed. This is the everyday type of observed relationship presented in the customary tabular form in which one of the series of measurements, in this case voltage, is permuted in ascending order of magnitude. TABLE 7.-TABULAR PRESENTATION OF RELATIONSHIP Table 7-a Table 7-b Voltage E | Current I Volume in Area in in Volts in Amperes | Cu. Cm. Sq. Cm. 3 O. O3 O. 9 o. 667 6 O. of I. 9 o. 528 9 O. II 3.9 O. 538 I 2 o. 15. 4 - 5 o. 778 I 5 O. I 9 4.6 o. 827 I 8 O. 24 4.6 O. 543 2 I O. 29 4.8 o. 792 24 O. 34 4.9 o. 694 2 O. 39 4.9 o. 694 3O O. 45 5. I o. 804 33 O. 50 6.6 o. 772 36 O. 55 7.8 o. 706 39 o.62 9.6 o. 750 2 O.69 II. 7 o. 496 45 o.76 I4.9 O. 59 I 48 o. 86 I6.2 o. 716 5 I O. 93 I7. 9 o. 771 I8.2 o. 489 I9. O o. 8 II I9.2 O. 792 I9.8 O.803 26.8 o. 664 44.8 o. 718 Table 7-5 gives the measurements of two quality charac- teristics of each of twenty-three different kinds of granular carbon. In this case the series of observed values of the DATA BY TABLES AND GRAPHS 65 volume of the pores is permuted in ascending order of mag- nitude. The corresponding customary graphical representations of such sets of data are presented in Fig. 25. 4 23 FIG. - 4.74 4, 76 4, 78 v.ALUEs of ex lo'9 || 24.—ONE GRAPHICAL PRESENTATION of PERMUTED SERIEs of DATA. In Fig. 25-a, there can be little doubt that the current is a function of the voltage E, although neither the tabular nor the graphical presentation gives the relationship quantitatively. !.O OO .4º O. 2 ſ F H. I | | I | |O 2O 3O 4.O 5C) VOLTAGE E IN VOLTS (a) O.9 O. 8 O. 7 O. 6 O. 5 O ſ O — e." * © e O O |- e ‘’ O O O O * O &P O }* O O l ſ | | O |O 2O 3 O 4 O 50 VOLUME IN CUBIC CENTIME TERS (b) FIG. 25.-ONE ForM of GRAPHICAL PRESENTATION of DATA of TABLE 7. In Fig. 25-6, there is a definite question as to whether or not the two characteristics are related at all. Now suppose we were to present in a similar way the 66 ECONOMIC CONTROL OF QUALITY distribution of 1,370 observed values of depth of sapwood given in Table 4 and also the relationship between depth of sapwood and depth of penetration. To do this would require an excessive amount of space. To get around this difficulty of presentation when the number of observations is large, customary practice calls for the grouping of the original data. 2. Presentation of Grouped Data We usually divide the range covered by a frequency dis- tribution of observations into something like thirteen to twenty equal intervals or cells, the boundaries of which are so chosen that no observed value coincides therewith, thus avoiding uncertainty as to which cell a given value belongs. The number of things having a quality X lying within a cell is termed the frequency for that cell; in a similar way, the ratio of the frequency of a given value of X to the total number m of observations is termed a relative frequency. The series of TABLE 8.—Distribution of DEPTH of SAP wood Cell Cell Midpoints Frequency Midpoints Frequency in Inches in Inches I . O 2. 3.4 I 5I I. 3 29 3.7 I23 I.6 62 4. O 82 I. 9 Io6 4. 3 48 2 . 2 I 53 4.6 27 2.5 I86 4.9 I4 2.8 I93 5.2 5 3. I I88 5.5 I frequencies and of relative frequencies constitute frequency and relative frequency distributions respectively. The dis- tribution of depth of sapwood can in this way be reduced to the form shown in Table 8. By thus grouping the original observations into cells, we secure a tabular presentation much simpler than that originally given in Table 4, but in the process we have slightly modified the original data. DATA BY TABLES AND GRAPHS 67 By grouping, we get an improved picture of the clustering of the observed values about a central value somewhere near the cell whose midpoint is 2.8 inches, as is shown in Fig. 26. In the first diagram the black dots represent ordinates pro- $500 e e e © Q 2O * O 3 O O O O I2OO O 3 O O. I 50H O O 900 H. O ta- O O' Q O or IOOH- 6OO Lal FREOUENCY OD DisrRIBUTION " ſ Q CUMULATIVE > O DISTRIBUTION -> | - O 3OOH- 2 O O' O 9- *———f * * , OL • *-*—— 15oof on 200ſ- 12OOH- tal —l 8 Isok 9 OO }- § o: 100F 6OO H. tal FREQUENCY CUMULATIVE (0. POLY GON POLY GON > 50 H. 3OOH- ID 2 OL 1–1–1–1–1–1–1–1–1 R OL *-*–1–1–1–1–1–1–1–1–1 i i t . 15OO 2OOT- 12OOH- (ſ) º }* 9 OOH- Q 150 ti- 9 tool. - 6OOH § FREO UENCY CUMULATIVE OD HISTOGRAM HISTOGRAM I) * ol 1–1–1–1–1–1–1–1–1–1–1–1–1. OL H––––––––––––––. l,O 1,9 2.8 3.7 4.6 5.5 I.O - 1,9 2,8 3.7 4,6 5.5 DEPTH OF SAP WOOD IN IN CHES DEPTH OF SAP WOOD IN INCHES FIG 26.-GRAPHICAL PRESENTATION of FREQUENCY DISTRIBUTION OF DEPTH OF SAP wood of TELEPHONE POLEs. portional to the corresponding cell frequencies, the ordinate for a given cell being placed at the midpoint of that cell. If we join these ordinates by a broken line, we get the frequency polygon. The method of obtaining the frequency histogram is clearly indicated by the figure itself. An ordinate in such 68 ECONOMIC CONTROL OF QUALITY graphical presentations is termed a frequency, meaning thereby the frequency of occurrence in the associated cell. We may plot as the ordinate at a given value of abscissa the total number of observations having a value equal to or less than that of the given value of abscissa. In this way we get the cumulative distribution, cumulative polygon, and cumu- lative histogram also shown in Fig. 26. These are often termed ogives. It is perhaps a matter of personal judgment depending upon the situation in hand as to whether the tabular or the graphical presentation of the frequency distribution of Table 8 is the more desirable. * Let us next try to present the data of Table 4 in such a way as to indicate whether or not there is any relationship between the two quality characteristics, depth of penetration Y and depth of sapwood X. In general, applying the same methods as those used above to obtain the reduced frequency distribution, we get the correlation table or scatter diagram of Fig. 27. The number of poles having values of depth of sapwood and depth of penetration lying within a given rectangle is printed in that rectangle. If we were to erect a parallelepiped on each rectangle with a height proportional to the number in this rectangle, the resulting figure would be a surface histogram. We might also construct a surface polygon in a manner analogous to that used in constructing the frequency polygon. What does the table or chart shown in Fig. 27 tell us about the relationship between the two variables therein considered? One thing is certain—the distribution of values of penetration in a given column corresponding to a given depth of sapwood depends upon the depth of sapwood. In other words, knowing the depth of sapwood, we have some information about the depth of penetration. We shall be content, therefore, to say for the present that these two qualities appear to be correlated and that, in general, the depth of penetration appears to be greater, the greater the depth of sapwood. Thus the table or chart of Fig. 27 does tell us something, but what it tells is qualitative and not quantitative. For example, it does not DATA BY TABLES AND GRAPHS 69 tell us how close a relationship exists between the two qualities. 2. 2 || 29 || 62 || O 6 || 153 || 86 193 || 8 8 || || 5 || ||23 82 48 4,6 4.3 4. O 2.7 3.4 3, I 2,8 2.5 |, 3 48 59 |, O 42 50 O. 7 - || 14 O. 4 | | l,O 1.3 1.6 1.9 2.2 2.5 2.8 3.1 3.4 3.7 4.0 4,3 4.6 4.9 5.2 5.5 × DEPTH OF SAPWOOD IN IN CHES - X FIG. 27.-TABULAR PRESENTATION of GROUPED DATA IN SCATTER DIAGRAM. 3. Choice of Cell Boundaries The choice of from thirteen to twenty cells is to a large extent empirical. Experience has shown that, when the data are grouped in this way, it appears possible to retain most of the essential information in the ungrouped data. To take a larger number of cells often confuses the picture and, in particular, emphasizes sampling fluctuations, the significance of which will be considered later. In general, other things being equal, the outline of the frequency distribution is more regular the smaller the number of cells. This is illustrated by the two frequency distributions of the data of Table 4 shown in Fig. 28. 70 ECONOMIC CONTROL OF QUALITY 4. Conclusion Both tabular and graphical presentations of original ungrouped data are cumbersome and often require a prohibitive amount of space, particularly when there are a large number of 2|O ſ rºs.<^*—23 CELLS | 8 O H. U) 5 O |-- 2 O |- 69 OO M-|-> 3 O }* O.5 |,O |.5 2.O 2.5 3.O 3.5 4.O 4.5 5.O 5.5 6.O DE P T H OF SAP WOOD IN IN CHES FIG. 28.—EFFECT of CLAssIFICATION on GRAPHICAL REPRESENTATION. observed values. Grouping of raw data materially reduces the space required and makes possible a better picture of the observed distribution whether in one or more dimensions, although the data in this form are not readily susceptible of causal interpretation. CHAPTER VII PRESENTATION OF DATA BY MEANS OF SIMPLE FUNCTIONS OF STATISTICS I. Simple Statistics to Be Used Table 9 presents for ready reference a list of those functions or statistics which we shall consider, the ones marked by an asterisk being the most important in the theory of quality control. * TABLE 9.—CoMMONLY Used FUNCTIONS or STATIstics Fraction ** Measures | Measures of | Measures | Measures of within Measures of & e tº g of Lopsidedness of Flatness | Relationship Certain Central Tendency ge & e & & º Dispersion or Skewness or Kurtosis|or Correlation Limits *Fraction | *Arithmetic mean *Standard| *Skewness | *Platness | * Correlation defective A deviation k 62 coefficient p Or r Maximum + Minimum Variance Correlation 2. or? ratio m Median Mean deviation Mode Observed range 2. Fraction p Defective or Non-Conforming This simple measure of quality was described in Chapter IV of Part II as the fraction of the total number of observations lying within specified quality limits. 71 72 ECONOMIC CONTROL OF QUALITY 3. Arithmetic Mean X as a Measure of Central Tendency By definition, the arithmetic mean X of n real numbers, X1, X2, . . . , Xi, . . . , Xn, is 2- 2- , 2, X, x_2^* + X3 + . . . tº + . . . t ºn tº & (II) 77 77 An approximate value for the mean is often obtained from the grouped data as indicated in Table Io which gives the 1,370 observed depths of sapwood grouped into I6 equal cells. The mean value obtained in this way will not, in general, be equal to that given by (II). For example, the mean value from the grouped data in Table Io is 2.914 inches, whereas the mean obtained from (II) is 2.900 inches. TABLE Io.—CALCULATION OF ARITHMETIC MEAN FROM GRouped DATA Mid-Cell Value Deviation * Observed in Inches in Cells from 5 | Frequency Ay X y I - O O 2 O I. 3 I 29 29 I.6 2 62 I 24 I . 9 3 Ioé 318 2 . 2 4. I 53 612 2.5 5 I86 93o 2.8 6 I93 I, I 58 3. I 7 I88 I,316 3.4 8 I 51 I,208 3.7 9 I23 I, Io'7 4 - O IO 82 82O 4.3 II 48 528 4.6 I 2 27 324 4.9 I3 I4. I82 5.2 I4. 5 70 5.5 I 5 I I 5 X e tº $ tº I,37O 8,741 XXy 874 141 = − = ––– = 6.380292 Xy I370 m = units per cell = o.3 inch Arithmetic mean X = 0 + m1a1 = 1.0 + 1.914088 = 2.914088 inches * The origin 6 is the mid-cell value of cell No, o, DATA BY SIMPLE FUNCTIONS OR STATISTICS 73 4. The Standard Deviation a as a Measure of Dispersion 2- Given a set of n real numbers, X1, X2, . . . , Xi, . . . , Xn, the standard deviation a of this set about its mean value X is, by definition, 72. $ºmsº 72. 72. X X,” A 2 X; X X;2 = i = 1 – 2—tº + X* = 1 = 1 — X*. (12) 77 77 The exact value of a can easily be obtained from (12) although this method of calculation introduces a prohibitive amount of work when the size m of the sample is large. For this reason as in the case of the average, we make use of the grouped data and calculate a as indicated in Table II. TABLE II.-CALCULATION OF THE STANDARD DEVIATION FROM THE GROUPED DATA g Deviation Observed Mºº in Cells from 5 Frequency Xy X”y X y I - O O O O I. 3 I 29 2. 29 I.6 2. 62 I 24 248 I. 9 3 Ioé 318 954 2 . . 4. I 53 612 2,448 2.5 5 I86 93o 4,650 2.8 6 I93 I, I 58 6,948 3. I 7 I88 I,316 9,212 3.4 8 I 5I I,208 9,664 3.7 9 I23 I, Io'7 9,963 4. O IO 82 82O 8,2Oo 4 - 3 II . 48 528 5,808 4.6 I 2 27 324 3,888 4.9 I3 I4 I82 2,366 5.2 I4. 5 7o 98o 5.5 I 5 I I 5 225 2 . . . . . . . . I,37O 8,741 65,583 m = units per cell = o.3 inch 1A11 = 2Xy == 874I. * 6.380292 Xy I37O XX 6 1A12 = * = º = 47.8708o3 pz = 1/12 – 141* = 7.162677 a = mus” = 0802895 inch 74 ECONOMIC CONTROL OF QUALITY Obviously, a small standard deviation usually indicates that the values in the observed set of data are closely clustered about the arithmetic mean; whereas, a large standard deviation indicates that these values are spread out widely about the arithmetic mean. For the time being it must suffice to picture the significance of this measure of dispersion somewhat after VA R J A B L E FIG. 29.- How THE STANDARD DEVIATION or INDICATES DIS PERSION. Two DISTRIBUTIONS DIFFERING ONLY IN STANDARD DEviATION. the manner indicated in Fig. 29 which shows two continuous distributions of the same functional form, differing only in standard deviation. 5. Skewness k The particular statistic which we shall use most extensively as a measure of the skewness of a distribution of n values of X is designated by the letter k and defined by the expression 7. X (X, -X)3 X x;3 3XX x;2 t = 1 i = } &mmº, i = 1 –H 2X3 7? 71 71 k = −. I = g3 y (13) X (X; – X)2 t = 1 77 where X is the arithmetic mean and a is the standard deviation DATA BY SIMPLE FUNCTIONS OR STATISTICS 7; of the n values of X. Of course, k may be either positive or negative. If the distribution is symmetrical, k is zero, but it should be noted that the condition k = o is not sufficient for - }* *A v Z / N -: / \ / \ º / \ \ 5 / \ 2 - / \ k = - 1 LL] \ 3 / \ ul º / k = O OC A \ sº / k = i / \ !---> / \ N !---> 2% N ^ _^ * ~ T.--T 1 l I [. ſ | ſ | ſ 1 * >1 ſ VARIABLE FIG. 30.-ILLUSTRATING USE OF k As A MEASURE OF SKEwNEss symmetry. Fig. 30 shows two continuous distributions of the same functional form, differing only in skewness. 6. Flatness + 82 The statistic 62 used as a measure of the flatness of the distribution is defined by the expression 72. 7? 7, X X;4 X X;3 X X;2 7? f = == i = -9 i = - X (X;—X)" , H––4x1=1--|-6X*H––3x" # = 1 71 º 77 77 7? 62 - := j (14) 70, 4 77 2 (X; –X)* O’ = 1 7. where the symbols used are those previously introduced. Fig. 31 pictures three symmetrical frequency distributions differing only in the degree of flatness. * Also called kurtosis. 76 ECONOMIC CONTROL OF QUALITY 7. Calculation of Statistics Let us see how simply the calculation of the four above- mentioned statistics may be carried out. For convenience we introduce a new term, the moment of a distribution. By definition, the jth moment, 19;, of a set of n values, X1, X2, . . . , Xi, . . . , Xn about the origin from which the values are measured is X X? 1H = =. ſº (15) i VARIABLE FIG. 31.-Illustrating Use of 82 As A MEASURE OF FLATNEss of DISTRIBUTION. Similarly, the jth moment of this same set of numbers about the arithmetic mean X is X (X; – X).j pj = −. (16) - 72 It may readily be seen that the formulas for standard deviation, skewness, and flatness may be greatly simplified by expressing the results in terms of the moments of the distribution, as shown in the lower part of the data sheet of Table 12. The recessary computations for finding the four statistics for the distribution of depth of sapwood are also shown in this data sheet. DATA BY SIMPLE FUNCTIONS OR STATISTICS 77 TABLE I2.-TYPICAL CoMPUTATION SHEET Subject Date 2/21/30 Depth of Sapwood Calc. by MBC in inches Checked MSH Cell Cell Deviation | Observed Frequency Mid- | Bound- in Cells | Frequency 3, X 3y2K2 3y2K3 392.4 III point ary from o, X 3y Per Cent O. 850 - I ... O O 2 O O O O O. I5 I. ISO I. 3 I 29 29 29 29 29 2 . I 2 I. 450 I. 6 2 62 I 24 248 496 992 4. 53 I. 750 I. 9 3 I O6 3.18 954 2,862 8,586 7. 74 2. O5 O 2 .. 2 4 I 53 6I 2 2,448 9,792 39, I68 II. I 7 2. 350 - 2.5 5 I86 930 4,650 23,250 I IG,250 I3.58 2. 650 2.8 6 I93 I, I 58 6,948 41,688 25 O, I28 I4. O9 2.95 O 3. I 7 I88 I,316 9, 2 I 2 64,484 45 I,388 I3. 72 3. 25C 3.4 8 I5 I I,208 9,664 77, 3 I2 618,496 II. O2 3. 550 —a– 3. 7 Q I 23 I, IO7 9,963 89,667 807,003 8.98 3. 850 4. O IO 82 82O 8,200 82, OOO 82O,OOO 5.99 4. ISO 4. 3 II 48 528 5,808 63,888 7 O2,768 3. SO 4. 450 4.6 I 2 27 324 3,888 46,656 559,872 I. 97 4. 750 4.9 I 3 I4 . I 82 2,366 30,758 399,854 I. O2 5. O50 5. 2 I4 - 5 7 o 98O I3,720 I 92,080 O. 36 5. 350 5. 5 I 5 I I5 225 3,375 50,625 O. O7 5. 650 X - I,370 8,74I 65,583 || 549,977 || 5,0I 7,239 X 8 m = units per cell = o. 3 - _ 232X 874I _ 1/11 = Txy" 1376 6.380292 _ 232* 65583 1/12 = >y T 1370 T 47.8708o3 _ 2yx* 549977 1A13 = xy T TI370 T 40I. 443066 _ 2934 50.17239 1A14 = 55 T Tºo T 3662. 218248 Alz = 1/12 – 1912 = 47.8708o3 – 4o. 708126 = 7. 162677 A43 = 1/43 – 31/411/12 + 21A113 = 4OI. 443066 – 9 I6.289 IO4 + 519. 45946 I = 4.613423 A4 = 1/44 – 4141 1/43 + 61/11° 102 – 31/114 = 3662. 218248 – I of 45. 295930 + 1 1692. 384081 – 4971 .454567 = 137. 85 I 832 X = 5 + m1a1 = 1.0 + .3(6.380292) = 2.91.4088 O = mugº = 0.3 (2.676318) = o. 802895 78 ECONOMIC CONTROL OF QUALITY A. Errors of Grouping.—It will be seen that all of the com- putations in the illustrative example make use of grouped data, thereby introducing a source of error. The question naturally arises as to whether or not an engineer should attempt to correct the moments thus obtained by some of the formulas, such as those of Sheppard, presented in almost every good text on statistical theory. - We shall consider three reasons why it seems likely that little is to be gained through the use of such corrections, at least in the class of problems considered in this book. These Te2SO11S alſe : (a) The actual limitations imposed in the development of the formulas for correcting the moments necessitate sharp dif- ferentiation between those distributions to which their applica- tion is justified and other distributions; and yet it is not feasible to formulate rules which can be applied intelligently to differentiate between these two classes of distributions without a full knowledge of the somewhat involved theory underlying the corrections. (b) The magnitudes of such corrections for the statistics are small, compared with the sampling errors of the statistics thus corrected, unless the sample size is very large, it being assumed that the interval of grouping is small compared with the maximum observed range of variation, as is the case when we use from 13 to 20 cells. Hence, in general, the corrections do not add much from the viewpoint of causal interpretation. (c) The corrected moment may in some cases differ more from the moment obtained from the raw data than does the uncorrected moment. As a case in point, the standard deviation of the 1,370 observed values of depth of sapwood is o.8O2555 inch as determined from the ungrouped data. The uncor- rected moment obtained from the grouped data is o.802895 inch; whereas the value of this moment corrected by Shep- pard's formula is o.7982II inch. Hence we see that in this example the correction factor does not correct. This situation may arise quite frequently, since the distribution of points within a given cell often does not satisfy the conditions tacitly DATA BY SIMPLE FUNCTIONS OR STATISTICS 79 assumed to exist in the applications of Sheppard's correction. Obviously, therefore, if one is to be sure that he has attained the correct moments for a given distribution, he must carry out the calculations of these moments from the ungrouped data. Since it is difficult to determine when the corrections should apply, since the corrections are usually small compared with the sampling errors of the moments, and since the cor- rections may not correct, it seems that little can be gained by applying the customary correction factors. B. Number of Figures to be Retained.—It will be noted that in the calculation of the statistics, the numerical work is carried out to more places than may often be used in the final form of presentation. The reason for doing this will become clear as we proceed, but one or two instances showing the necessity for such a procedure may not be out of place at this point. In the problem just considered, suppose that we wish to determine the error of the average. In general, this will be expressed in terms of the observed standard deviation a which ſº e e Or in turn has its own error customarily taken to be --, where 277 n is the number of observations. Since the number of figures which we wish to retain in the average depends upon the error of the average, we must know this error before we can decide how many figures to retain. The calculation of this error, however, involves the use of the average itself. Hence we must carry enough figures in the average during the process of calculation of its error so that the final number of figures retained in the average will not be influenced by the number of figures retained in the calculation of the standard deviation. It is obvious that the same line of reasoning applies in determining how many figures to retain in the standard deviation. In the general case, starting with a series of observed values, our interpretation of the data involves the use of certain statistics expressed as symmetric functions of the data. Before we can tell definitely how many figures to retain at a 8O ECONOMIC CONTROL OF QUALITY given stage of the calculation, we must have completed all the calculations. Obviously we cannot carry an indefinitely large number of figures. The detailed calculations carried out in this book will serve to show what we have found to be satis- factory practice. It does not appear feasible, however, to lay down simple, practical, and infallible rules. 8. Measures of Relationship As engineers, we are accustomed to think of two or more things as being related when we can express one of them as a mathematical function of the others. However, in the scatter diagram, Fig. 27, showing the observed values of depth of sapwood X and depth of penetration Y, we see that for a given value of X there are several values of Y so that these two quantities do not appear to be related in a functional way; although there does appear to be some kind of relationship between them. The knowledge of the depth of sapwood gives us some information about the depth of penetration. To measure this kind of relationship, we make use of the correlation coefficient. By definition the correlation coefficient r between n pairs of values of X and Y is X, X,Y, ++= — XY 7? 7"xy = & (17) Ox0 y The method of calculating r is illustrated in Table 13. We shall see later that the value of r must lie between + 1 and – I. The significance of r must be developed as we proceed. * 9. Other Statistics Let us first consider measures of central tendency other than the arithmetic mean. By definition, an average of a series of n values of a variable is a number greater than the least and less than the greatest when all of the values of the DATA BY SIMPLE FUNCTIONS OR STATISTICS 8 I TABLE 13.-METHod of CALCULATING Cork ELATION CoEFFICIENT A = Depth of Sapwood. Y = Depth of Penetration (1) (2) (3) (4) (I) (2) (3) (4) (1) (2) (3) (4) A Y 711 ni YY A y 711 n LXY A Y 711 ni XY o. 7 I O. 7O || 3. I O. 7 | Io 2 I 7O || 4. O || 3.4 5 68. Oo I ... O I I . OO I ... O || 22. 68.2O 3. 7 I I4.8o I. 3 | 40 | 161.2O . 3 || O. 4 I . S. 2. 1.6 || 42 | 208.32 || 4-3 || || 9 || 4 I7.20 o. 7 | I 5 13.65 I . 6 212. o. I. 3 || 4 22.36 9 || 3 4. I . O | I 2. I 5.60 2.2 24 | 163.68 1.6 || 7 48. I6 I 3 I I .69 . 2.5 6 46.5o I 9 || 7 57. Ig r) 8 2.8 7 6o. 76 2. 2 | 6 56.76 O. 4 sº I , 2. 3. I I 9.61 2.5 7 75. 25 o. 7 || II : ; 2.8 || 4 48. 16 I . O 2 . SO I. 3 # : 38 || 3:4 || 0-7 3 7. I4 3. I 5 66.65 1.6 12.8% I. O | I 5 || 5 I OO 3. 4 || 3 43.86 * 5 e I. 3 || 29 || 128. I 8 3. 7 || I I 5.9 I I. 3 36 88. 92 2 . 2. 27 2OI. 96 I 6 5 36.8o I.6 | I4 || 42.56 2.5 II | 93.5o I. 9 || 3 26, 22 I . 2. r) 2.8 I2 | I I.A.. 2 is sº sº 9 7. 22 4 - 24 2. 2 || 3 3o. 36 3. I 2. 2.I. ob 2.5 I II. So 2 . O. 4. I o. 88 3.4 2 23. I2 2.8 3 38.64 o. 7 || II 16.94 3. I | 3 42.78 I ... O 2. 2.4O || 3.7 | O. 7 I 2.59 3.4 2 3I. 28 I. 3 || 48 || I37.28 I . O || IO 37. OO ...] I 17.3% I.6 || 39 || 137.28 I 3 | I 3 62.53 4.6 r) 36.86 I. 9 || IO || 41.8o I.6 || 2 | | 124.32 m cº * 2 . O. Q 9.68 I . 9 : I68.72 || 4.9 || 1 , o | 1 4.9C 2 . 2 2 227. 92 1.6 22 . ( 2. 2.5 O. 4 I I . OO 2.5 II | IOI. 75 I . 9 º §§ O. 7 || I4 24. 5C 2.8 7 72. 52 2 . J. I Io. 78 I. O | 5o I25. OO 3. I 4 || 45.88 2.5 2 24. 50 ; : 59 #3: 3.4 4. 5O. 32 2.8 || 2 27. 44 I . 34 I30. OO 3. I I I 5. I 9 I. 9 || IQ 92.25 || 4.o o. 7 2. 5.60 3. 7 || 2 36.26 2 . ). 7 38.5o I ... O 2. 8. Oo 4 - 3 || I 2.I. of 2.5 2. I 2.5o I. 3 IO 52. OO & 6 I.6 | Io 64.oo || 5.2 I.O I 5.2O ... . . . . . || || || 3 || 3 || || 3 || | | #1. I. 3 || 5 I | 185.6 2.2 I 5 I 32. OO 3. 7 I I9. 24. . O4 2.5 I2 I2O. oO 4. O I 2O. 8o I.6 || 45 | 201.60 e 6 o 2.8 I4 | I 56.8o 4 . I 23. 92 I. 9 i. ; : 3. I r) 24.8o 2 . 2 | I I IO . g * ** * * * 2 . I I 3. 7 S. 2.5 | I 2 84. OO 5.5 5 3.75 2.8 2 I 5.68 n = 1,370 XmixY = 6,765.77 X Y = 4.637654 Xnix Y === 4.938;18 Oxo y = or 498779 X M -* tºº, ºx-xy 85.18 – 4.6376 r = = 4.93 4.037 * = o 60320. OxOry O. 498779 82 ECONOMIC CONTROL OF QUALITY variable are not equal and equal to the common value of the variable when all of the values of the variable are equal. Therefore, the arithmetic mean is only one of an infinite number of measures of central tendency. Typical means often used in maximum X + minimum X 5 characterizing data are the median, O and mode. Naturally, we may expect the different kinds of averages of a series of numbers to differ among themselves. Just as an example, we give below four averages for the series of fifty-eight observed values of the charge on an electron. Median = 4.785 × Io-49 e.s.u. Max. + Min. 2. Mode Arithmetic mean 4.775 × Io-19 e.s.u. 4.779 × Io-" e.s.u. == 4.780 × Io-10 e.s.u. Next, let us consider some measures of dispersion, skew- ness, and flatness other than those previously given. A measure of dispersion very commonly used in engineering work is the mean deviation p defined for the case of n values of X by the expression 72, X |X. — X| where, as usual, the symbol | | represents the absolute value of a quantity. In the same way, any even moment of a dis- tribution about its mean is a measure of dispersion, as is any odd moment of absolute values of the deviations from the mean. Hence, there is an indefinitely large number of possible measures of dispersion of this kind. Furthermore, if we turn to any standard text on statistical theory, we shall find other kinds of measures of dispersion, such as symmetric ranges, of which there is also an indefinitely large number. In the same way, we may set up an unlimited number of different measures of skewness and flatness. Obviously, there- DATA BY SIMPLE FUNCTIONS OR STATISTICS 33 fore, we need to have some general principle to guide us in choosing measures of such characteristics of a distribution of data as the central tendency, dispersion, skewness, and flat- In CSS. One basis of choosing between two statistics as a measure of a characteristic of a distribution is the difference in the amount of labor involved in their calculation. As a case in maximum X + minimum X point, such measures as the median, TX 3 and mode, can readily be determined by observation of the observed frequency distribution; whereas, the calculation of the arithmetic mean involves considerable labor. It is believed, however, that the cost of the manual labor involved in the analysis of engineering data is for the most part a very small per cent of the cost of taking the data. If we can get more information out of one measure than we can out of another, the cost of analysis will not, in general, be a deciding factor. Casting about for some more fundamental basis of choice, we take note of the fact that it is usually desirable to have a statistic which is an algebraic function of the data. It is obvious that these functions must be symmetric since they must be independent of the order in which the data were taken. It follows from algebraic theory that the chosen func- tions must be expressible in terms of what are generally known as sum functions, because all symmetric functions are SO expressible. Now, the sum functions are defined as S1 = X1 + X2 + . . . -- X; + . . . -- Xm S2 = X1* + X22 + . . . -- X:2 + . . . -- X.nº tº. (e (e e o * • ? - (19) Sj = Xī’ + X2' + . . . -- X: + . . . -- Xn' * . - Obviously, . is the jth moment 1; of the distribution about the origin. 84 ECONOMIC CONTROL OF QUALITY. . . The statistics X, o, k, 32, and r satisfy the condition of being symmetric functions of the data, but still we must try to find out if they are the most useful symmetric functions. In the remaining chapters of Part II, we shall justify the use of these five statistics to the extent of showing that they go a long way towards expressing the total amount of information contained in a set of data. - CHAPTER VIII BASIs For DETERMINING How To PRESENT DATA I. The Problem Let us consider again the distribution of the 1,370 observed values of depth of sapwood. So far as this or any similar set of data is concerned, we assume that one observation contributes just as much information as any other in the same set. The total information is given by the observed distribution. If, then, we are to present the total information, we must give the original frequency distribution. For reasons already con- sidered, however, we find it desirable to condense the original data insofar as possible by calculating certain statistics. In the previous chapter we showed how to effect this reduction and illustrated the method by application to the distribution of depth of sapwood. The information contained in this dis- tribution, reduced to the form of statistics, is given in Table 14. TABLE I4.—INFORMATION IN FoRM of STATISTIcs Average X = 2.91.41 inches Standard Deviation g = O.8029 inch Skewness k = O. 2407 inch Flatness 82 = 2.6870 inches Number of Observations n = 1,370 If the statistics of Table 14 actually contain the total information in the original series of observations, it should be possible to reproduce this distribution from these statistics. Obviously, it is not possible to do this, and therefore the statistics do not contain all of the information. However, they do con- tain a surprisingly large percentage, as we shall now see. 85 86 ECONOMIC CONTROL OF QUALITY Table 15 gives the results of two attempts to reproduce the original distribution from the observed statistics. The second row is the distribution obtained from the average and standard deviation alone, while the third row is that obtained using, in addition, the skewness of the original distribution. TABLE 15.-SHowING How MUCH INFORMATION Is Contained IN A FEw SIMPLE STATISTIcs Cell Midpoint. . . . . . . o. 4|o. 7|I. o! I. 3|I. 6] I. 9|2. 2 2.5|2, 8|3. I |3.4|3. 7|4. O|4. 3|4. 614.9/5. 25.5 Observed Frequency. o o 2| 29| 62|IOG|I53|I86|I93|I88|I5 I |I23| 82| 48| 27| I4|| 5 || I Normal Law Fre- Cille Il Cy . . . . . . . . . . . I | 5 || I2] 27| 53| 92||138|I 79|2O2|Io9|I 7 O|I27 82| 46|| 23| IO 3 || I Second Approxima- tion Frequency. . . . O || 0 9| 25| 55| 99|I49|I89|2O7|193|I59|II 6| 77| 46|| 25| I3| 6 || 2 That the approximate or theoretical distribution obtained through the use of the average X, standard deviation g, and skewness k is closer to the observed distribution than is that obtained through the use of only the first two of these statistics can be seen quite readily from Fig. 32. DEPTH OF SAP WOOD IN IN CHES © OBSERVED POINTS 2OO I- N — SECOND APPROX|MATION \ — — — NORMAL LAW 15 O H. Uſ) uſ ~! O Cl Ul- O IOO H. OC Uij CO > I) 2. 5 O H. O -- ºr O L I ſ I I ſ ſ S O O.5 i.O 1.5 2.O 2.5 3.O 3.5 4.O 4.5 5.O 5.5 FIG. 32.-SIGNIFICANCE OF Aver AGE, STANDARD DEVIATION, AND Skewness. DETERMINING HOW TO PRESENT DATA 87 The surprising thing is that a knowledge of the average and standard deviation alone enables us to reproduce SO closely the observed distribution in this case. Here, the approximation is so good that it is somewhat doubtful whether or not, from the viewpoint of presentation alone, we can attach any practical significance to the increase in the amount of information given by the introduction of the skewness over that given by the average and standard deviation alone. In fact, engineers are usually interested in knowing only the number of observations lying within certain relatively large ranges, such as the average X plus or minus two or three times the standard deviation a. Table 16 presents the observed percentages of the 1,370 observations lying within these ranges together with those estimated from a knowledge of the average and standard deviation. A knowledge of k as here used adds nothing to the precision of our estimate of the number of observations lying within these or any other ranges symmetric in respect to the average. TABLE 16.-PERCENTAGE of OBSERVATIONs LYING witHIN PARTICULAR RANGES Range Range Range Range A = o. 67450 X == a X == 20 A =E 30. Estimated, Per Cent. . . . . . . 5O.OO 68.27 95.45 99.73 Observed, Per Cent. . . . . . . . 47.45 66. 57 95.9 I 99.93 Difference, Per Cent. . . . . . . 2. 55 21.70 o. 46 O. J.O In the next few paragraphs we shall see how these simple statistics often enable us to approximate very closely the original distribution. In general, we shall find that the infor- mation contained in statistics calculated from moments higher than the second depends to a large extent upon the nature of the observed distribution; therefore, these statistics are somewhat limited in their usefulness. The really re- markable thing is that so much information is contained in the average and standard deviation of a distribution. 88 ECONOMIC CONTROL OF QUALITY The specific problem to be considered is: Given a series of numbers, Xi, X2,..., Xi, ..., Xn, representing an observed distribution of some quality characteristic X such as any of those previously discussed, let us try to find some function f(X, X, o, k, 82) of X and the four statistics calculated from the observed distribution such that the integral ſº X, o, k, 82)d X (2O) of this function from X = a to X = b gives approximately the total number of observed values lying within this same interval. When the approximation is good, we can say that the statistics contain practically all of the total information in the original distribution. In fact, as already noted, we can say that these statistics contain most of the information of practical en- gineering value when the approximation X--za. & ſ f(X, X, o, k, 82)dX (21) *X – 2a: is good, where, as before, the values of z with which we are usually most concerned are O.6745, I, 2 and 3. Common sense tells us that the degree of approximation in a given case will depend upon the function f. Of course, it is desirable to be able to estimate the amount of information contained in the statistics independent of the function f. For reasons which will be considered later, we find that under the state of control of manufactured product the function f which is best in the majority of cases is the same for most quality characteristics. Hence, what we shall do is to show how much information is contained in these statistics for this limiting type of distribution function which is approached as we approach the state of control. We shall then review the work of the Russian mathematician, Tchebycheff, which makes it possible for us to see how much of the total information is contained in the average and standard deviation of a distribu- tion independent of its functional form. DETERMINING HOW TO PRESENT DATA 89 2. Statistics to be Used when Quality is Controlled When the number n of measurements of some quality X have been made under the conditions of control, we find in general that the function f in (2O) can be assumed to be one or the other of the following two forms without introducing practical difficulties: 2 11 3C f(x) = ov2. cº, (22) Or o f(x) = – *|- (-; ), (23) O V 27ſ. where x = X – X. Under the conditions of control, it may then be assumed that the integral of either one or the other of these two functions over a given range should give approximately the number of observed values within the corresponding range, particularly when the number n of observed values is comparatively large. We need, therefore, tables of values of the integrals of these functions for n = I. The integral of (23) is ſº-ſº-ſº + º-º-º-º-º: = F(z)-kF2(z), (24 where Fi(z) is the integral of (22), and z = * Tables A and B Or give the functions F(z) and F2(z) respectively. Now we are in a place to see how the approximations given in Table 15 were obtained. The method is illustrated in detail in Tables 17 and 18 derived from approximations (22) and (23) respectively. Corrected moments were used in Tables 17 and 18 and Fig. 39. We have already noted that k contains some information not contained in the average and standard deviation in the sense that the use of all three gives the closer of the two approx- imations to the observed frequency distribution of depth of sapwood. If, however, we are interested in the number of observed values within a symmetrical range about the observed 90 ECONOMIC CONTROL OF QUALITY I ſ—we TABLE A.—VALUEs of Fi(z) = −7– e dz v/or O z Fi(z) z | Fl(z) || 2 | Fi(z) z | Fi(z) z | Fi(z) z Fi(z) z Fi(z) .oO | .oOOO | .45 | . I'737 | . 9o .316o |I.35|. 4II5 |I. 8ol .464I 2.25|.4878 |2.70 | . 4966 ..or .oO4o |.46 | . I'773 | . 91 | .3186|I.36 .4131 |I. 81| .4649 |2.26|.4881 |2.7I | . 4967 .O2 | . Oo3O | .47 | . I 808 .92] .3212 |I. 37 .4147|I. 82|. 4656|2.27 .4884 |2.72| . 4968 .O3 | . of 20 |.48 |. I844 .93| .3238||1.38|.4162 |I.83| .4664 |2.28 .4887|2.73|.4969 .O4|.oi6O | . 49 |. I88o .94 .3264 |I.39| .4178 |I.84 .4671 |2.29| .4890 |2.74|. 4970 .o5 |. O2OO |. 5ol. 1915 . .95|.3290 |I.4ol. 4193 |I.85| .4679 |2.30|.4893 |2.75|.4970 .o6 | .oz.39 |. 51 | . I950 | .96 .3315|I.4I | .4208 |I.86|.4686|2.31 | .4896 |2.76|.4971 . O7 | . O279 |. 52 | . I985 | .97 .334o |I.42] .4222 |I.87] .4693 |2.32 .4899 |2.77|,4972 .o8 | . OSI9 |. 53 |. 2020 | .98|.3365 |I.43| .4237 |I.88] .4700 |2.33 .490I 2.78|,4973 .og | . OS59 |.54|. 2054 | .99 .3389 |I.44 .4251 |I.89 .4706|2.34 .4904|2.79| . 4974 . Io .ogg3 |. 55 .2089 |I.ool .34I4|I.45|.4265 |I. 9o .4713 |2.35|.4906|2.8ol. 4975 . I I | . O438 |. 56 | .2123|I. oil .3438 |I.46|.4279 |I. 91 .4720 |2.36|.4909|2.81|.4975 12|. O478 |. 57 | .2157 |I.O2] .3462 |I.47 .4292 |I.92] .4726|2.37|,491 I |2.82|.4976 . I 3 |.o.517 | .58|. 2191 |I.O3] .3485 |I.48|.4306|I.93| 4732 [2.38|.4914 |2.83|.4977 . I4 oš57 |. 59 . .2224|I.O4] .3508 |I.49 .4319|I.94| 4738|2.39| .4916 |2.84|. 4978 . I5 oš96 |.6o .2258|I.o.5|.3532 |I. 5ol .4332 |I.95] .4744 |2.4ol. 4918|2.85| . 4978 . I6 |. O636 |.61 | .2291 |I. O6] .3555 |I.5I] .4345 |I. 96) .475o 2.41 | . 4920 |2.86|. 4979 . I7 | . O675 |.62 | .2324 |I.O7] .3577|I. 52 .4358|I. 97 .4756|2.42|.4923 |2.87 .4980 . I 8 |. O7I4 |.63 | .2357 |I.o8 .3599 |I. 53| .437O |I.98|.4762 |2.43| . 4925 |2.88|.498o . I9 |. O754 |.64 .2389 |I. Og|.3622 |I. 54 .4382 |I.99|.4768|2.44 . 4927 |2.89|.4981 .20 | . O793 |.65 .2422 |I. Iol .3644 |I. 55] .4395 |2. ool .4773 |2.45|.4929 |2. 9ol .4982 .2I o832 | .66 | .2454 |I. I I | .3665 |I. 56|.44O6|2. OI! .4778|2.46|.4931 |2.91 .4982 .22 | o87I | .67 .2486 |I. 12 .3687|I. 57 .4418|2. O2] .4783 |2.47] .4933 |2.92] .4983 .23 .og Io .68 .2518|I. I.3| .3708 |I.58|. 4430 |2. O3] .4788 |2.48|.4935 |2.93| .4983 .24 | .og49 |.69 | .2549 |I. I4 -3729 |I. 59| .444I 2.04] .4793 (2.49|.4936|2.94 .4984 .25 | .og&7 |.70 | .2581 |I. 15|.3749 |I.6ol .4452 |2.05|.4798 |2.5ol. 4938|2.95 .4984 .26|. IO26 |.7I | .2612 |I. 16 .377O |I.6I .4463|2.06|.48o3|2.5I] .4940 |2.96|.4985 . 27 | . Io94 | .72 | .2643 |I. 17|.3790 |I.62 .4474 |2. O7] .4808 |2.52|.4942 |2.97 .4985 .28 |. I IO3 |.73 | .2673 |I. 18|.38IO |I.63| .4485 |2.o&|. 4813 |2.53 .4943 |2.98|.4986 .29 |. I I4I |.74 .2704 |I. I9| .383o |I.64 .4495 |2.09|.4817|2.54|.4945 |2.99| .4986 .30 | . I 179 |.75 | .2734 |I.20). 3850 |I.65] .4506|2. Io .4822 |2.55] .4946|3.oo! .4987 .31 | . I217 | .76 .2764 |I.21|.3869 |I.66|.4516|2. II | .4826|2.56 .4948 |3. Iol . 4991 .32 |, 1255 |.77 | .2794 |I. 22| .3888 |I.67 .4526|2. I2] .4830 |2.57 .4949 |3.20. . 4993 .33 |. I293 |.78 .2823 |I. 23| .3907 |I.68|.4535 2. I3] .4834 |2.58|.4951 |3.30|.4995 .34 |. I331 |.79 .2853 |I.24 .3925 |I.69| .4545 |2. I4] .4838|2.59 .4952 (3.40|.4997 .35 | . 1369 |.8o .2882 |I. 25] .3944|I. 70' .4555 |2. I5] .4842|2.60 . 4954 |3.5o .4998 .36 |. 1406 | .81 | .291 I |I. 26|.3962 |I. 71 | .4564 |2. I6|.4846|2.61 .4955 |3.60|.4999 .37 |. 1443 |.82 | .2939|I. 27 .398o |I. 72] .4573 |2. I7] .485o 2.62|. 4956 |3.70|.4999 .38 |. 1481 |.83 | .2968|I. 28, .3997 |I.73| .4582 |2. I8|.4854|2.63|.4958|3.8o|.5ooo .39| . I 518 |.84 .2996 |I. 29 .4O15|I.74 .4591 |2. I9|.4858 |2.64 . 4959 |3. 9ol. 5ooo .40 | . I 554 |.85 .3024|1.30 .4032 |I.75|.4599|2.2O .4861 |2.65|.4960|4.ool .5ooo .4I | . I 591 |.86 .3051 |I.31|.4O49 |I.76|. 4608 |2.21 | .4865 |2.66 : 4961 .42 1628 .87 | .3079 |I.32 .4066|I.77|,4617 |2.22|.4868|2.67|,4962 .43 |. 1664 .88 .3106 |I.33| .4083 |1.78] .4626 2.23 .4872 |2.68|.4963 .44 |. 1701 | .89 .3133|1.34 4099 |I.79 .4633 |2.24 .4875 (2.69|.4965 DETERMINING HOW TO PRESENT DATA 9I I – 14 22 TABLE B-VALUEs of F2(z) = − [I – (1 – zº)e^** 6V2T z F2(z) || 2 | F2(z) || 2 | F2(z) z F2(z) z F2(z) || 2 | F2(z) || 2 | F2(z) .oO | .oOOOO | .45|. OI857 | . 9o .oš806|I.35|.o8848 |I.8ol. O9597 |2.25|.o8798 |2.70 | . O7742 .OI | .oOOOI | .46|..or 933 | .91|.oš894 |I.36|.o8890 |I. 81|.og S90 |2.26| o8774 |2.71 | . O7722 . O2 |. OOOO4 .47 O2OI I | . 92 oš98o |I. 37 . O393O |I. 82|. O9584 |2.27| o8749 |2.72 . O7702 .O3 | .oOOO9 |.48 |.O2O89 | . 93 . O6066|I.38|.o897O |I. 83|. O9576 |2.28|.o8724 |2.73|. O7682 .O4|.oOOI6 |.49 |.O2.168 .94 .06152 |I.39|. O9008 |I.84|.ogj68|2.29|.o8699 |2.74|.O7663 .o.5 | .oOO25 | .50 |.O2248 .95|.06236|I.40 . O9045|I.85|.og559 |2.30|.o8674 |2.75|. O7644 . O6|.oOo36|.5I .O2329 .96|.o6320|I.41|. O908o |I.86|. O9549 |2.31|.o865o 2.76|. O7625 .O7| .oOO49 |. 52 |.O24II | . 97! .off.404 |I.42|.og II 5|I.87 .ogg39|2.32|.o8625 |2.77|. O7606 .o8|.OOO64 .53|.O2494 | .98|.o6486|I.43|.og I48|I.88|.o9527 |2.33 o86oo |2.78 . O7588 .og | .oOo31 |.54|.O2578 .99 .06568|I.44 .ogI8o|1.89 .095.16|2.34|.o8575 |2.79|. O7569 . Io .oOogg | .55 |.O2662 |I.ool off649 |I.45|.og2II |I. 9o .oggo3|2.35|.o855o |2.8ol. O7551 . I I .oOI2O |. 56|.O2748 |I.or|.o6729 |I.46. .og24I |I. 91, .og490 |2.36|.o8525 2.81 |. O7534 I2 | .oOI43 | . 57|.O2833|I.o.2|.o6809 |I.47 .og269 |I. 92 .og477|2.37|.o85ool 2.82|. O7516 I3|.oO167 |.58|.O2920 |I.o.3|.o6887 |I.48 .og296 |I.93|.og463 |2.38|.o8475 2.83|. O7499 I4 .oOI94 |. 59 |. O3OO7 |I.O4|. O6965 |I.49|. O9322 |I. 94 .og448 |2.39| o845o 2.84|. O7482 . I 5 | .oO222 |. 60 | . O3095 |I.o;|. O7042 |I. 5o .ogg|47|I.95|.og433|2.4o o8426|2.85|. O7465 . I6|.oO253 |.61 |.o3183 |1.06| of 118|I.5I] .ogg|7|I|I.96|.og417|2.4I o84OI 2.86|. O7448 . I7] .oO285 |.62 |.o3272 |I.o/|.07.193 |I. 52|.og394 |I.97|.og4OI 2.42|.o8376|2.87|. O7432 . I 8 |. Oo319 |.63 |.o3361 |I.o8|.o'7267 |I. 53|. O94I 5|I. 98|.ogg'84 |2.43|.o8352 |2.88|. O74I6 19|. Oo355 |.64|.o3450 |I.ool.oz.34o |I. 54 .og435|I.99|.og366 2.44 o8327|2.89 .07400 .2O | .oO392 || 65||.o354O |I. Io . O7412 |I. 55|.og454 |2.ool. O9349 |2.45|.o83O3 |2. 9ol. O7384 .2I .oO432 |.66|.og631 |I. II] .oj483 |I. 56 . O9472 |2. or .0933O |2.46|.o8279 |2.91 |. O7369 . 22 |. OO473 |. 67 | . OS72I |I. 12|.o'ſ 552 |I. 57|. O9489 |2. O2 . O9312|2.47 o8255 2.92|. O73.54 .23|.oOSI6 |.68|. O3812 |I. 13|. O7621 |I.58|. O9505 |2: o3|. O9293 |2.48|.o8231 |2.93|. O7339 .24|.Oo361 |.69|.o3904 |I. I4. O7689|I. 59 .og SI9 |2. O4|.og273 |2.49|.o82O7|2.94|.o/324 .25 | .oO607 |.70 |.o3995 |I. 15|.o?756|I.6o .ogg|33 |2.05|.og253 |2.5ol.o&I83|2.95|. O7309 .26|| .oO656 |. 71 || O4086 |I. 16|.o'7822 |I.61|.og S46|2.06 .og233 |2.5I] ..o8159 |2.96|. O7295 .27| .oO705 |.72 |.o.4178 |I. 17|.o?886|I.62|.og;57 |2.07 .og213|2. 52|.o8136|2.97 . O7281 .28|.oO757 |.73|.O4270 |I. 18|.oj950 |I.63|.ogs 67|2.08 .og Ig2 |2.53|.o8II2 |2.98|. O7267 .29| .oO8IO |.74 |. O4362 |I. 19|.o8or 2 |I.64|.og577|2.09|. O917O |2.54|.o8089 |2.99|. O7254 -3ol.oO865 .75 |.O4453|1.20 o8073 |I.65|.og;85 |2. Io .09149 |2.55|.o8066||3. Col. O7240 .31 .oOo.21 |.76|. O4545 |I.21|.o8133 |I.66 . O9592 |2. II] .og I27 |2.56|.o8043 |3. Io . O7II.8 .32 | .oOo/9 |.77|.o.4637 |1.22|, o8192 |I.67|.og S99 |2. I2] .og Io; [2.57| o802O |3. 20' . O7016 .33|.oro38 |.78|. O4728|1.23 o825o |I.68|.og604 |2. 13|.ogo82 |2.58|. O7998 |3.30|. O6933 .34|.orogg|.79 |. O482O |I.24 o8306|I.69|.og608 |2. I4] .ogoóo 2.59|.o7975 3.40|. O6866 .35 | . of 161 |.8o oA91 I |1.25|.o8361|1.70 .ogóI2 |2. I5] .ogo37|2.6o|. O7953 |3.5ol of 813 .36|..or 225 | .81 |.ošoo2 1.26|.o84.16||1.71|.og614|2. 16|.ogo14|2.61 |. O7931 |3.60|. O6771 .37|..of 290 |.82 |.o;og3 |1.27 o8468|I. 72|.ogó16|2. I?|.o8991 |2.62|. O7909 |3.70|.oč739 .38|.oſ 356 |.83 |.o;183|1.28 o8520 |I.73|.og616|2. 18|.o8967 |2.63|. O7888 |3.8o|. O67.14. .39|..or 424 |.84|.o;274 |1.29|.o8571 |1.74|.og616|2. 19 .08943 |2.64|. O7866|3. 9ol offö96 .40 |..or 493 |.85 |.o;363|1.30|.o862o 1.75l.og615|2.20 oš919 |2.65|.07845|4.ool offö83 .4I of 564 |.86|.o;453 |1.31|.o8668|I. 76 .ogó13|2.21|.o8895 |2.66|. O7824 .42|, or 635 | .87|.o;542 |1.32|.o8715 |I. 77 .ogóIo|2.22 o887I |2.67|. O78o3 .43 | . of 708 .88 .o;631 |I.33|.o876O |1.78 .ogó06|2.23|.o8847 |2.68|. O7782 .44|..or 782 | .89 |.o;719 |1.34|.o8805 |1.79 .ogó02 |2.24 o8823 |2.69|. O7762 92 ECONOMIC CONTROL OF QUALITY TABLE I?.-DISTRIBUTION OF DEPTH of SAP wood CALCULATED FROM (22) n = 1,370 Y = 2.9 I4088 Subject Date 2/21/30 Depth of Sapwood Calc. by MBC a = O. 7982 II in inches Checked MSH Cell Cell Pºº- 2. Differ Fr Approxi- |Observed Mid: Bound: fºx (...) | F(.) | “... q., le mate | Fre- point ary 3. Oſ Cl * |Frequency quency o. 25 | 2.664I 3.3376 o. 4995 O. 4 O. OOIO I .4 I | o. 55 2.3641 2.96.18 o. 4985 o. 7 O. OO33 4.5 5 o. 85 2.064I 2.5859 o. 4952 I ... O o. oo&7 II . 9 I 2 2. I. I 5 I. 764I 2.2 IOI o. 4865 I. 3 o.OL 98 27. I 27 29 I .45 I .464I I. 8342 . o. 4667 I.6 O. O390 53.4 53 62 I. 75 I. 1641 I. 4584 o. 4277 I. 9 o. of 72 92. I 92 Ioé 2. O5 o. 864I I. o825 o. 3605 2. 2. O. IOO4 I37.5 138 I 53 2.35 | O. 564I o. 7067 o. 26ol 2.5 o. 1305 || 178.8 I 79 I 86 2.65 o. 2641 –o. 3309 || O. 1296 2.8 o. 1476 | 202.2 2O2. I93 2.95 || O.OS 59 |+o. O45o O.OI8o 3. I O. I45 I I 98.8 I99 I 88 | 3.25 | O. 3359 o. 4208 o. 1631 3.4 o. 124o | 169.9 I7o I5 I — 3. 55 o. 6359 o. 7967 o. 2871 3.7 o.O924 | 126.6 127 I 23 3.85 | O. 9359 || I. I725 | O. 3795 4... O O. O597 8I. 8 82 82 4. I5 | I. 2359 | I. 5483 o. 4392 - 4.3 O. O337 46.2 46 48 4.45 | I. 5359 I . 9242 | O. 4729 4.6 - o.OI64 22.5 23 27 —| 4.75 I. 8359 || 2.30Oo o. 4893 4.9 o. oO7O 9.6 IO I 4 5. OS 2. I359 || 2.6759 o. 4963 5.2 O. OO25 3.4 3 5 5.35 | 2.4359 || 3. O517° o. 4988 5.5 O. OOO9 I . . I I 5.65 | 2.7359 || 3.4275 o. 4997 X o. 9992 | 1,368.9 | 1,370 I,37O § TABLE 18.-DISTRIBUTION of DEPTH OF SAPwood CALCULATED FROM (23) m = I 370 Subject Date 2/21/30 X = 2.91.4088 or = .7982 II Depth of Sapwood Calc. by MBC in inches k = .244925 Checked MSH Cell Cell Deviation Ob d ; Boundary from x z(x/or) F1(z) F2(z) +kF2(2) F(2) + kF2(2)| Difference | Frequency | #. O. 85 2. O64I 2.5859 O. 4952 O. O.799 O. O.I.96 O. 5 I48 I ... O O. OO65 9 2 I. IS I. 764I 2. 2 IOI O. 4865 o, o&89 O. O2 I 8 O. 5083 I. 3 O. Or 8I 25 29 I. 45 I.464I I. 8342 O. 4667 O. O958 O. O.235 O. 4902 I. 6 O. O.399 55 62 I. 75 I. I64I I. 4584 O. 4277 O. O924 O. O.226 O. 4503 I. Q O. O7I9 99 IO6 2. O5 O. 864 I I. O825 O. 3605 O. O729 O. OI79 0.3784 2. 2 O. IO84 I49 I 53 2.35 0. 564I o. 7 o'67 O. 26OI O. O4O6 O. O.O.99 O. 27 OO 2.5 O. I378 I 89 I86 2.65 O. 264I O. 3309 O. I 296 O. OIOS O. O.O.26 O. I322 2.8 O. ISOI 2O7 I93 2.95 O. O.359 O. O450 O. OI8O O. OOO3 O. OOO I O. OI79 3. I O. I4I 2 I93 I 88 3.25 O. 3359 0.4258 O. I63 I O. OIG4 O. O.040 O. I 59 I 3.4 O. II 60 I59 ISI 3. 55 O. 6359 O. 7967 o. 287 I o. o.488 O. OI2O O. 275 I 3. 7 O. O.85 O I IG I 23 3.85 O. 9359 I. I 725 O. 3795 O. O79 I O. O.I.94 O. 36OI 4. O O. O560 77 82 4. I5 I. 23.59 I. 5483 O. 4392 O. O945 O. O.23 I o. 416I 4. 3 O. O336 46 48 4.45 I. 53.59 I. 9242 O. 4729 O. O947 O. O.232 O. 4497 4.6 O. Or 84 25 27 4. 75 I. 8359 2. 3000 O. 4893 o. O&67 O. O2 I 2 O. 468 I 4. 9 O. O.O.9 I I 3 I4 5. O5 2. I 359 2.67.59 O. 4963 O. O779 O. O IQ I O. 4772 5. 2 O. OO40 6 5 5. 35 2.4359 3. OSI 7 o. 4988 O. O7I 8 O. O I 76 O. 48 I2 S. 5 O. OOI7 2 I 5. 65 2. 7359 3.427.5 O. 4997 O. O684 O. OIG8 O. 4829 2 O. 9977 I,370 I,370 94 ECONOMIC CONTROL OF QUALITY average, it follows from (24) that the skewness k does not add to this information because the integral of (22) over a symmetrical range is identically the same as the integral of (23) over the same range. In passing, we should note that the function (22) is the familiar bell-shaped normal law curve whose significant charac- SO.OOOOO Co OF AREA WITHIN O it O.6745 of 68.268.94% OF AREA WITHIN O 3: I or 95.44998% OF AREA WITH IN O t 2 or 99.7300.2% OF AREA WITH IN O + 3 or -3 Of – 2 Oſ - |O –O.674.5 C, O O.6745C, iO, 2O 3Cſ FIG. 33.-NoFMAL LAw CURVE. teristics are shown in Fig. 33. The function (23) will be referred to as the second approximation. 3. Why the Average X and Standard Deviation a are always Useful Statistics Let us consider the case where nothing is known about the distribution of observed values. To what extent are we justified in assuming that the average, standard deviation, skewness, and flatness contain significant information? We have already seen that the amount of information given by these statistics of value in reproducing approximately the original distribution, depends upon the nature of the original distribution as reflected in the form of function f that would be required to satisfy the condition that its integral DETERMINING HOW TO PRESENT DATA 95 over a given range should be approximately equal to the number of observed values within this same range. However, even when nothing is known about the condition under which the distribution was observed, we find that the average and standard deviation enable us to estimate within limits which are quite satisfactory for most purposes, the number of obser- vations lying within any symmetrical range X + za, where z is greater than unity. In fact, the proportion of the total number of observed values within any such limits is always I © greater than 1 – 3. This follows from a general theorem, the 2, proof of which can be framed in the simplest kind of elementary mathematics, as we shall now see. Tchebycheff's Theorem.–Given any set of n observed values expressible by the frequency distribution of m different values, Xi, X2, . . . , Xi, . . . , Xm p177, p271, • * - 5 piń, • * - 5 pm?! where pin represents the number of values of Xi, then *-* X pin X; 77, 2 pin * = 1 * = 1 and 2 pin(X; – X)2 1 = 1 2 pin 1 = 1 Let Pan denote the number of values of X such that x = (X – X) does not exceed numerically za where z > 1, and n – Pan denote the number of values of x that do exceed zo. We may write - o” = 21px;” + 22p;xi”, where 21 denotes summation for all values of x; which do not 96 ECONOMIC CONTROL OF QUALITY exceed za and X2 denotes summation for all values of xi which do exceed zo. Since all values of pixi” are either positive or zero, o” = 22p;xi”. Obviously, therefore, o 2 > Xopizºo” since all values of xi included in the summation 22 are greater than zo. But 22pi = 1 - P2. Hence a 2 > (I – P2)2°o”, | Or I - (I – P2)22, , (25) ( – PJ & #, 2, 2 and P. - I -, (26) 2. We see that no matter what set of observed values we may have, the number of these values Pam lying within the closed 4-E, & I range X == Zo is greater than ( * ..). whereas the number 2, (I – Pe)n lying without this range is less than #. 4. Importance of Skewness k and Flatness 32 Given a set of any n real numbers X1, X2,..., Xi, . . . , Xn, what does a knowledge of the skewness k and flatness 82 for this set of numbers really tell us independently of any assump- tion as to the nature of the distribution of the numbers as was made in deriving the theoretical distributions in Table 15? To get at this question, let us assume that the skewness k is equal to zero. Obviously, for a distribution to be symmetrical, it is a necessary condition that its skewness be zero. If this condition were also sufficient, it would be possible to say of the DETERMINING HOW TO PRESENT DATA 97 set of numbers given above that they were symmetrically distributed about the arithmetic mean value, and hence that there were just as many on one side of the mean as there were on the other. This would oftentimes be really worthwhile information. It can readily be shown, however, that the condition k = o is not sufficient for symmetry. For example, the dis- tribution A : 2. — I y: I I6 I6 6 # I satisfies the condition that its skewness is zero, although it is obviously not symmetrical about its mean value X = o. In fact, it is far from being symmetrical as are many others which may be found by empirical methods. In this particular instance, instead of finding the set of numbers equally divided on either side of the average, we find sixteen on one side and twenty-three on the other. Hence we must conclude that a knowledge of k in itself does not present very much information. In a similar way it can be shown that a knowledge of 32 in itself does not present any very useful information about the distribution of a given set of n numbers. These results are of considerable importance because they show that the tabulation of moments higher than the second for the purpose of summarizing the information con- tained in a set of data is likely to be of little value unless there is also given some function involving these statistics, the integral of which between any two limits gives an approximate value for the observed frequency corresponding to these two limits. In the general case, therefore, where one wishes to summarize an extensive series of observations which may not satisfy the condition of control, it is necessary to give a satisfactory function of this character to be used in interpreting the sig- nificance of the tabulated statistics from the viewpoint of presentation of the total information contained in the original set of data. Such functions are usually termed theoretical frequency distribution functions, and from the viewpoint of 98 ECONOMIC CONTROL OF QUALITY presentation of an observed set of data, it would appear that the one to be used is usually that one which satisfies best the condition described in Paragraph I of this chapter. - 5. Conclusion We may divide observed distributions into two classes— those that have and those that have not arisen under controlled conditions. For distributions of the first class, the three simple statistics, average X, standard deviation o, and skew- ness k contain almost all of the information in the original distribution. For those of the second class the most useful statistics are the average and standard deviation. These contain a large part of the total information in the original distribution, at least in respect to the number of observations lying within symmetrical ranges about the average. CHAPTER IX PRESENTATION OF DATA To INDICATE RELATIONSHIP I. Two Kinds of Relationship Two kinds of relationship call for consideration: mathe- matical or functional, and statistical. * Functional Relationship.–If for each value of some variable A a given law assigns one or more values to Y, then we say that Y is a function of X and write Y = f(X). As a simple example, we may take Y = c(X - a) + b. sº A/ ºs/ / 4 (a) (b) FIG. 34.—GRAPH of FUNCTION Y = c(X – a) + b SHowING SIGNIFICANce of PARAMETERs a, b, AND c. Obviously, the graph of this function is a straight line passing through the point X = a, Y = }. The arbitrary constants a, b, and c in this function are called parameters. If we fix the values of a and b, and give to c all possible values, we get a pencil of lines through the point (a, b). Fig. 34-a shows such a pencil 99 IOO ECONOMIC CONTROL OF QUALITY through the point (1, 2). In a similar way, if we fix the value of c and assign arbitrary values to a and b, we get a family of parallel lines. Fig. 34-8 shows such a family for c = 3. This simple example illustrates a general principle that should be kept in mind, viz., that the expression of a func- tional relationship involves two things: I. The form of the functional relationship. 2. The specific values of the parameters in that relationship. Thus, in the problem just considered, the form of the function is linear since Y varies directly as X. How it varies is fixed by the values of the parameters a, b, and c. Statistical Relationship.–If for each value of some variable A a given law assigns a particular frequency distribution of values of Y not the same for all values of X, then we say that Y and X are statistically related. Two variables statistically related are said to be correlated. If we let .zd KdV represent the frequency of occurrence of values of X within the interval X to X + d X simultaneously with values of Y within the interval Y to Y + d'Y, the func- tional relationship z = f(X, Y) is said to characterize the statistical relationship between A and Y. One important statistical relationship which will often be considered in further discussions is the so-called normal frequency function in two variables X and Y, I ––– (i. * – 9. */ 2, - eT3TFF) (; + i. *::::), (27) 2Tozgºv/ -- 7-2 where x = X – X and y = Y — Y. This is the familiar bell- shaped frequency surface shown in Fig. 35. Obviously, five parameters X, Y, or, gy, and rare involved in (27). Our interest at present is centered in the fact that the characterization of a DATA TO INDICATE RELATIONSHIP IOI statistical relationship involves two things—form and specific values of parameters—as did the characterization of functional relationship. FIG. 35.--THE NoFMAL SURFACE. 2. Observed Relationship In our causal explanation or interpretation of data we assume that both functional and statistical relationships exist. In fact it is one of the fundamental objects of experi- mental investigation to determine these relationships or physical laws, as they are customarily called. This practical problem involves, in most instances, the formulation of the law from a study of the observed data, including both the functional form of the law and the estimate of the parameters in the law. Taking the simplest case of relationship between two quality characteristics X and Y, it is obvious that our formulation' of the law and our estimate of the parameters must be based upon an observed set of, let us say, n pairs of simultaneously observed values of the two characteristics. In other words, the total information is tied up in these m pairs of values. Suppose that we are studying the relationship between IO2 ECONOMIC CONTROL OF QUALITY two physical quantities, such as the length L of a rod and the temperature 0 at which this length is measured, or the distance s that a body falls starting from rest and the time t that it is falling. One object of such a study is the expression of the law of relationship. For example, we often assume that the empirical law relating the length and temperature of a rod of given material is linear, or, in other words, that the length varies directly with the temperature; i.e., L = Loſ I + o-0), where Lo is the original length of the rod, and o. is the parameter indicating rate of increase with temperature. In a similar way, we say that the law relating s and t in the case of a freely falling body is s = #at”, where a is a parameter. Having decided once and for all that the law in question is such and such, it remains for us to discover the best values of the param- eters, as is illustrated by these two simple problems. A statement of the law and estimates of the parameters in that law is the common method of summarizing data indicating relationship. However, even in the simple case where we believe that a functional relationship exists, it is a difficult matter to determine what this functional relationship likely is; and, having once decided what function to assume, we must choose one from among the many different possible ways of finding estimates of the required parameters. In other words, the problem of presenting data in this way is to a large extent indeterminate even when the assumed relationship is functional. It goes without saying that the indeterminateness becomes even greater when the relationship assumed to exist is statistical. To emphasize what has just been said, let us try to find the relationship between the current through and the voltage across a carbon contact from the data given in Table 7. In this case there is no a priori basis for assuming the form of the law of relationship. If, however, we assume that it is functional and parabolic in form, or, in other words, if we assume that the current Y is related to the voltage X in the following way, Y = ao + al X+ ag X*, DATA TO INDICATE RELATIONSHIP IO3 we must find, from the data, estimates of the three parameters ao, ai, and a2. If we had a universally accepted method of finding these parameters under these conditions, the problem of presenting relationship would be quite simple indeed. As we have already said, however, there are many different ways of estimating these parameters, four of which are: I. Direct substitution of observed values. 2. Graphical method. 3. Method of least squares. 4. Method of moments. The details of the methods of estimating the parameters in these different ways are given in standard treatises on curve fitting. It will serve our purpose here to consider merely the variability in some of the results obtained by these different methods. Two of the several possible sets of values for the parameters that can be obtained by direct substitution are those in the equations - Y = – O.OIOOO + O.OI333X + O.OOOOOX*, and Y = o.o.9000 — o.oO1672 + O.oOo;6X*. Each of the following equations contains one of the infinite number of possible sets of values for the parameters obtainable by the particular method indicated. I. Graphical Method Y = – O.O2446 + O.OI225A + O.OOOI2X*, 2. Method of Least Squares 1 Y = O.OO809 + O.OO967 x + O.OOOI6X*, 3. Method of Moments? Y = O.O.2649 + O.OO83IX + O.OOOI8-Rº. * This equation was obtained by minimizing the vertical deviation of a point from the curve of fit. Obviously, this is only one of an infinite number of different ways in which the minimizing process could be carried out, by choosing different distances to minimize. We customarily minimize one of the three distances, vertical, horizontal, or perpendicular. r * * We may use any three moments. The first three are usually chosen. IO4 ECONOMIC CONTROL OF QUALITY Obviously, any one of these equations is supposed to summarize the data of Table 7 in respect to relationship. It is apparent, however, that the details of this summary depend upon the choice of the method of calculating estimates of the parameters. If in this case a different law of relationship is assumed to exist, the values of the parameters supposed to contain the information in the original set of data may be expected to be different from those given above. The difficulties of expressing relationship in this simple problem are multiplied many fold when the relationship is statistical instead of functional. In the light of these considerations, it becomes apparent that the problem of presenting essential information in respect to relationship is a complicated one and that a complete discussion of the subject is beyond the scope of the present text. What we shall do in the remainder of this chapter is to consider the significance of the correlation coefficient as a measure of relationship, because we shall find it to be a satis- factory measure in most of the problems with which we have to deal. 3. Information Given by the Correlation Coefficient 1 A. Let us assume that we have n simultaneously observed pairs of values of two quality characteristics X and Y. As a specific case, let us consider the observed set of sixty pairs of values of tensile strength and hardness previously given in Table 3 and shown graphically in Fig. 36. It may be shown that the line of best fit to such an array of points obtained by the method of least squares * through minimizing the squares of the vertical deviations of these points from this line is O’ y = rº, (28) where x = X – X, and y = Y – Y, the symbols X, Y, or, gy, and r being expressed in terms of the n observed pairs of values * It will be found helpful to read Chapter IV of Mathematical Statistics by H. L. Rietz in connection with the remainder of this chapter. * Throughout the remainder of this chapter, a line of “best” fit is always to be taken in the least square sense. DATA TO INDICATE RELATIONSHIP IO5 of X and Y. In the same way, the equation of the line of best fit obtained by the method of least squares through minimizing the horizontal deviations of the points from this line is given by the equation x - y (29) Similarly, the line of best fit obtained by minimizing the squares of the perpendicular deviations of the points from the line of fit is given by the equation I y = — :* — a y”) — V(a * – a y?)? + 4r°ozºa. * x. (30) y / B/ / / A / O / % 9.” / /* A. 2" /* D / 22 ,' 2. 2' / e 2' O 2 * . . .';… . . O / Z” //2 e O O O ſº 22 X O •2%. O O 2 / •2 / O © O ,” O O * 2° 2',4 O o” " A O 2^ / e” 2^ / O 2^ / O 2' O /* ſ / A - MINIMIZING THE PERPENDICULAR DEVIATIONS / B — MINIMIZING THE X DEVIATIONS / C - MINIMIZING THE y DEVIATIONS FIG. 36.-LINEs of FIT DERIVED FROM A KNow LEDGE OF X, Y, or, oy AND r. Ioé ECONOMIC CONTROL OF QUALITY Equation (30) with a positive sign before the radical gives the line of worst fit. Having summarized the information in the sixty pairs of values of tensile strength and hardness in the form: Average Tensile Strength Y in psi 31869.4 Average Hardness X in Rockwells 69.825 Standard Deviation oy of Tensile Strength in psi = 3962.9 Standard Deviation or of Hardness in Rockwells = II.773 Correlation Coefficient r = o.683 we may write down without further work the equations to the three lines of best fit just mentioned. They are y = 229.904×, 3 = O.OO2O29y, y = 492.837.x. These are shown graphically in Fig. 36. In Figs. 36 and 40 the variables are expressed in terms of their respective standard deviations, and the units of the scales are made equal. B. If, in a scatter diagram such as that showing the rela- tionship between depth of sapwood and depth of penetration, we plot the averages of the column and row arrays, we get some such result as that indicated in Fig. 37. The line of best fit to the averages of the columns when each squared deviation is weighted by the number of points in the corresponding column is given except for errors of grouping by (28); similarly, except for errors of grouping, the line of best fit to the averages of the rows is given by (29). These two lines are called re- spectively the lines of regression of y on x and of x on y. It is shown in elementary texts on statistics that, if all of the standard deviations in the column arrays are equal, then for linear regression each is equal to the standard deviation sy of the observed points in the scatter diagram about line (28), where * Jy = gºv/ * 7.2. (31) * When this condition is satisfied, the distribution of y is said to be homoscedastic. DATA TO INDICATE RELATIONSHIP Io'7 With this same restriction, if all of the standard deviations in the row arrays are equal, then it follows that each is equal to the standard deviation ºr of the points about line (29) and is given by the expression Ja: = gºv/ – r2. (32) O AVERAGES OF Y ARRAYS O AVERAGES OF X ARRAYS O LINES OF RE GRESSION 234 }=ºH.º: }* º O l | l l | ſ l | | O | 2 3. 4. 5 6 DEPTH OF SAPWOOD IN INCHES — X | | | FIG. 37.-LINEs of REGRESSION. Under these conditions, it follows from what has just been said and from Tchebycheff’s theorem that the fraction of the total number of points in the scatter diagram within the band y =E 23 y = rºx + ŽJy - (33) O y e I will be greater than I – H. 22 If this scatter diagram has been obtained under conditions of control or, in other words, if the distributions in the row IO8 ECONOMIC CONTROL OF QUALITY and column arrays are approximately normal, the number of points within such a band will be approximately that derived from the normal law integral. Fig. 38 shows such a band for the 1,370 pairs of values of depth of sapwood and depth of penetration for the case z = 3. Under controlled conditions, this band should include approximately 99.7 per cent of the 4.6 4,3 4, O 3, 7 3.4 3. ] 2,8 2.5 2.2 O.7 O, 4 l,o 1.3 1.6 19 2.2 2.5 2.8 3, 3.4 3.7 4.0 4.3 4.6 4.9 5.2 5,5 DEPTH OF SAP W OOD IN | N C H E S – X, FIG. 38.-LINE of REGRESSION AND 99.7 PER CENT LIMIts. I,379 points. We find that it actually includes 99.1 per cent of the observed values, even though the data do not rigorously meet the condition of control. What has just been said concerning the band about the line of regression of y on x holds good in a similar way for the corresponding band about the line of regression of x On y. DATA TO INDICATE RELATIONSHIP IO9 C. If we rewrite the equation (27) of the normal surface in the form I x2 2trazgºv/ſ – rº 2. F é 2 , (34) we see that all values of x and y for a constant value x, of x lie on an ellipse defined by the equation O I 3.2 * 27-3. (i. + *: * ...) = x1°. (35) I — 7-2 Oract’ Oy Oa:0 y By revolving the original axes through an angle of such that 27-0 way tan 26 = −3, (36) Oſa: — Oy the equation of this ellipse for any value of x becomes 2 axiº + by 1% = x*, (37) I I + I N I I \ 2 + 4r” q = — smº * * - - - - 2(I — rº)|\gg” gy” ozº a y” azºo,” |’ I I I I I \* , 47.2 ,-----|(º). NG-P), +| 2(I - r?) gº a y” ozº gy” ozºo y” Hence the semi-axes of any ellipse are where and vº. and §, (38) respectively. When the observed frequency distribution in two dimensions has been obtained under controlled conditions and sometimes even when the conditions have not been controlled, the number nx within the ellipse x is given approximately by the integral X 2 ſ e-Éxdy = 1 – e X*. (39) O - I IO ECONOMIC CONTROL OF QUALITY TABLE C Fraction Fraction Fraction Fraction X” Outside Inside Outside Inside x? 2–3 x” I – e – 3 x* e - #x” I – e – 3 x* O. I O. 95 I:229 o.O4877I O. 9ooo O. IOOO o. 2107 O. 2 o. 904837 o.O951.63 O.80Oo O. 2COO o. 4463 O. 3 o,860708 O. I39292 o. 7500 o. 25OO O. 5754 O. 4 o,818731 o. 181269 o. 7000 o. 3OOO O. 7I 34 O. 5 o. 778801. o. 221 199 o. 6ooo O. 4OOO I.O217 o.6 o. 740818 o. 259182 o. 5oCO o. 50CO I .3863 o. 7 o. 704688 o. 295.312 o. 4COO o. 6000 I. 9326 o. 8 o. 670320 o. 32968o O. 3OOO o. 7000 2.408o O. 9 o,637628 o. 362372 o. 25OO o. 75oo 2.77.26 I ... O o. 606531 o. 393469 O. 2COO o. 8ooo 3.21.98 2 ... O o. 367879 o. 6321.21 O. IOOO o. 9000 4.6052 3. O o. 2231.30 o. 776870 O. OSCO O. 95oo 5.99.I 5 4.. O O. I35,335 o. 864665 O. OIOO O. 99CO 9. 2 IO4 5. O o.o8208; O. 91791 5 o.OO3O O. 997O II .8.194 6. O o.O497.87 O. 9502I 3 O. OO2 O. 9973 II .8290 7. O o.O3OI 97 O. 9698o3 8. O o.OI8316 o, 981684 9. O o, or I Io9 o. 988891 IO. O o.OO6738 o. 993262 II ... O o.OO4087 O. 9959 I 3 I 2 ... O o.OO2479 O. 997.52 I I 3. O o.OOI 503 o. 998497 I4. O o.OOO912 o. 999988 I 5. O O. OOO553 O. 999447 I6.o o.OOO335 o. 999665 I7.o o.OOO2O3 O. 999797 I8. O o.OOOI 23 o. 99.9877 I9. O o.OOOO75 O. 999925 2O ... O O. OOOO45 O. 999955 From Table C we can read off the value of this integral for a large range of values of X*. Fig. 39 illustrates the method of constructing 50 per cent and 99.73 per cent ellipses for the distribution of 1,370 pairs of values of depth of sapwood and depth of penetration. Observation shows 49.9 per cent and 99.12 per cent within these ellipses. DATA TO INDICATE RELATIONSHIP III A = Depth of Sapwood in inches Y = Depth of Penetration in inches n = 1370 2rozoº; - tan 20 = — = 2. O X = 2 91.4088 or = o. 7982 II al Il 20. a.” – gº.” 4393; Y = I. 59I46o a y = o. 624872 2a: = 67°42' 32" r = o. 6032OI o: = 33° 51' 16" I | +++ N I 1 \* , 4r? | t = — I — —H – — = - *-* = I. I QIOA.I 42 2(I * rº) az” ov” a z* & ov” ozºa w” 9I94 2 2 I I I I I 4? & = - - - - - | = [ . .2O II axi” -- by 1% = x* Let x* = 1 .3863 or 1 — e-#x* = o. Sooo Let x* = II. 8290 or I — e – #x * = O. 9973 x = I. I774 X = 3.4393 –– = 1.078 -** = o $114 -*= = *.ico, -** = 1 38 v: - 1.97°4 V, -o v: -3.15° V = 1.49 I. 191941x1* + 5.3OI 130y1* = 1.3863 I. I91941x1* + 5. 3OI 130y1* = II. 8290 4,6 '4,3 4, O 3.4 3. I 2.8 2,5 2.2 1.9 2,2 2.5 2, 8 3, 1 42 || 5 O | 4 3.4 3.7 4,0 4.3 4,6 DEPTH OF SAP WOOD IN IN CHES — X 4.9 5.2 5.5 5.8 FIG. 39.-ILLUSTRATION of METHOD of FINDING 5o PER CENT AND 99.7 PER CENT ELLIPSES FROM THE DATA II 2. ECONOMIC CONTROL OF QUALITY Similar calculations of the correlation ellipses for the sixty pairs of simultaneously observed values of tensile strength and hardness previously discussed give the results shown graphically in Fig. 40. In this connection the line of best fit is that obtained by minimizing the perpendicular distances of the points from the line. 3.5 r- y / 3.O / / -\ 2,5}- &/ S 2” +/ 2" 3. 2' 2,OH g/ & 4. o AS & & $º 33 _{0 º G © 2 9& QS CŞ zºN Še/ zºº $o gº & J.OH- & 2 tº SZ *@*\ sº O 2% 5 3. O : •ey &s O C. O.5 H. / f O 2’ O. 2. O O 5 o!- O ~~~!" X O 5 O PER CENT 2% º O • 27°. ./- ELL | PSE º, -o.5H & 2 2// 12 e O O . O 2' 4. º 2 O ~7 }- -] O }* _2 / &P O & 2^ Ağ / 99.73 PER CENT ^ cº, / 2' * • A -2. OH- 2^ S$ g L^ / -2.5 H. / -3,OH. / / - 3.5 f 1/ | | | | | I | l | I ſ | -3.5 -30 -2.5 -2.O –1.5 —l.O -O.5 O O.5 !.O 1.5 2.O 2.5 3.O 3.5 H A R D N ESS - FIG. 40.-INFORMATION GIVEN BY Aver AGE, STANDARD DEVIATION, AND CoRRELATION CoEFFICIENT. The striking thing about the illustrations considered in this paragraph is that, under certain conditions, a knowledge of the five statistics X, Y, or, gy, and r gives us so much of the total information contained in the raw data. If r be the correlation coefficient between any given set of m pairs of values Xi Yi, X2Y2, ..., X;Y, ..., Xn Yn of any DATA TO INDICATE RELATIONSHIP II.3 two variables X and Y, it is interesting to note that r* = 1 is both a necessary and sufficient condition that the set of points lie on the line (28), because sy = O only when r = == I. In this case sº is also zero and the two lines of regression (28) and (29) coincide. In other words, r* = I is a necessary and sufficient condition that Y be a linear function of X. If rº is approx- imately equal to unity, it is not necessary that all of the points lie near the line of regression although a majority of them do. We must know something about the nature of the scatter before we can interpret r in this case. 4. Relationship between Several Qualities What has been said about the relationship between two quality characteristics can easily be extended to the case of several. We shall consider here only the use of the correlation coefficient in determining the plane of best fit and the location of the observed points in a band about this plane for the case of three variables. *g Let us assume that we have n sets of simultaneous values of three variables X, Y, and Z. Let X, Y, Z, or, oy, oz, rºy, rye, and raw be the arithmetic means, standard deviations, and correlation coefficients respectively. It may easily be shown that the plane of regression of z on x and y, when x = X – X, y = Y - Y, and z = Z – Z, is given, except for errors of grouping, by the following ex- pression 2 = a + & + cy, (40) where - a = O, O2 (raz - ryzrºy) & = § I oz(I - ray”) > (4 ) 7°/2, — At C = oz( Q/2 ſay? zº). (42) oyſ I — ray”) These equations show that a knowledge of averages, standard deviations, and correlation coefficients gives us the * Obviously rev = rvº, etc. II4 ECONOMIC CONTROL OF QUALITY information required to construct such a plane. As an illus- tration, Table 19 gives these statistics for the sixty sets of values of tensile strength, hardness, and density previously given in Table 3. TABLE 19.-INForMATION of TABLE 3 GIVEN IN TERMs of SIMPLE StATIstics Density X in Hardness Y in Tensile Strength Z gm/cm.” Rockwells in psi Arithmetic Mean. . . . 2.6785 69.825 31,869.4 Standard Deviation.. o.og86 II. 773 3,962.9 rzy = o. 616 ry2 = O.683 rzz = o. 657 Substituting these values in (40) we get z = I 53Io.35 x + I 50.988y. The standard deviation oz.yr of the points from this plane is given approximately by I 7-yz raz % Tyz I facy *zz ray I 2)” - == 2,638.5 psi. (43) oz.ja: = Oz (I – ray The graphical representation of the plane was given in Fig. I4. Under conditions of control the number of points within the band formed by the two parallel planes spaced at a distance zozyr on either side of the plane of regression should be approx- imately given by the normal law integral, Table A. Naturally we can duplicate the above discussion for the planes of regression of y on 2 and x and of x on y and z. Equation (43) enables us to measure the scatter of the . observed points in Fig. I4 from the plane of regression shown therein. It is of interest to compare the standard deviation ozyr with the corresponding standard deviations say and $2. * The numerical result given in (43) is obtained by using more decimal places than shown in Table 19. Cf. Paragraph 7, Chapter 7, Part II. DATA TO INDICATE RELATIONSHIP II 5 measuring respectively the standard deviation of the points from the line of regression of z on y and z on x. It is easily verifiable that the equations of these two lines of regression are O'2 z = rſ2+y = 229.964y, Oy and O2 2 = raz--& = 26,418.993.x. O'a, It also follows that /— Jzy = oz V I — ryz” - 2,893.98 psi J2a: = O(2^\/ I — raz” - 2,987.028 psi. Both of these standard deviations are larger than azºry given by (43), the relative magnitudes being represented by the lengths of the lines in Fig. I4-d. and 5. Measure of Relationship—Correlation Ratio Given any set of n pairs of values XI Y1, X2Y2, . . . , X;Y., . . . , XmYn, another useful measure of relationship is the correlation ratio myº of Y on X. By definition $1,” 2 = T – Tlya: I 2 Oy where 31,” is the mean square of deviations from the means of the arrays of y's. The correlation ratio may of X on Y may be defined in a similar manner. It is shown in elementary texts that the square of the correlation ratio must lie between O and I and satisfies the expression I → mycº > r2. The condition that myº.” = I is sufficient to prove that the variable Y can be expressed as a single-valued functional rela- 1. Cf. Rietz, loc. cit. II6 ECONOMIC CONTROL OF QUALITY tionship of X, and that the condition myº — rº = o is satisfied if and only if the regression of y on x is linear. Since the square of the correlation ratio can never be less than the square of the correlation coefficient, it follows that r is zero if myº” is zero. However, the condition that r = o, does not necessarily mean that my: = O. Furthermore, it should be noted that the correlation coefficient r may be zero even though Y is a function of X. Rietz 1 has shown that this is true, for example, when Y = cos \X. 6. Measure of Relationship—General Comments From the viewpoint of presentation of information to show statistical relationship, it is necessary to do more than simply tabulate statistical measures such as the correlation coefficient and correlation ratio.” It will be recalled that a similar state- ment had to be made in respect to the interpretation of moments of a frequency distribution higher than the second. In contrast with this situation, however, we have seen that the average and standard deviation of the distribution contain a large amount of the total information given by that distribution independent of its nature. Of course, the knowledge of these * “On Functional Relations for which the Coefficient of Correlation is Zero,” Quarterly Publications of the American Statistical Association, Vol. XVI, September, 1919, pp. 472-476. * Incidentally, it should be noted that both the correlation coefficient and the correlation ratio are only measures of certain characteristics of correlation defined in the first paragraph of the present chapter. In other words, the frequency distribu- tion functions of the x arrays of y's need not all be alike and hence there may be definite correlation although r = o and myr = O. A case in point is the scatter dia- gram of numbers shown below and typical of an indefinitely large number which might be constructed. I 4. i . 2. I DATA TO INDICATE RELATIONSHIP II 7 two statistics gives us perfectly definite information about an observed set of n pairs of values of any two variables X and Y. Thus we can say: (A) If r2 = 1, then y is related to x by a linear function. (B) If nº = 1, it follows that Y is a function of X or that Y = f(X). (C) The regression of y on x is linear if and only if my.” – rº = O. However, other values of r and m do not give us such positive information. For example, if r2 = O, it does not necessarily follow, as we have already seen, that there is no correlation between Y and X. Similarly, if myº = o, then r = O, but if r = o, it does not necessarily follow that myº = O. Moreover the conditions 7.2 = I myº == I 9 o myº — r" = O do not necessarily tell us much about the correlation between Y and X. We have seen what a useful tool the correlation coefficient r is under certain conditions. We must have been struck, how- ever, with the interesting fact that neither r nor any other measure of relationship gives a fraction of the total information definable within certain limits irrespective of the nature of the relationship, a condition that is satisfied by the average X and standard deviation a of an observed distribution. In other words, no matter whether we express the relationship as functional or statistical, the significance of a given parameter is in the present state of our knowledge dependent upon the form of the relationship, whereas certain information is given by the average X and standard deviation a of a frequency distribution independent of the form of the distribution, and this is made useful through the Tchebycheff theorem. PART III Basis for Specification of Quality Control A Statement of the Necessary and Sufficient Conditions for the Speci- fication of a Controlled Quality CHAPTER X LAws BASIC TO CONTROL I. Contro/ We like to believe that there is law and order in the world. We seek causal explanations of phenomena so that we may predict the nature of the these same phenomena at any future time. As stated in Part I, a phenomenon that can be predicted, at least within limits associated with a given probability, is said to be controlled. Prediction only becomes possible through the acquisition of knowledge of principles or laws. 2. Exact Law By an exact 1 law we shall mean a rule whereby we can predict with a high degree of precision the future course of some phenomenon. An illustration will serve to clarify this definition. If we impose an electromotive force E sin of upon the simple circuit, Fig. 41, with inductance L, capacity C, and resistance R, the current i at any time t is given by the solution of the differential equation R E SIN out L ſidt H di E sin of = L- + Ri + dt C FIG. 41.-ExAMPLE of CoNTROLLED PHE- The current through this NoMENON OBEYING AN EXACT LAW. circuit is, therefore, a simple example of a controlled phenomenon obeying an exact law, in this case a differential equation. C 1 Of course no physical law is exact in the rigorous mathematical sense. The significance of this term as here used will become clear as we proceed. IT). I I22. ECONOMIC CONTROL OF QUALITY Of this same character are the numerous laws of physics and chemistry, such as Newton's laws, Fermat's principle, Maxwell's equations, the principle of least action, and so on. Naturally, the control of quality of manufactured product involves the use of all known exact laws of this character. These laws alone, however, are not enough to insure control because, as we have already noted in Part I, the variability in quality often is unexplainable upon the basis of known exact laws. We say that such variations are produced by unknown or chance causes. If then we are to secure control of quality of product, we must make use not only of exact laws but also of laws of chance, sometimes termed statistical laws. Perhaps the basic law of this character is the law of large numbers. 3. Law of Large Numbers If we flip a coin, either the head or the tail must come up. If we repeat the experiment again and again, we find that there is a certain constancy in the nature of the results obtained and that this constancy appears to be independent of whether you flip the coin or whether I flip it; whether the coin is flipped in some far-off country or at home. From every corner of the world, we get evidence of a certain constancy in the experimental results; i.e., it appears that the observed ratio of the number of times that a head comes up to the total number of throws approaches in a certain sense a constant value for a given coin. This kind of experience is, however, not limited to coin throwing; and, as a result, the following general principle is accepted as a law of nature: Whenever an event may happen in only one of two ways, and the event is observed to happen under the same essential conditions for a large number of times, the ratio p of the number of times that it happens in one way to the total number of trials appears to approach a definite limit, let us say p, as the number of trials increases indefinitely. Symbolically we may state this law in the form Ls p = p, (44) 77 – CO LAWS BASIC TO CONTROL I23 where Ls stands for what we shall term a statistical limit,' which differs from a mathematical limit in that we do not reach a number no of trials such that, for all values of n greater than mo, the ratio of the number of times an event happens to the number of trials differs from some fixed value by less than some previously assigned small quantity e. We shall call this limiting value p an objective probability, and we shall assume that this objective probability of an event happening under the same essential conditions may be used in the same mathematical sense as we use measures of a priori probability in the mathematical theory of probability. Mathematical or a priori probability is usually defined in some such way as the following: If an event can happen in a definite number n of mutually exclusive ways, all ways being equally alike, and if m of these ways be called favorable, then the ratio " is the a priori probability of the favorable 7? event. For example, in the tossing of a coin the number n of ways in which the event may happen is considered to be two—head or tail. If the turning up of a head is taken as favorable and if the two ways the event may happen are equally likely, the a priori probability of a head is #. In a practical case, we never know whether or not the ways an event may happen are equally likely; often we do not even know the number m of ways. Hence we cannot calculate the a priori probability of an event. Assuming the existence of an a priori probability p of an event, the best we can ever hope to do is to adopt some estimate p of this probability which may not and, in general, will not be the true objective value p. Obviously, the concept of a priori probability is not the same as that of a statistical limit. Furthermore, even though an a priori probability of an event does exist in an objective sense, it is not necessary that even an infinite sequence of trials will lead to the establishment of this a priori probability that can be accepted in a rigorous logical sense. On the other hand, if we knew in a given case that an objective a priori * See Fig. I of Appendix II as an illustration of the way p approaches a statistical limit. I24 ECONOMIC CONTROL OF QUALITY probability did exist, it appears that we would most likely have faith that the more observations we took in determining an empirical measure of this objective probability, the better our estimate would become. In general it appears that we must believe that estimates of probabilities derived from large samples are, in the long run, better than those derived from small samples. In other words, it is perhaps reasonable to believe that our best estimates of a priori objective prob- abilities are those values which we determine through large samples. So far as the present book is concerned, a priori probabilities and probability distributions will be characterized by a bold-faced notation wherever necessary for the sake of clearness. Whether we think of these as statistical limits or simply as mathematical entities should not influence to a marked extent their practical significance in that in any case the important thing to note is the way in which estimates of these probabilities represented by the regular symbols are actually derived from the data." A slightly more extended form of this law of large numbers is as follows: If we make a series of n measurements Xi, X2, . . . , Xi, . . . , Xn of some quality characteristic X in such a way that each measurement is made under the same essential conditions, the ratio p of the number of times that an observed value X will be found to lie within any specified range Xr to X, to the total number n will approach a statistical limit p as the number m is increased indefinitely. A still more general statement of this law is: If we take a series of m samples of n measurements, * Y Xii, Xiz, . . . , Xii, . . . , Xin X21, X22, . . . . Xzi. . . . , X2n . . . . . . . f.” (45) Xmi, Xm2, . . . , Xmi, . . . , Xmn . 1 It will be found helpful to read, in this connection, the discussions of the defini- tions of statistical limit and probability found in such books as Fry's Probability and Its Engineering Uses; Coolidge's Probability, and Rietz's Mathematical Statistics. LAWS BASIC TO CONTROL 125 in such a way that each one of the m samples is drawn under the same essential conditions, and if we let 6 be a symmetric function or statistic of the n values of X in a sample of size n, the ratio p of the number of times that the observed value of 6 will be found to lie within the range 01 to 02 to the total number m of samples will approach a definite statistical limit p as the number m of samples is increased indefinitely. Functions of this type are termed statistical laws. To control quality we must make use of both exact and statistical laws. 4. Point Binomial in Relation to Control If p is the mathematical or a priori probability of the occurrence of an event or success and q is the mathematical or a priori probability of the non-occurrence of the event, it readily follows that the probabilities of O, I, 2, 3, . . . , i, . . . , n occurrences of the event in n trials are given by the suc- cessive terms of the point binomial (q + p)”. It also follows that: Average number of successes = pn. (46) Standard deviation of number of successes = V/pan. (47) We are now in a position to consider evidence in justification of our assumption of the existence of the law of large numbers. 5. Evidence of the Existence of the Law of Large Numbers A. Tossing a Coin or Throwing Dice.—Experience shows that, if we throw what appears to be a symmetrical coin or die a very large number of times, the statistical limit of the ratio of the number of heads to the total number of throws of the coin is #. Similarly, if the occurrence of 1, 2, or 3 on a symmetrical die be termed a success, the statistical limit of the ratio of the number of successes to the total number of throws of the die is #. If then our previous assumptions are * See any elementary textbook on probability. I26 ECONOMIC CONTROL OF QUALITY justified, we should expect 1 to find the relative frequencies of occurrence of O, I, 2, 3, . . . , n successes in a large number of throws of n dice to be given by the successive terms of the point binomial (; + #)”. We may make use of some of the experimental results obtained by throwing n dice a large number of times to see how closely the observed frequency distribution of successes checks that of the point binomial. The second column of Table 20 gives the observed relative frequencies of O, I, 2, 3, . , twelve successes in 4,096 throws of twelve dice.” The third column of this table gives the mathematical probabilities, or, in other words, the successive terms of the point binomial (; + #)12. - A little observation shows that the second and third columns reveal a striking agreement. In other words, it appears that TABLE 2.0.-RELATION BETween MATHEMATICAL PROBABILITIES AND ExPERIMENTAL RESULTs Number Observed Mathematical || Number Observed Mathematical of Relative Probability of Relative Probability Successes | Frequency p (; + #)” Successes | Frequency p (; + 3).” O O. OOOO O. OOOT). 7 o. 2068 O. I934 I o.OOI7 o.OO29 8 O. I3O9 o. I 208 2. o.OI46 o, or 61 9 o.O627 O. O537 3 o.O483 O. O537 IO o.OI73 o, or 61 4. O. IO 5o o. I2O8 I I o. oO27 o.OO29 5 o. 1785 O. I 934 I 2 O. OOOO O. OOO2 6 O. 2314 o. 22.56 the rule of procedure followed in calculating the mathematical probabilities in this particular case leads to a close prediction of the experimental results. We return in Part VI to consider more critically the closeness of check between the mathematical probabilities and the observed relative frequencies. *Strictly speaking, we know that the conditions of symmetry are not satisfied by actual coins and dice, hence the statement here made is only approximately true. * These data are given in An Introduction to the Theory of Statistics, by G. Udny Yule (8th ed.), p. 258. LAWS BASIC TO CONTROL 127 B. Sampling Experiment.—If we were to draw a series of m chips with replacement from a bowl containing a large number of similar chips each marked with a given number, common experience leads us to believe that the observed relative fre- quency of the occurrence of a given number would approach as a statistical limit the relative frequency of this number in the bowl as the number of trials increased indefinitely. It follows that, if we were to draw a series of n chips with replace- ment and then a series of say 2n chips, the observed frequency distribution of numbers in the sample of 2n chips should approach closer to the actual frequency distribution of numbers in the bowl than should the observed frequency distribution of say only n chips; or, in general, the larger the number in the sample, the closer, in the statistical sense, should be the approach of the observed frequency distribution of the sample to the true distribution in the bowl. The results of the fol- lowing experiment give evidence that such a prediction, made upon the assumption of the existence of the law of large numbers, appears to be justified. Successive samples of 5, Io, 20, IOO, and 1,000 chips were drawn with replacement from a bowl in which the frequency distribution of the numbers on the chips in the bowl was that indicated in the upper left-hand corner of Fig. 42. The observed relative frequency distributions of numbers for the samples of different size are also shown in this figure. We witness the smoothing out of the distribution with increase in the size of sample as is predicted upon the assumption of the law of large numbers. C. Distribution of Number of Alpha Particles.—In 1910, Rutherford and Geiger observed the distribution of frequencies with which o, I, 2, . . . , m alpha particles struck a screen of constant dimensions in successive equal intervals of time. The objective probability of a particle striking the screen as estimated from this experiment is O.O46; and, assuming that this can be used as a mathematical probability in a point * “The Probability Variations in the Distribution of a Particles,” Philosophical Magazine, Series 6, Vol. XX, 1910, p. 698. I28 ECONOMIC CONTROL OF QUALITY binomial (7+p)" where q + p = I, we get the smooth fre- quency distribution shown in Fig. 43. The agreement between the observed relative frequencies and those calculated from the point binomial is further justification for our belief in the law of large numbers. | { | syst Em of causes sample of 5 | || || | || —“a SAM P L E OF 1 O SAM PLE OF 20 SAMPLE OF IOO SAMPLE OF IOOO FIG. 42.-TYPICAL ExPERIMENTAL EVIDENCE FOR LAw of LARGE NUMBERs. D. Macroscopic Properties of Matter.—We might be willing to agree that there appears to be a close agreement between what was observed under A, B, and C and that which was predicted upon the assumption of the existence of the law of large numbers, and yet we might not appreciate the full extent to which this law is basic to our modern conceptions of physical LAWS BASIC TO CONTROL I29 and chemical laws. Perhaps our best justification for belief in this law comes from study of the macroscopic properties of matter expressed in terms of its microscopic properties. For example, we believe that a gas is made up of a large number of molecules dancing about in a way characterized O. PARTICLES BEING E MITTED UNOER A CONSTANT SYSTEM OF CAUSES 6OOr- e OBSERVED POINTS O - THEORETICAL DISTRIBUTION (o.954+o.o.46)64 O 1. ſ ſ ! I 1–1 I _l =*-a-—1 O 5 YO 15 2O 25 3O N U M BER OF O PARTICLES IN INTERVAL FIG. 43.−FREQUENCY DISTRIBUTION of ALPHA PARTICLEs. by the Brownian motion previously considered. For a single molecule the properties of greatest importance are perhaps those of position, velocity, and mass. In most practical applications, however, we do not interest ourselves so much in these as we do in the properties of a group of molecules, such as pressure, viscosity, temperature, and entropy. Now, it is shown in elementary texts on kinetic theory that these four properties are statistical in nature and result from a I3O ECONOMIC CONTROL OF QUALITY state in which the law of large numbers applies with great precision. For example, it is shown in discussions of kinetic theory that the pressure p of a gas containing v molecules, each of mass m, is given in terms of the root mean square velocity v/º by the expression p = #m v2. - (48) Thus we see that the pressure of a gas is a statistical average dependent upon the law of large numbers for its constancy and yet under constant temperature conditions we know that the pressure remains constant within the precision of our measurementS. In a similar way, we find the law of large numbers playing an important rôle in the discussion of Brownian motion, the fluctuation in density of a fluid, the distribution of velocities of electrons emitted from a hot filament, the distribution of thermal-radiation among its different frequencies, rates of diffusion and evaporation, rates of thermal and electrical conduction, rate of momentum transfer, rates of thermal and photo-chemical reactions, and so on indefinitely. Upon the basis of results such as indicated under A, B, C, and D, we make the following assumption: There exist in nature systems of chance causes which operate in a way such that the effects of these causes can be predicted after the manner just indicated, by making use of customary probability theory in which objective probabilities in the limiting statistical sense are substituted for the mathematical probabilities. Stated in another way, we assume that there are dis- coverable constant systems of chance causes which produce effects in a way that may be predicted. 6. Controlled or Constant System of Chance Causes . The unknown causes producing an event in accordance with the law of large numbers will be called a constant system of chance causes because we assume that the objective prob- ability that such a cause system will produce a given event is independent of time. LAWS BASIC TO CONTROL I3 I In other words, a cause system is constant if the phe- nomenon, produced thereby satisfies the conditions charac- terized by either (44) or (45). 7. Meaning of Cause As human beings, we want a cause for everything but nothing is more elusive than this thing we call a cause. Every cause has its cause and so on ad infinitum. We never get quite to the infinitum. In this sense there must always exist a certain amount of topsy-turviness about the world as we perceive it. All that we can do is to find certain practical rules or relationships among the things which we observe. In doing this, we introduce a lot of terms which we cannot explain in the fundamental sense, but which we use to great advantage as, for example, mass, energy, electron, and so on. Under these conditions we go ahead undaunted and introduce theories as to how these things are related, even though we do not know what these things are that we talk about. As an example, we have theories of light, but we do not know what light is. In some ways it acts like a wave, in others like a corpuscle. From our viewpoint, the justification of the use of either the wave theory or the corpuscular theory of light is that it helps one to attain the desired end. So, in the simple theory of control, we talk about causes even though we do not know what a cause really is any more than we know what light or electricity is. Nevertheless, when we apply control theory, as we do in this book, it is just as easy to get a “feeling” for what we mean by cause in a specific case as it is to get a feeling for what we mean by light when we talk about it. 8. Variable System of Chance Causes All systems of chance causes are not constant as two simple examples will serve to show. Fig. 44 shows the fluc- *An interesting discussion of cause and effect will be found in W. E. Johnson's Logic, Vol. III, treating of the logical foundations of science, and published by the Cambridge University Press, 1924. - I32 ECONOMIC CONTROL OF QUALITY tuations 1 in general business conditions over the period from 1919 to 1928. Similar curves could be given for the fluctuations in market prices of individual commodities or stocks. It is well recognized that the causes of such fluctuations are, for the most part, unknown. The general belief is, however, that variations of this character show distinct trends and possibly cyclic movements—the existence of either rules out the con- stancy of the cause system. Y2O r- o! N") NORMAL A. Rºy: - I O H. : - 3 O I ſ | | | | | | | _l 1919 1920 1921 1922 1923 1924 1925 1926 1927 1928 YEARS FIG. 44.—GENERAL BUSINESS COMPARED witH NorMAL. Fig. 45 shows the growth in the number of Bell-owned telephones in the United States from 1876 to 1928. Similar curves of growth could be given for sales of almost all com- modities, such as radio sets, electric washing machines, per- fumes, automobiles, and so on indefinitely. Always in such curves there are certain irregularities introduced by chance causes. In fact, the causes of such growth in a particular case are usually unknown, although they certainly do not exhibit the characteristics of a constant system. * Weber, P. J., “An Index of General Business Activity,” Bell Telephone Quarterly, April, 1929, pp. 124-131. LAWS BASIC TO CONTROL I33 9. Statistical Laws Constant systems of chance causes give rise to frequency distributions, often called statistical laws. One such is the law of mortality, and another is the law of distribution of dis- * | 6 r- 68|O| 2 º--ºw-}* 4 ſº am me • * * * = - - - - - - - - * * * = - * * * * * * * FIG. 45.-NUMBER OF TELEPHONES IN THE BELL SYSTEM. placements of a particle under Brownian motion, both of which were mentioned in Part I. Another well-known example is Maxwell's law of dis- tribution of molecular velocities, cv.2 dy = Ae 4 duz dwy duz, (49) * It will be noted that a frequency distribution as here used is in the sense of an objective law of distribution whereas, in Part II, it was simply introduced as a function such that its integral over a given range is a fair approximation to the observed number of observations falling within that range. I34 ECONOMIC CONTROL OF QUALITY where dy is the probability of a molecule having a velocity v with components lying within the respective ranges ºr to vº + dor, vy to vy + doy, and v2 to v2 + doz; and where 4 and c are constants for a particular kind of molecule in a given state. We may transform this law into one which gives us the prob- ability dy that a molecule will have a speed between v and v + do. By so doing we get cv.2 dy = Be T2 vºdo, (50) where B is a constant different from A. The constants A, B, and c in these equations can be determined experimentally for a gas under given conditions and these laws may then be used to predict either the number of molecules having an x, y, or 2. component within given limits or a speed v within a given range. Equation (50) may be stated in terms of the root mean square speed V2 in the following way % 30% dy = *(...) e 203 v2dy. (50-a) 2try” Using the value 461.2 meters per second at zero degrees centi- grade determined from (48) for the root mean square speed of an oxygen molecule, we get the distribution of speeds of one thousand oxygen molecules given in Table 2.I. TABLE 21.-DISTRIBUTION of SPEEDs Meters per Number of Meters per Number of Second Molecules Second Molecules O— I OO I3− I4 4OO-5OO 2O2–2O3 IOO–2OO 8I— 82 500—600 I 5I-I 52 2OO–3OO 166–167 600–700 9I- 92 3OO–400 2I4–2 I 5 700 76–77 Fig. 46 shows schematically the shape of this distribution * Data taken from Meyer's Kinetic Theory of Gases. LAWS BASIC TO CONTROL I35 curve and the relationship between the mean speed 5, root mean square speed Vº, and modal speed 5. The mean speed v = 424.9 meters per second Root mean square speed V2 = 461.2 meters per second Most probable speed 5 = 376.6 meters per second Obviously, if the quality of a product is controlled in the sense that the fluctuations therein obey the law of large numbers and hence some statistical distribution law, we must know N | V - - V. V VV2 FIG. 46.-A STATISTICAL LAw—ONE ForM of MAxwell's LAw For Oxygen MoLECULEs. this law in order to predict how many pieces of product will have qualities lying within given limits. To be of use in this as in any other problem, statistical theory must provide us with statistical distribution laws. It is but natural, therefore, that attempts should have been made to discover and tabulate all such laws. As early as 1756 a law of error was proposed, and in quite rapid suc- cession other simple laws of error were suggested. 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CIJE， ODI ±， ±（√( 5，0 ſ, æ:Sp(OLIQºII8J, SIQ•ßgI↓ SIG SJLOĢIĀLĀCI ȘI ORYELO ，O SAYI 3 BR3ſtræ0 || 608T ºrrºIBA||30－0， S º U 8，881 a 3，87* p(OI！Oſnae gÝ Iºſraeſ (x%‘o-(x}4'o ºtrºp-5 *TWACHO， SI VOIJL Oſnaeſ ºg LLYŁ3：30 I38 ECONOMIC CONTROL OF QUALITY An attempt was made to apply the normal law to many observed distributions, but it was soon found to be un- satisfactory in a majority of problems. This situation gave rise to an active search for more general laws, some of which are indicated in the last six rows of the table in Fig. 47. Two of these general laws should be briefly considered here as we shall have occasion to refer to them in one way or another. One is that of Pearson represented by the dif- ferential equation avg (62 +3) I dy 3. Io92 — I 261 — I 8 - - - - 9 v'6, y (51) 3 dx o'(482 – 331) ++ 81 (62 +3) 232 – 351 – 6 .2 Io92 – 1261 – 18 IO62 – 1261 – 18 Logº-Tag Tº where y is the relative frequency function of the deviation x from the arithmetic mean, 31 is the square of the skewness k, a is the standard deviation, and 32 is the measure of flatness. This general law obviously gives rise to several special laws depending upon the functional form of the solution of (51). In turn the form of the law depends upon the values of 31 and 62, as illustrated in Fig. 48. The upper part of this figure shows some of Pearson’s laws fitted to observed data, the corre- sponding values of 31 and 32 being given at the bottom of the figure. It is shown in elementary treatises on frequency curves that some of the laws [solutions of (51)] are valid for whole areas in the 3132 plane; whereas others are valid only for points lying on a certain curve; still others only for one point as is the normal law which corresponds to the point 81 = O, 32 = 3, as is readily seen by substitution of these values in (51). Pearson and his followers claim that these laws have been found to cover practically all cases coming to their attentIOn. LAWS BASIC TO CONTROL I39 TYPE a TYPE b TYPE C T YP E cl T YP E e TYP E f | O wf º •J e k 8 - & 6 H Q- € e C 4 H e 9 4-b § ta e i e ] 2 L hal ſ ! | 1 | | | 1 | | O I.O 2.O 3.O 4.O 5.O 6.O /3 FIG. 48.-RELATIONSHIP BETween PEARson Types AND 8, AND 82. I4O ECONOMIC CONTROL OF QUALITY The other important general law is the Gram-Charlier series ſo-º-º-º-º-º-º-º) + #(or * ..) – I 52 + IOz” — zº) 5! oro I |J. +(8–13.4%)(-is-sº-sºº). e | (52) I _2% e where do(z) = V.” 2, and z = . By taking enough terms 2T and using the proper parameters, this law may be made to fit almost any frequency distribution. Io. Exact and Statistical Laws—A Comparison Perhaps the most important characteristic difference be- tween an exact and a statistical law is that the former states something that is true for a single thing or event, whereas the latter states something that is true on the average or in the long run. The exact law applies to the individual thing, whereas the statistical law applies to a group of the same kind of things. - In general we like to think that exact laws apply under conditions where the physical phenomena are quite well understood, as is true for the current through a simple circuit discussed at the beginning of this chapter. In a similar way, we think of statistical laws as applying where the details of the phenomena are not so thoroughly understood. Between these two apparent extremes lies that great body of facts or data which have not been explained in terms of either of the two kinds of laws just considered; yet even here we find rules or laws which make possible a kind of prediction. Two illus- trations will serve to clarify this statement. We have already called attention to the problem of the economists in forecasting business conditions. There are companies devoting all their time to forecasting. In general, LAWS BASIC TO CONTROL I4 I they claim to have discovered a way of breaking down a time series, such as that shown in Fig. 44, into four parts: (a) Trends, (c) Seasonals, (b) Cycles, (d) Erratic Fluctuations. An outline of the technique involved in such a study is given in most of the elementary books on business statistics. A rule for forecasting developed in this way is sometimes called a law although most people would, to say the least, probably insist on calling it an empirical law. To refer to it as an em- pirical law, however, is somewhat misleading, because any law, insofar as it is derived from experience, is empirical. This point we shall have occasion to emphasize again and again as we proceed. Perhaps the best that we can say is that the degree of empiricism is greater in this case than it is in the case of the so-called exact or statistical laws already considered. Such rules as are used in business forecasting have to do in general with data, the causal explanation or interpretation of which is not thoroughly understood. In other words, here, as in the case of statistical laws, the phenomena themselves are to a large extent attributable to chance or unknown causes. It should be noted, however, that here probability theory does not apply directly because the conditions for the law of large numbers do not hold. This point has been emphasized by Persons." In other words, probability theory does not apply simply because a phenomenon is attributable to chance causes. Let us next consider the phenomenon of growth which comes nearer to being reduced to an exact law than does that of customary economic time series. The literature on this subject is very extensive. Fig. 49 shows the forecast of the population growth of the United States.” It is interesting indeed to see 1 Persons, Foster, and Hettinger, The Problem of Business Forecasting, Houghton Mifflin & Co., New York, 1924. * Raymond Pearl, The Biology of Population Growth, Alfred Knopf, New York, 1925. This book includes an appendix with 165 references. I42 ECONOMIC CONTROL OF QUALITY how closely the observed points fall on this logistic curve, the equation for which is = I 97.27 tº: ( ) y = } + 67.322-0% 53 By means of this law, Pearl predicts the future course of population growth to the year 2 IOO, at which time the popula- tion is to be approximately 197,000,000. The general law of growth y = d + I + gar-Haº-Haº-H . . . -- ana." (54) 2OO F-- – - — — — — — — — — — — — — — — — — — —- - -----, UPPER Li MIT OF PRESENT CYCLE 197. 274 - T e’ | 75 H. 2^ Af / / 15 O H. / IOO H. 75 H. 25 H. O L- 1 ==== E = t = FI I l | ſ l I l | 1 2O 4O 6O 8 O 2O 4 O 60 8 O 2O 4 O 60 8 O 2O 40 60 89 17OO | 8 OO | 9 OO O YEAR FIG. 49.-Forecast of PopULATION GRowTH of THE UNITED STATEs. 2 IOO is shown by Pearl to be applicable to a large number of dif- ferent kinds of populations, and for this reason it may be claimed that the law is less empirical than the laws used in forecasting business conditions. It would perhaps be generally agreed, however, that this law of growth is more empirical than Newton's laws of motion. If we were to observe the growths in population for a large number of pairs of fruit flies, we could expect upon the basis of the work of Pearl and others, that these growths LAWS BASIC TO CONTROL I43 would vary about the law of growth. It seems reasonable to believe that we would find a statistical distribution at any point along the line as indicated in Fig. 50. Such a phenomenon is of interest because it suggests the possibility of the use of probability theory in predicting the deviation from this line— something that economists in general feel cannot be done in connection with economic forecasts. The causal basis for this frequency distribution might be set up after the manner in which hereditary influences are explained by Whittaker and Robinson. They assume that i T | M. E. FIG. go.-STATISTICAL DISTRIBUTION AT ANY PoinT IN A LAw of GRowTH. the chest measure of an individual, for instance, is the result of a very great number of chance causes present in the heredity and environment of the individual. This suggests a type of law derivable upon a causal basis similar to that involved in the study of chemical kinetics. The growth curve under these conditions may be thought of as an exact law, and the distribution about this curve at any point may be thought of as a statistical law. In other words, the general law of growth may be a combination of exact and statistical laws. This suggests another viewpoint in respect to the so-called exact law which is worth considering briefly. As an illustration of an exact law, we have used the dif- * The Calculus of Observations, Blackie & Son, Ltd., London, p. 167. I44 ECONOMIC CONTROL OF QUALITY ferential equation relating the current in a circuit to the inductance, capacity, and resistance of that circuit. The cur- rent, even though it appears to be continuous, is really a flow of a number of discrete units of charge or electrons. Thus, if we could see what is actually taking place when the current appears constant, we would likely find that the number of electrons per second passing a given point is not constant. The apparent constancy is, as in the case of the pressure of the gas, the result of the law of large numbers. Hence we see that our exact law is, in the last analysis, statistical in the sense that the current is a phenomenon obeying the law of large numbers. It should also be noted that all exact laws are sub- ject to statistical laws of error about which we shall hear more as we proceed. II. Summary From what has been said in this chapter, it seems reasonable to draw the following conclusions: A. It is not feasible to make pieces of product identical one with another. Hence a controlled product must be one of variable quality. B. To be able to say that a product is controlled, we must be able to predict, at least within limits, the future variations in the quality. C. To be able to make such predictions, it is necessary that we know certain laws. D. These laws may be exact, empirical, or statistical. Exact laws are generally stated in terms of the differ- ential equations of physics and chemistry. Statistical laws are the frequency distributions arising from the very general law of large numbers. All other laws are empirical. The technique of finding and using exact and statistical laws is better established than that of finding and using what we term empirical laws. CHAPTER XI STATISTICAL CONTROL I. Conditions for Control If there is a causal orderliness in events and phenomena as we postulate, then it follows that, to one with perfect knowledge, everything is predictable and therefore controlled. However, for practical purposes the quality of product is controlled only to the extent that we know the laws that make prediction possible. For one to be able to say that a phenom- enon is controlled, it is necessary and sufficient that he know the laws which make prediction possible. In practice, however, we must start with an observed set of data representing the fluctuations in some phenomenon and try to determine from these whether or not the product is controlled. Such a procedure involves, as do all scientific attempts to discover natural laws, logical induction in that we must employ some such argument as this: Since the observed fluctuations are such as might have occurred provided the phenomenon obeyed such and such laws, then it follows that these laws do control this phenomenon; whereas all that we are rigorously justified in saying is that these laws may control this phenomenon. For this reason we perhaps never can say that the behavior of a phenomenon in the past is sufficient to prove that the phenomenon is controlled by a given set of known laws. All that we can ever say is that experience has shown that such behavior appears to be sufficient. Furthermore it is a significant fact, as we have seen in the previous chapter, that empirical laws do not make possible the prediction of erratic fluctuations upon the basis of prob- ability theory. If product is controlled only in this empirical I45 I46 ECONOMIC CONTROL OF QUALITY sense, it follows that we cannot obtain the economic advantages discussed in Part I. For this reason it is desirable to attain the state of statistical control in which the natural law of large numbers makes prediction possible. 2. Necessary and Sufficient Conditions for Statistical Control We shall assume that the necessary and sufficient condition for statistical control is that the causes of an event satisfy the law of large numbers as do those of a constant system of chance causes. If a cause system is not constant, we shall say that an assignable cause of Type Z is present. Assignable causes of this type in an economic series are such things as trends, cycles, and seasonals; and in a production process, they are such things as differences in machines and in sources of raw material. Stated in terms of effects of a cause system, it is necessary that differences in the qualities of a number of pieces of a product appear to be consistent with the assumption that they arose from a constant system of chance causes. We say appear because, as is always the case in trying to find a law controlling a phenomenon, we can never be sure that we have discovered the law. Obviously such appearance is not sufficient in the logical sense although it must be in the practical SCI1S62. . 3. Necessary and Sufficient Conditions—Continued Let us see how the law of large numbers gives a basis for determining from the observed fluctuations in a phenomenon whether or not it is statistically controlled. For this purpose let us consider the practical problem presented in Part I, Chapter II, Paragraph 2. If this product is statistically controlled, there is an objective probability p that a piece of this product will be defective. It follows, as we have seen in our previous discussion of experi- mental evidence for the existence of the law of large numbers, that the observed fractions defective in successive samples of STATISTICAL CONTROL I47 size n should be clustered or distributed about the value p = p in accord with the terms of the point binomial (q + p.)”. Graphically this means that, if we take the observed values of the fraction defective p as ordinates and a series of numbers corresponding to a sequence of samples of size n as abscissae, the observed fractions should be distributed about the ordinate p after the manner indicated schematically in Fig. 51. The frequency distribution of values of p observed in an infinite sequence of samples of size n should be some curve 0– O tal > O 5 : P O til O - © 2 Q H. O H - - - - - - - - - - - - - - - 3. : H t—l —l- | I I l | ſ I ſ 1—1 1– | | ſ O 1 2 3 OO RELATIVE FREQUENCY FIG. 51.-SchEMATIC of OBJECTIVE ConDITION. such as that indicated at the right of the figure. This is the picture of what happens in this very simple case deduced from the postulated law of large numbers. - The practical problem involves induction instead of de- duction. We start with a sequence of observed values of the fraction defective, and from this we try to determine whether or not the quality as measured by fraction defective is sta- tistically controlled. As indicated in Part I, the method of attack is to establish limits of variability of p, represented by the dotted lines parallel to the line p = p in Fig. 51, such that, I48 ECONOMIC CONTROL OF QUALITY when a fraction defective is found outside these limits, looking for an assignable cause is worth while. How to establish these limits is the question of utmost importance, because it must be satisfactorily answered if statistical control of a production process is to be a practical objective. Experience like that presented in Part I leads us to believe that it is feasible to establish workable rules for setting these limits. These rules will be presented in Part VI. For the present we shall confine our attention to a consideration of some of the fundamental problems which must be considered in the establishment of a scientific basis for setting such limits. - A. Obviously, it is not possible to observe an infinite sequence in order to discover the objective probability p even though it exists and is discoverable in this way. In practice, therefore, we must substitute some experimentally determined value for the objective value p. B. Assuming for the sake of argument that in some manner we have found the true objective value p, it follows from what has previously been said that, no matter how we set the limits about the line p = p (so long as they are not outside the limits of the frequency distribution at the right of Fig. 51), some of the observed fractions will fall outside these limits. Therefore, if we look for trouble in the form of assignable causes of Type I every time an observed fraction falls outside these limits, we shall look a certain number of times even though none exists. Hence we must use limits such that through their use we will not waste too much time looking unnecessarily for trouble. C. The fact that an observed set of values of fraction defective indicates the product to have been controlled up to the present does not prove that we can predict the future course of this phenomenon. We always have to say that this can be done provided the same essential conditions are maintained, and, of course, we never know whether or not they are maintained unless we continue to experiment. If experience were not available to show that a state of statistical equilibrium once reached is usually maintained, we could not attain most STATISTICAL CONTROL I49 of the economic advantages of Part I. Evidence of the type given in Figs. 6 and II seems to justify our belief in the con- stancy of the condition of statistical equilibrium when it is once attained, subject to the limitation that there is no a priori reason for believing that an assignable cause has entered the production process. CHAPTER XII MAXIMUM CoNTROL I. Maximum Control Defined The object of industrial research is to establish ways and means of making better use of past experience. To do this it is essential that research reveal natural laws. The ideal goal sometimes pictured for research is complete knowl- edge of all the laws of nature so that one could predict the future course of all phenomena. The belief in the existence of such a goal rests upon the assumption of a causal orderliness of the universe. If a manufacturer could tell what the quality of each piece of product is going to be, or, more generally, if we could predict exactly the future course of a phenomenon, then we could say that this quality or phenomenon exhibited maximum control. This amounts to assuming that, with perfect knowl- edge of the universe, it would be possible to obtain exact control of quality of product because the element of chance fluctuation in quality could be removed. It is important to note, however, that such a goal is neither feasible nor economic. To emphasize this point, let us take a very simple illustration. All of us are perhaps willing to admit that it is not feasible to find the causes which control the course of a single molecule of a gas. It is also reasonable to believe that there is a state reached in the control of quality beyond which it is just as foolish to try to go as it is to try to find the causes of the motion of a given molecule. Suppose, however, that we did have knowledge which would enable us to set down the differential equations of motion of a system of molecules. Assuming that one could solve I5o MAXIMUM CONTROL I5I these, a little calculation shows that he would have to live something like Iolº years to set down his results for only a thimbleful of molecules at room temperature even though he worked 12 hours per day. Obviously the results of such perfect knowledge would not be usable in an economic sense. In other words, it is believed that there is a limit beyond which it is not economically feasible to go in trying to eliminate chance fluctuations. Common sense guides us in setting conditions to be satisfied by a cause system in a state of maximum control. If one were ill and were told by his physician that there were likely a very large number of causes of his illness, he would feel more discouraged about his condition than he would if he were told that there was only one cause. This follows because it is customarily found to be difficult to ferret out and assign a single cause of illness when there are several unknown causes. What has just been said is true subject to the limitation that each cause produces practically the same effect as any other. Naturally, if one of the causes is known to produce a pre- dominating effect, a person will feel that there is greater likeli- hood of his being able to find this cause than if each of the causes produces the same component effect. This kind of experience leads us to postulate that it is not feasible to explain in terms of specific causes those phenomena which are attribut- able to a very large number of causes such as the throw of a head on a coin, the motion of molecules, the daily fluctuations in the price of a stock, hereditary influences, and so on. Therefore maximum control for our purpose will be defined as the condition reached when the chance fluctuations in a phenomenon are produced by a constant system of a large number of chance causes in which no cause produces a pre- dominating effect. However, in order that these conditions for maximum control may be of practical use, they must be expressed in terms of the effects of the causes. This is obviously necessary because we cannot find out anything about the causes except through their effects. We shall soon discover that serious I 52 ECONOMIC CONTROL OF QUALITY difficulties are involved in trying to set up necessary conditions for maximum control in terms of the distribution of effects of a constant cause system. 2. Characteristics of Maximum Control—Molecular Phenomena At first thought one might expect to find that the dis- tribution of displacements of a particle undergoing Brownian motion should be characteristic of maximum control. Since, as previously noted, this distribution is normal and corresponds to the point (o, 3) in the 3132 plane (Fig. 52), one might be led to ask if there is an objective point of maximum control. A O O.5 I.O 1.5 O ! I U & 2 3 Q-s—POINT OF MAXIMUM CONTROL 2 4° * 4 * FIG. 52.-Is THERE AN OBJEctive Point of MAXIMUM ContRol? As we have already seen, however, the distribution of molecular velocities is not normal even though this distribution obviously arises under a condition of maximum control to the same extent as does the distribution of displacements. This fact alone is sufficient to show that there is not an objective point of maximum control. 3. Necessary Conditions for Maximum Control–Simple Cause System - Let us assume that there are a finite number m of inde- pendent causes, C1, C2, . . . , Ci, . . . . Cm, and that the resultant effect of these causes is the sum of their individual effects. - MAXIMUM CONTROL I 53 In one case let us assume that these m causes produce effects 31, W2, . . . . Wix . . . ; Wm. respectively, with probabilities 1213 p25 tº º tº 3 Pi, tº º o 5 pm. In the other case let us assume that the probability of the ith cause (i = 1, 2, . . . , m) producing a contribution x in the interval x to x + dx is fi(x) dx. A little consideration shows that such systems may be said to exhibit maximum control when: pi = p) º = f;(x) * and m large. (55) wi = xj m large, Obviously the first set of conditions gives rise to a dis- continuous distribution, the ordinates of which are the terms of the point binomial (q + p)” where the effect of each cause is assumed to be unity. As we know, such a distribution is smooth and unimodal. Hence smoothness and unimodality are necessary conditions for maximum control in terms of effects for this simple discontinuous cause system. It is readily shown for the point binomial that 2 _ (? - ?)” and b. = 3++*. (56) pqm pqm From these equations we see that no matter what the values of p and q are, the values of 31 and 32 approach the normal law values o and 3 respectively as m becomes large. This state of affairs is shown graphically in Fig. 53. Hence we see under what conditions the distribution of effects for such a simple cause system approaches normality, characterized by 31 = O and 32 = 3. Of course, the condition that 31 = O and 82 = 3, although necessary for normality, is not sufficient. To one not accustomed to think of distribution functions 61 I54 ECONOMIC CONTROL OF QUALITY .x: !.O i O. 5 FOR ALL VALUES OF p, V3" = o AT INFINITY 4 i FoR ALL VALUES OF p, /32 =3 AT INFINITY | | 1 —l |O |OO 1,000 10,000 VALUES OF m Fig. 53–Conditions UNDER WHICH Distribution of EFFECTS APPROACHES NormaLITY. MAXIMUM CONTROL I 55 in terms of 31 and 32, Fig. 54 is of interest because it gives two binomial distributions fitted by theoretical curves. In the one case p = q = } and the number m of causes is 16. In the other case p = O.I., q = O.9, and m = IOO. This figure illustrates O.2O ſ • (0.5 + 0.5) 16 NORMAL LAW O. I O H. i O.O5 H. | | | | l | O 2 4. 6 8 | O | 2. |4 RESULTANT EFFECT X O. : 4 r- • (o. 1 + o-9) 199 SECOND APPROX|MATION O. I O H. O.O.8 H O, O6 H. | O,O4 pºss O.O2 H. ſ O 2 4. 6 8 |O | 2 | 4 || 6 | 8 20 22 RESULTANT EFFECT X FIG. 54.—APPROACH To NormaLITY witH INCREASE IN NUMBER of CAUses. the rapid approach to normality with increase in the number of causes irrespective of the value of p. * * For the continuous cause system, it may be shown that Bl _ B2 – 3 3 = , and 32 = ... + 3, (57) | Subject only to limitations not met in practice. See for example, Romanovsky, V., “On the Distribution of an Arithmetic Mean in a Series of Independent Trials,” Bulletin of the Russian Academy of Science, 1926. I 56 ECONOMIC CONTROL OF QUALITY where 61 and 32 represent the distribution of the resultant effect of the operation of m continuous causes and B, and B2 represent the cause function f(x). From (57) we see that no matter what the distribution function of a cause is, the dis- tribution function of the resultant effect will approach nor- mality as the number m of causes increases indefinitely. The rate of approach to normality, however, is much more rapid than we might at first expect, as we shall see in Part IV in our discussion of the distribution function of the arithmetic mean. 4. Necessary Conditions—Some Criticisms That chance causes produce equal component effects is obviously not a necessary condition for maximum control, although the discussion of the previous paragraph is thus limited through (55). Thus, in our previous reference to the difficulty of ferreting out a cause of illness from among many causes, it was not necessary to impose the restriction that the causes should produce equal effects. On the other hand, Some restrictions must be placed on the relative magnitudes of the effects as well as upon the number of effects in order that it appear reasonable that one cause may be separated from the others. For example, few of us, strictly speaking, are ever ill from a single cause, and yet we know that causes of illness are findable. It is perhaps enough to insure feasibility of discovery of a cause that the effect of this cause be large compared with the resultant effect of all others. It is not possible, however, to say how large the effect of one cause must be in respect to the resultant effect in order that it be discoverable. Hence we cannot write down explicit require- ments to be fulfilled by a cause system in order that it represent the state of maximum control. However, so long as one cause does not produce an effect greater than the resultant effect of all the others, it seems reasonable to believe that considerable trouble will be expe- rienced in discovering this cause when there are a large number of other causes. With this restriction on the relative mag- MAXIMUM CONTROL I 57 nitudes of component effects, the distribution of resultant effects may be shown to approach normality as the number of causes is increased indefinitely subject to limitations of no practical interest. Perhaps this fact gives credence to a some- what widespread popular belief that normality is a limiting condition approached whenever the number of causes is large. Before too much significance is attached to this fact we must recall that, as shown in the second paragraph of this chapter, normality cannot be, rigorously speaking, a necessary condition for maximum control. From a practical viewpoint we are most concerned with the need for sufficient conditions for maximum control. We want to be able to say that, since the distribution of observed effects of a chance cause system is of such and such nature, therefore the cause system is in the state of maximum control. Neglecting for the present the limitations of all inductive Inferences of this type, let us see if approximate normality is a sufficient condition for maximum control. That this condition is in itself not sufficient can easily be seen by looking at Fig. 55. Here we have two identical normal curves (broken curves) with their averages separated by one and one-half times the standard deviation of either. The result of compounding these two distributions is shown by the black dots. The smooth solid curve is a normal one fitted to the resultant distribution. Suppose now that product comes from two sources, the corresponding qualities being distributed normally as shown by the broken curves. Obviously we could not readily detect the existence of the difference between the two sources by an examination of the resultant curve assuming normality to indicate maximum control. The possibility of such a situation arising in practice, however, is precluded, if we apply the test for maximum control only in those cases where we have first assured ourselves that the data exhibit statistical control. For these reasons it is believed that approximate normality of an observed distribution arising under controlled conditions may be taken as indicating that the cause system is in a state I 58 ECONOMIC CONTROL OF QUALITY of maximum control. On the other hand, the fact that an observed distribution is not approximately normal is not sufficient evidence that the phenomenon is not in the state of maximum control. Some may argue that there exists a general law charac- teristic of the state of maximum control. Suppose then that we make such an assumption. In practice we would always try to fit the observed distribution with this general law; and, having successfully done this, we would argue that the phe- nomenon exhibited maximum control. Since one can fit almost 8 O r- O RESULTANT CURVE O O NORMAL LAW CURVE Uſ) — — — TWO NORMAL CURVES Z 9 60 - H. qſ & __ - U 2- - S ,” SS Uſ) N / OD / \ \ ô 40 H. Z / N LL / X O y / \ / N CC N dO 20 H. / N 3 / N Z 2^ N 2" ^ 22° t - T Y ~ O | -- ~ | l | | ſ | I 1 * ~ * —l * O 2 4. 6 8 | O | 2 |4 | 6 VARIABLE X FIG. 55.—Evidence THAT NorMALITY Alone Is Not SUFFICIENT ConDITION For CoNTROL. any distribution by taking enough terms in a general law such as the Gram-Charlier series, the conclusion that the phe- nomenon exhibits maximum control is foreordained. For this reason it does not appear that much is to be gained by such a teSt. 5. Some Practical Conclusions It appears that there is no characteristic of an observed distribution which in itself is sufficient to indicate a state of maximum control. If, however, the effects appear to have arisen under controlled conditions and at the same time exhibit normality, there is good reason to believe that a state MAXIMUM CONTROL I 59 of maximum control of the cause system has been reached. The occasions when these two conditions are satisfied, however, are so rare that the test is of little utility. We have also seen that normality of a distribution is not a necessary con- dition for maximum control. When a phenomenon has been shown to exhibit control, we have likely gone about as far as we can in detecting the existence of assignable or discoverable causes by standard tests. Our experience shows that after assignable causes of Type I have been found and eliminated, the observed dis- tribution is usually smooth and unimodal. Furthermore, most distributions exhibiting control have been found to be sufficiently near normal to be fitted by the first two terms of the Gram-Charlier series previously referred to as the second approximation (23). PART IV Sampling Fluctuations in Quality A Discussion of the Sampling Fluctuations in the Simple Statistics Used in the Control of Quality CHAPTER XIII SAMPLING FLUCTUATIONS I. Sample One dictionary definition of sample is: “A part of anything presented as evidence of the whole.” Thus, the people living in New York City constitute a sample of those living in the United States. The top layer in a barrel of apples is a sample of those in the barrel. The fish taken from a lake are a sample of those in the lake. The instruments inspected from the product of a given day constitute a sample of that day's product. In each of these instances, the whole of the thing sampled is finite in the sense that there is a finite number of people in the United States, apples in a barrel, and so on. We may, however, think of any one of these samples as a sample of the whole of the possible number of things which the same cause system could produce if it continued to function indefinitely. In this sense the product for a given period is a sample of that which can be produced by the same manu- facturing process. Millikan's measurements of the charge on an electron are a sample of the indefinitely large number of measurements that can be made by this method. On the one hand, we are interested in what the sample tells us about a finite lot or number of things. On the other hand, we are interested in what the sample tells us about the cause system producing the sample—in this sense all our experience is a sample. Thus the data used in establishing natural laws is a sample from the possible infinite set of data that these laws could give. 2. Sampling Fluctuations Even though produced under essentially the same con- ditions, no two things are identical in the sense that no two I63 I64 ECONOMIC CONTROL OF QUALITY apples on the same tree are identical. The differences, as we have said, are attributed to the effects of chance or unknown causes. If we look at one thing after another produced under presumably the same conditions, we find that the quality varies from piece to piece. Such variations are called sampling variations or fluctuations. These sampling variations may be produced by either variable or constant systems of chance causes. As seen in Part III, there is reason to believe that we may find and eliminate variable chance causes, but not those of a constant system in which there is no predominating cause. Hence we must always have sampling fluctuations in the quality of product. However, if produced by a constant system, they are controlled sampling fluctuations in that they can be pre- dicted by well-established probability theory. 3. Simple Illustration of Sampling Fluctuations Let us start our study of sampling with an experiment in which 4,000 drawings of a chip from a bowl were made with replacement; that is, after drawing a chip, it was replaced and thoroughly mixed with the others before another was drawn. In the bowl there were 998 circular chips on each of which there was a number. Forty chips were marked o, 40 were marked – O.I., 40 were marked + O.I., and so on as shown in Table 22. Before replacing a chip in the bowl, the number was recorded. The 4,OOO observed values are given in Table A, Appendix II. In this experiment we have as near an approach as is likely feasible to the condition in which the law of large numbers applies' since, to the best of our knowledge, the same essential conditions can be maintained. The differences between successive numbers drawn are beyond our control. Dividing the observed values into four sets of 1,000 each, we get the four grouped frequency distributions of columns 3, 4, 5, and 6 in Table 23. Column 2 gives the corresponding distribution in the bowl. * Cf. Paragraph 3, Chapter X, Part III. SAMPLING FLUCTUATIONS I65 TABLE 22.-MARKING ON 998 CHIPs for SAMPLING ExPERIMENT Marking Number || Marking Number || Marking Number || Marking Number on Chip of on Chip of on Chip of on Chip of A Chips A Chips A Chips X Chips –3.o I — I. 5 I 3 O ... O 4O I. 5 I3 –2.9 I – I .4 I 5 O . I 4O I 6 II —2.8 I — I. 3 17 O. 2 39 I. 7 9 –2.7 I – I - 2 I9 O. 3 38 I . 8 8 —2.6 I – I . I 22 O. 4. 37 I. 9 7 –2.5 2. - I - O 24 O. 5 35 2 . O 5 –2.4 2. —o. 9 27 o. 6 33 2 . I 4. –2.3 3 —o. 8 29 o. 7 3I 2. 2. 4. — 2 .. 2 4. —o. 7 3 I o. 8 29 2. 3 3 — 2 . I 4. —o.6 33 O. 9 27 2.4 2. — 2 ... O 5 —o. 5 35 I. O 24 2.5 2 – I. 9 7 —o. 4 37 I . I 22. 2.6 I — I 8 8 —o. 3 38 I . 2. I9 2.7 I —1.7 9 — O. Q. 39 I. 3 17 2.8 I — I .6 II — O. I 4O I. 4 I 5 2.9 I 3. O I TABLE 23–GROUPED FREQUENCY DISTRIBUTIONS IN SAMPLING ExPERIMENT Observed Distributions Cell Distribution Midpoint in Bowl Sample No. 1 Sample No. 2 Sample No. 3 |Sample No. 4 –3.o 3 5 I Q 2. –2.5 9 9 I4 IO 9 — 2 ... O 28 36 2 29 25 — I. 5 65 55 5 I 2. 49 — I . O I Q. I I 23 II 3 I 24 II 2 —o. 5 I74 I65 187 I8 I I9I O I98 2O3 I95 I8O 2O4. O. 5 I74 172 176 I69 I 82 I . O I 2. I I23 I 25. I QC I23 I. 5 65 68 71 67 64 2 . O 28 3I 3I 32 25 2.5 9 8 8 II I 2 3. O 3 2. 4 3 2. I66 ECONOMIC CONTROL OF QUALITY As is to be expected, no two of the observed distributions are the same, and no one of them is the same as that in the bowl. In fact the differences between these five distributions are quite marked as is evident from their graphical presen- tations in Fig. 56. The differences look much like those previ- * 15 O }* |OOH- UN | V E R S E 5 O }* O hº –––1—1—1–1—1–1–1– i -3.O -2.O —l, O O l,O 2.O 3,0 NUM BER ON CHIP 2OOT- Pºme (ſ) Q- IL - º 5 150 Ui- 9 too. }* § SA M P L E SA M P LE CD NO. : NO, 2 > 5 OH- - T) 2 OL | hº M 2OO º (ſ) 0- T 150 H. hºme O 5 | OOH }* Cº. SA M P LE SAM PLE # NO. 3 NO. 4. > 5 OH- }* r) 2. OL l—1–1–1–1–1–1–1–1 l 1—1–1–1–1–1 I I —” º I | | I | -3,O -2.O –I.O O I.O 2. O 3,O -3,O -2,O –|, O O I.O 2.0 3,O NUMBER ON CH | P NUM BER ON CHI P FIG. 56.-SAMPLING FLUCTUATIONS UNDER ContRoLLED ConDITIONs. ously shown in Fig. I9—so much so, in fact, that one might hesitate to say that the distributions in Fig. 19 reveal any evidence of lack of statistical control, although, as we shall soon see, an assignable cause was present in that case. Hence we see that we may be misled if we depend upon the qualitative appearance of deviations to indicate the presence of an assign- able cause. What we need in such a case is some quantitative measure of the deviation of the distribution in a sample from that in the bowl to be used as a basis for detecting lack of control. SAMPLING FLUCTUATIONS 167 4. Sampling Fluctuations in Simple Statistics We shall use simple statistics such as the average X, standard deviation g, skewness k = V/3, and flatness 82 for expressing quantitatively the differences between the observed distributions. For example, columns 2 to 5 of Table 24 give the observed values of these statistics for the four observed distributions of Table 23. We see how the observed distribu- tions differ quantitatively in respect to these simple statistics. Column 6 of Table 24 gives, for comparison purposes, the values of these same statistics for the distribution in the bowl. TABLE 24.—OBSERVED VALUES of STATIstics For DISTRIBUTIONS GIVEN IN TABLE 23 Observed Distributions Distribution Sample Sample | Sample Sample in Bowl No. I No. 2 No. 3 No. 4 Average. . . . . . . . . . . . . . O. OOI 5 o.O445 |-o. OO60 o.O365 O Standard Deviation. . . . I.O2.19 I. OOI9 I.O.317 O. 97.39 I. OO7o Skewness. . . . . . . . . . . . . —o. O903 –o. of 26 O. O631 o.Oo38 O Flatness. . . . . . . . . . . . . . 2.92.57 2.9904 2.7996 3. O757 2.93O2 Instead of performing such an experiment to determine how samples differ, we try to predict such variability in the prob- ability sense. To do this, we must find the distribution func- tions of averages, standard deviations, and other statistics in samples of size n drawn from the distribution in the bowl. Usually this is a complicated mathematical procedure, as we shall soon see. Therefore, to begin with, we shall take a simple example in which the distribution functions can be derived by elementary arithmetic. 5. Simple Problem in Prediction of Sampling Fluctuation— Problem of Distribution Suppose that there are just four similar chips in a bowl, and that these are marked I, 2, 3, and 4 respectively. Suppose that samples of 4 are to be drawn with replacement. The I68 ECONOMIC CONTROL OF QUALITY problem to be considered is the prediction of sampling fluctu- ations in the simple statistics. Since the number of ways of choosing r things from m things where each of the r things may be any one of the m things is m”, it follows that a sample of four may be chosen in 44 = 256 different ways. Obviously not all of the 256 samples will be different. A little study will show that the different possible samples are those given in Column I of Table 25, and that the number of ways in which these may be drawn are as given in Column 2. The corresponding dis- tributions of statistics X, o, k, and 32 can now be set down as in the last four columns of this same table. The frequency distributions of these and certain other statistics are shown graphically in Fig. 57. It is of interest to note that the method of finding the distributions in Fig. 57 is purely an analytical one involving simple arithmetic. One sets down all of the possible samples of size four that can be drawn from the bowl, and then finds the averages, standard deviations, and other statistics for this set of possible samples. If we assume that the sampling fluctuations in the statistics of samples drawn from such a bowl satisfy the law of large numbers, it follows from evidence given in Part III that the observed distributions of statistics in samples of size four may be expected to approach as statistical limits the respective & * This involves the assumption that similar in the phrase “similar chips” has the significance of the phrase “equally likely” so often used in probability theory. It seems reasonable to believe, however, that “equally likely” is a concept which has significance for the external world rather than for mathematics. On this point it will be of interest to read “Probability as Expressed by Asympototic Limits of Pencils of Sequences,” by E. L. Dodd, published in the Bulletin of the American Mathematical Society, Vol. 36 (1930), pp. 299–305. For example, he says: “In pure mathematics, the word probability may be taken to signify simply the ratio of the number of objects in a subset to the number in the set, so long as discrete or arithmetic probability is being considered. It is, indeed, as far outside the field of mathematics to determine whether two events are equally likely as to determine whether two bodies have the same mass. Even in the applications, the rôle of pure mathematics is merely to count expeditiously the elements of sets and subsets, or, more generally, to determine certain measures of sets, which are believed by competent judges to depict adequately situa- tions in the external world.” SAMPLING FLUCTUATIONS TABLE 25.-SIMPLE PROBLEM IN DISTRIBUTION THEORY Number Sample º:... y Oſ Å. 62 Occurs I I I I I I . OO O o ) 2222 I 2 . OO O O Indeterminate 3333 . I 3. OO O O 4444 I 4. OO O O III 2 4. I. 25 O. 433O I. I 547 2.3333 III.3 4. I. 5o o. 8660 I. I 547 2. 3333 III.4. 4. I. 75 I. 2990 I. I 547 2. 3333 222 I 4. I. 75 O. 433O – I. I 547 2 . .3333 2223 4. 2.25 O. 433O I. I 547 2. 33.33 2224 4. 2.5o o. 8660 I. I 547 2.3333 333 I 4. 2.5o o. 8660 – I. I 547 2. 3333 3332 4. 2.75 O. 433O – I. I 547 2. 3333 3334 4. 3.25 O. 433O I . I 547 2. 3333 444 I 4. 3.25 I. 2990 – I. I 547 2. 3333 4442 4. 3.5o o. 8660 – I. I 547 2. 33.33 4443 4. 3.75 O. 433O – I. I 547 2. 3333 II 2.2 6 I. 5o o. 5000 O I . OOOO II.33 6 2 . OO I. OOOO O I . OOOO II.44 6 2.5o I. 5oCO O I . OOOO 2233 6 2.5o o. 5000 O I . OOOO 2244 6 3. OO I. OOOO O I . OOOO 3344 6 3.5o O. 5000 O I . OOOO II 23 I 2 I. 75 o. 8292 O. 4934 I.6281 II 24. I 2 2 . OO I .2247 o. 8165 2 . OOOO II.34 I Q. 2.25 I .2990 o. 2138 I .2798 22I 3 I 2 2 . OO o. 7071 O 2 . OOOO 22I4 IQ 2.25 I o897 o. 652O 2.0970 2234 I 2 2.75 o. 8292 O. 4934 I. 628.I 33 I2 I 2. 2 . Tº o. 8292 -O. 4934 I. 6281 33 I4 I 2. 2.75 I.o897 —o.6520 2.0970 3324 I 2 3. OO o. 7071 O 2 . OOOO 44I 2 I 2 2.75 I .2990 —o. 2138 I .2798 44I 3 I2. 3. OO I .2247 —o. 8165 2 . OOOO 4423 I 2 3.25 o. 8292 -o. 4934 I. 628.I I 234 24 2.5o I. II 8o O I .6400 I7o ECONOMIC CONTROL OF QUALITY distributions of these same statistics shown in Fig. 57. In general, the prediction of sampling fluctuations in statistics of samples of size n drawn from a distribution such as that in the bowl requires the knowledge of the distribution functions of these same statistics. Observed fluctuations may or may not have in them component effects of variable chance causes. 6. Relation of Sample to Universe Let us now examine the relationship between some of the simple statistics for the universe (Fig. 57-a) and the averages or expected values of the distributions of these same statistics. For example, Column I of Table 26 gives the values of some of the simple statistics of the universe, and Columns 2 and 3 give the corresponding expected values for samples of size four and Co respectively. TABLE 26.-RELATION OF SAMPLE TO UNIVERSE & Sample Sample | Correction Standard Universe e g n = 4 77 = Co Factor | Deviation Average. . . . . . . . . . . . . . . 2. 5000 2.5oCO 2.5oCO Median. . . . . . . . . . . . . . . 2. 5000 2. SCOO 2.5oCO Root Mean Square Deviation . . . . . . . . . . . I. I 18O o. 91.78 I . I I 8o I. 218 I O. 3755 Mean Deviation. . . . . . . I . OOOO o. 8o36 I . OOOO 1. 3826 O. 4052 Skewness k. . . . . . . . . . . . O O O Flatness 82. . . . . . . . . . . . I .6400 I 7.562 I .6400 The important thing to note is that the expected value of a given statistic in samples of size n is not necessarily equal to the value of this statistic for the universe so long as the sample size n is a finite number. Suppose now that the statistics of the universe are unknown although the functional form is known. We see that, if we wish to estimate a given statistic for the universe from that for a sample of size n, a correction factor is required. Two such factors are given in Table 26 for the case in hand. Another interesting point is that a statistic of the universe may be estimated from the same or other statistics of a sample. SAMPLING FLUCTUATIONS I71 yº 6O r- ö : I ſt 4 O H. til Y - Of Lu J H. 2O H. Cº. ti- | O O ſº | | - | | | O | 2 3 4. O | 2 3 4. UN | V E RSE A V E RA GE (a) (b) 6 O P- |2O r– ö - z 40 H. 80 H. til ID Cſ Lu }* }* OC 2O 4.O ti- O - O , I | , O | 2 3 4. O | 2 3 4 M E D I A N MAXIMUM + M | N | M UM (C) 2 (d) 6O I- 6O T- O : 40 lºse 4O H. ul I) § # 20 F 2O H. ti- O L i | O L 1 ſl | | | | O O.5 |.O 1.5 O O. 5 |.O 1.5 STANDARD DEVIATION ME AN DEVIATION (e) (f) 12O r- 6O T- ö 2 8O mº 4.O ºms til D Cſ Lu 4O !- - 2O H. OC ti- •l L. Lll | | | | | | O - —l O +! O |.. O 1.5 2.O SKEW NESS F LATN ESS (9) (h) Fig. 57–DISTRIBUTION of STATIstics For SAMPLEs Shown in TABLE 25. 172 ECONOMIC CONTROL OF QUALITY Thus either I.218.1 times the standard deviation of a sample of four or 1.3826 times the mean deviation of a sample of four may be used as estimates of the standard deviation of the universe (57-a). The standard deviations of these esti- mates, however, are not equal. We say that one is more efficient than the other. As a measure of this efficiency 1 we take the ratio of the squares of the respective standard devi- ations. For the simple case under consideration the efficiency te . (o.4o K2)? of the root mean square estimate is (o,4952)” – I.1644. (o.3755)* It is suggested that the reader start with some simple universe other than the one used in this chapter and find for this chosen universe the distributions of the four simple statistics for some sample size. By such a procedure, one easily discovers that the distribution function of a given statistic involves a sample size n and depends upon the func- tional form of the universe. It is also discovered that, in general, the correction factors required to go from the expected value of a statistic in a sample of size n to the same statistic of the universe depend upon the nature of the universe and upon the size of the sample. In other words, we come in this way to see that the problem of interpreting a sample involves the specification of the universe and the determination of the distribution function of a given statistic in samples of a given size drawn from this universe. 1 This measure of efficiency is defined as follows: The standard deviation of the mean of m, corrected root mean square deviations (in samples of four) is o.375.5/v/m. while the standard deviation of the mean of m2 corrected mean deviations in samples of four is o.4052/v/m2. If these two standard deviations are to be equal, we must have O. 3755 O. 4052 7721 , V mº Hence the efficiency of the root mean square deviation is m2 (o. 4052)* E = — = —. m1 (o. 3755)* SAMPLING FLUCTUATIONS I73 7. The Problem of Determining the Allowable Variability in Quality from a Statistical /iewpoint If the quality X of a product is statistically controlled, the probability dy that a unit of this kind of product will have a quality X lying within the range X to X + d x is expressible as a function f of the quality X and m' parameters, or formally dy = f(X, \, \2, . . . , Mi, . . . , Mm')d X. (58) We have seen that samples of size n drawn from such a product exhibit sampling fluctuations. These fluctuations may be measured quantitatively in terms of some statistic 0 of the samples, such as average, standard deviation, etc. For each such statistic there is some relative frequency distribution function fe(0, n), representing the distribution of possible values of the statistic 0 in samples of size n drawn from the universe (58). It follows that the probability dye of an observed value of the statistic 0 falling within the range 0 to 0 + de is given by the rela- tionship dye = fo(0, n)de. (59) In general, the distribution functions of the universe and of the statistics may be either continuous or discontinuous. Thus, in Paragraph 5 of this chapter we considered in detail the distribution functions of several statistics for samples of four drawn from a discontinuous universe. Later we shall consider distribution functions for continuous universes. An allowable variability in quality will be defined as one that may reasonably be classed as a sampling fluctuation, or, in other words, one that may reasonably be attributed to the effects of a constant system of chance causes. In the next two chapters we shall consider in some detail the nature of the frequency distribution functions characterizing sampling fluctuations in Some of the simple statistics previously introduced. CHAPTER XIV SAMPLING FLUCTUATIONS IN SIMPLE STATISTICs UNDER STATISTICAL ContROL I. Method of Attack In this chapter we shall assume that the universe of possible effects of the cause system is known, and that the sampling fluctuations obey the law of large numbers. Distribution functions of statistics basic in the theory of control and in the establishment of quality standards are discussed in sufficient detail to make clear their use throughout the remaining chap- ters of the book. Only those points are discussed which have been found helpful in answering practical problems of the following type: A. How shall we determine when quality is statistically controlled? How shall we establish standards of quality? How shall we establish allowable limits in design? . How shall we establish allowable limits of variation from standard quality? - How shall we select a representative sample of product? How large a sample shall we take? The reader primarily interested in such questions may wish to turn immediately to those sections outlining the answers which have been found satisfactory in practice. He will find, however, that these questions, like many of those confronting us every day, do not permit of answers which can be considered as final. One common question will suffice as an illustration of what is meant: What should a child learn I74 F LUCTUATION S—UNDER STATISTICAL CONTROL 175 in school? No one knows the answer to this question, and yet we must adopt an answer in the form of an established cur- riculum. For the most part, we have faith that students of education having a knowledge of the fundamental difficulties involved in getting the answer will be able to make progress in that direction. Similarly, one interested in the answers to the several questions stated in the previous paragraph will find that some parts of the following discussion which at first appear abstract and impractical may actually prove to be the most helpful in the establishment of fundamental principles upon which to base production methods. Starting with the assumption that the universe of possible effects of the controlled system of chance causes is of the form y = f(X, \1, M2, . . . , Mi, . . . , \m'), we shall need to know the probability P that a statistic of a sample of size n produced by this constant system of causes will fall within the range 01 to 62 given formally by the integral - 62 6 1 We shall find that the distribution function of the statistic depends upon the function f of the universe of effects of the cause system, and that the distribution functions of even the simple statistics are unknown except for a very limited number of forms of the function f. In fact, we shall find that for the most part the distribution functions of the simple statistics are known only when the distribution function f of the possible effects of the cause system is normal. Since, however, the normal function involves the assumption that the variable X may extend from – CO to + Co., and since we do not know of any quality X which rigorously satisfies this condition, we see that the theoretical frequency distribution functions which we are to use never can represent practical conditions rigorously. In this same connection, much of the 176 ECONOMIC CONTROL OF QUALITY theory is based upon the assumption of continuity of the observable values of the quality X; although this can never be attained in practice because of inherent limitations in our measuring instruments. Experimental results obtained by sampling under controlled conditions are introduced to in- dicate, in a more or less practical way, the significance of the two limitations just stated. Even when the distribution function fo(0, n) of a statistic 6 is not known so that we cannot calculate the probability P that 6 will lie within a given range, the results of comparatively recent theoretical work enable us to obtain quite satisfactory estimates of the probability P, provided we know the expected value 6 of 6 in samples of size n and the standard deviation ge of 6 measured about the expected value 6. Oftentimes we know the moments of a distribution function, although we do not know the functional form. The work of Tchebycheff referred to in Part II makes it possible for us to say that the probability Pige that an observed value of 9 will fall within the limits 6 + tre satisfies the inequality I Ploe X. I 2. where t is not less than unity. We may also use Tchebycheff's theorem to advantage when the indefinite integral of the dis- tribution function is unknown even though the function is known. Comparatively recent work has given us the expected values and standard deviations of most of the statistics which now appear to be useful in quality control work. Furthermore, these expected values and standard deviations are known for discrete and finite universes of the type which we have to deal with in practice. Hence, we have available for use a certain amount of theoretical work which is immediately applicable to commercial conditions, and which enables us to state at least a lower bound to the probability associated with a symmetric range about the expected value of a statistic. FLUCTUATIONS-UNDER STATISTICAL CONTROL 177 Recently Camp 1 and Meidell 2 have shown that the prob- ability Pro satisfies the inequality I > _ –- Pigo - I 2.25!” provided: (a) The distribution function fe(6, m) of the statistic 6 is unimodal with a modal value é coinciding with the expected value 6. (b) The distribution function foſé), n) of the statistic 6 is monotonic on either side of the modal value. Hence it follows that if we can show that the distribution function of the statistic satisfies the Camp-Meidell conditions, we can estimate the probability associated with a symmetric range about the expected value within closer limits than we can if we know nothing whatsoever about the form of the dis- tribution function of the statistic. In certain instances it is sufficient for practical purposes to be able to show that the modal value is approximately equal to the expected value, and that the distribution function is monotonic about the mode. In this connection, it might be noted that the Camp-Meidell relation applies strictly to a continuous function, although it may easily be shown that this limitation is of no practical significance in the cases where we make use of this theory. Experimental results are introduced wherever necessary to bridge over gaps in available theory. These same experi- mental results will be used extensively in the remaining chapters of the book wherever we consider the problem of interpretation of a sample. In our discussion we shall use bold-faced type to in- dicate the parameters and functional form of the universe of effects of the cause system and also the expected values, standard deviations, and other functions derived from known distribution functions of statistics. The regular italic notation * Camp, B. H., “A new Generalization of Tchebycheff's Statistical Inequality,” Bulletin of the American Mathematical Society, Vol. 28, 1922, pp. 427–432. * Meidell, M. B., “Sur un problème du calcul des probabilités et les statistiques mathématiques,” Comptes Rendus, Vol. 175, 1922, pp. 806-808. 178 ECONOMIC CONTROL OF QUALITY will be used for the corresponding observed characteristics of a sample as indicated in Table 27. TABLE 27.-NotATIONs For UNIVERSE AND SAMPLE Universe Sample Distribution. . . . . . . . . . . . . . . . . . . f(X, A1, A2, . . . , \m') f(X, \, \, . . . , Am) Fraction Defective or Fraction within Given Limits. . . . . . . . . . p p Average. . . . . . . . . . . . . . . . . . . . . . . X X Standard Deviation. . . . . . . . . . . . . Or Or Skewness. . . . . . . . . . . . . . . . . . . . . . k =V8, k = V 31 Flatness. . . . . . . . . . . . . . . . . . . . . . . 32 62 2. Fraction Defective 1 That the fraction defective should play an important rôle in modern production is at once apparent when one con- siders that so many quality measurements are made with a go-no-go gauge. It is but natural, therefore, to consider first the nature of the sampling fluctuations in this fraction under controlled conditions. º The distribution function for the observed fraction defective p or fraction found between any two specified limits X, and X2 in samples of size n drawn from a controlled product of any functional form whatsoever is given by the terms of the point binomial (q + p)”. (60) The expected value p, modal value j, and standard deviation op of this distribution function are given by the following relationships: ” = p (61) * sºmº p p = (62) op * The derivation of the formulas cited in this paragraph are given in almost any elementary text on statistical theory. * Of course the modal and expected values of p are not always equal. However, the difference is too small to be of any practical importance in most applications. FLUCTUATIONS-UNDER STATISTICAL CONTROL 179 In these relationships p is the probability that a constant cause system will produce a defective piece of product. We see at once that the first distribution function (60) that we have chosen is not normal. In fact, it is not even continuous. As pointed out in Part III, however, the point binomial theoretically can be approximated quite closely by the ordinates (or appropriate areas) of a normal curve of the same mean value and standard deviation as the point binomial, provided p is approximately equal to (I — p) and m is very large. We saw in this same connection, however, that the approximation is quite good when p = I — p even if n is no greater than 16; similarly when p = 0. I and n is no greater than IOO. This gives us, therefore, some idea of the degree of precision which we can expect to attain by assuming that the distribution of the observed fraction defective p is normal. Since the modal and expected values of p may be con- sidered equal, and since the discrete distribution can be quite accurately fitted by a function satisfying the Camp-Meidell requirements, it follows that the Camp-Meidell inequality may be assumed to give a close approximation to the lower bound of the probability associated with any symmetrical range about the expected value p. Knowing the standard deviation of p, we may make use of the normal law integral to calculate the probability that an observed fraction p will fall within any two limits p1 and p2, provided the values of p and n are such that the normal law is a satisfactory approx- imation. If the conditions are such that we cannot use the normal law, we may always make use of this value of p and its standard deviation in establishing limits with probability bounds in accord with the Tchebycheff inequality. 3. Average—Normal Universe Perhaps the arithmetic mean is used in engineering work more often than any other statistic to express the central tendency of a group of data. We shall therefore consider next the fluctuation of this statistic in samples of size n drawn from a normal universe. It is a simple matter to show that I8O ECONOMIC CONTROL OF QUALITY under these conditions the distribution of the average X is normal with a standard deviation ºr where o is the standard - 77 deviation of the universe. So long, therefore, as we are dealing with samples from a known normal universe, it is a very simple matter to obtain from Table A the value of the probability that an observed average will fall within any two arbitrarily chosen limits. Hence, from a theoretical viewpoint, we need give no further consideration to the distribution of the average of a sample from a normal universe. It is of interest, however, to see how closely experimental results may be expected to check the theoretical ones, even though we cannot, for reasons previously cited, experiment with samples drawn from a strictly normal universe. Perhaps we cannot duplicate the conditions under which we should expect to find agreement between theory and practice more closely than by drawing chips from a bowl in the manner described in the previous chapter. Obviously, the distribution in the bowl is discontinuous and does not extend to either side of the average beyond three times the standard deviation; whereas a normal distribution is con- tinuous and extends to infinity in both directions. It is of interest, therefore, to note how closely the observed distribution of 1,000 averages of four, Fig. 58, approaches normality. The data of Table A, Appendix II, were divided as indicated into I,000 groups of four each. 4. Average—Non-Normal Universe Even for so simple a statistic as an average, we do not know the distribution function when the universe is not normal." We do, however, know the moments of this dis– tribution function in terms of the moments of the universe. * For exceptions see “On the Means and Squared Standard Deviations of Small Samples from any Population” by A. E. R. Church, Biometrika, Vol. XVIII, pp. 321-394, 1926, and “On the Frequency Distribution of the Means of Samples from Populations of Certain of Pearson's Types,” by J. O. Irwin, Metron, Vol. VIII, pp. 5I-IO6. FLUCTUATIONS-UNDER STATISTICAL CONTROL I 8 I As in the case of averages from a normal universe, the expected value of averages is the average X of the universe. Similarly, go º e sº ſº tº º º O' the standard deviation oz of this distribution is equal to v/n 77 where q is the standard deviation of the universe. With this information we are in a position to apply Tchebycheff's theorem. We may do better than this, however, because it is known that the skewness kx and the flatness B2x of the distribution 2OO r- O © OBSERVED DISTRIBUTION - NORMAL DISTRIBUTION (ſ) 2 H. OC Lil (ſ) § 100 - ti- O CC. # > 50 H. Im) 2. O L. ſ | ſ | ſ I ſ | l | | - 1.5 - I.O —O.5 O O.5 I.O 1.5 AVERAGE X FIG. 58.-ExPERIMENTAL EVIDENCE THAT THE DISTRIBUTION of Aver AGEs of SAMPLES OF SIZE m DRAWN FROM AN EXPERIMENTALLY NorMAL UNIVERSE Is NoFMAL. of averages are given in terms of the corresponding functions of the universe by the following expressions: k kx = −7=, * T V. ( 63) By-º-º: 2, X 77 3 From (63), we see that, if the sample size n is made large enough, no matter what the skewness and flatness of the universe are, the skewness and flatness of the distribution of averages of samples of size n approach normality as charac- I82 ECONOMIC CONTROL OF QUALITY terized by the values o and 3 respectively. It remains for us to show that, even for comparatively small values of n, the distribution of averages may be considered to be normal to a high degree of approximation, thus making possible the use of the normal integral, Table A, in establishing sampling limits. Again we shall appeal to the use of experimental data. Tables B and C of Appendix II give the results of 4,000 draw- i | O O smº. 5 O }== O º –2,4 -I, 2 O dº 1.2 2.4 UNIVERSE AVERAGE X RECTANGULAR UNIVERSE 2OO O O g| 55 150H O or E < §: TooH > u Žá so O O L *-i-I-1–1–1 ſ -1.2 -0.8 -0.4 O O.4 O.8 1.2 1.6 UNIVERSE AVERAGE X RIGHT TRIANGULAR UNIVERSE FIG. 59.-UNIVERSES AND DISTRIBUTIONS OF Aver AGES FROM RECTANGULAR AND RIGHT TRIANGULAR UNIVERSEs. ings with replacement from each of the universes, rectangular and right triangular, described in Table 28. Fig. 59 gives the observed distributions of averages of 1,000 samples of four for each of the two experimental universes. To show how closely these observed distributions actually approach nor- mality, we have drawn smooth normal curves having expected values and standard deviations determined from theory upon the basis of our knowledge of the universes. The closeness of fit is striking and illustrates the rapid approach of the FLUCTUATIONS-UNDER STATISTICAL CONTROL I83 distribution to normality as the sample size is increased. Such evidence, supported by more rigorous analytical methods beyond the scope of our present discussion, leads us to believe that in almost all cases in practice we may establish sampling TABLE 28.-MARKING on CHIPs for ExPERIMENTAL UNIVERSEs Rectangular Universe Right Triangular Universe Marking Number | Marking Number || Marking Number | Marking Number on Chip of on Chip of on Chip of on Chip of A Chips X Chips A Chips A Chips –3. O 2. O. O Q. – I. 3 4O o. 7 2O –2.9 2. O. I 2 — I . 2 39 o. 8 I9 — 2.8 Q. O. Q. Q — I . I 38 O. 9 I8 –2.7 2. O. 3 2. – I . O 37 I . O 17 –2.6 2. O. 4 2. —o. 9 36 I . I I6 –2.5 2. O. 5 Q —o. 8 35 I . TX I 5 — 2.4 Q o. 6 2. —o. 7 34 I. 3 I4 –2.3 2. o. 7 Q. —o. 6 33 I .4 I3 — 2 . ). 2. o. 8 2 —o. 5 2. I. 5 I 2 — Q . I 2. O. 9 Q —o. 4 3I I 6 I I — 2 ... O Q. I ... O 2 —o. 3 3O 1.7 IO – I 9 Q I . I Q – O. 2 29 I . 8 9 — I .8 2. I 2. 2. – O . I 28 I. 9 8 —I. 7 2 I 3 2. O ... O 27 2 ... O 7 — I.6 2. I.4 2 O . I 26 2 . I 6 — I 5 2 I. 5 2. O. 2. 2. 2. 2 . 5 — I.4 2. I 6 2. O. 3 24 2. 3 4. —I .3 2. I. 7 2. O. 4. 23 2.4 3 — I . ). 2. I. 8 2. O. 5 22 2.5 Q. — I . I 2. I. 9 Q o. 6 2 I 2.6 I ... O 2 2 ... O Q , 9 2. 2 . I 2. ... 8 2. 2 . ) 2. .7 2. 2. 3 2. .6 2 2.4 2. . 5 2. 2.5 2. 4 2 2.6 2 ... 3 2. 2.7 2. . 2 2. 2.8 2. , I 2. 2.9 2 3. O 2. I84 ECONOMIC CONTROL OF QUALITY limits for averages of samples of four or more upon the basis of normal law theory. 5. Standard Deviation—Normal Universe The distribution function of the standard deviation has been studied by “Student,” Pearson,” and Fisher.” They have shown that the distribution function of the observed standard deviation a for samples of size n may be expressed in terms of the standard deviation o of the universe in the fol- lowing way: ºn — 2 mor2 e 2a"da. (64) 77 2 O' n — 3 m — 1 2 2 (ii)" 2. e We note at once that the distribution of q is asymmetric, although it approaches symmetry as the size n of the sample increases. Although we have the distribution function in this case, we do not have a table of its integral as we have for the normal law. Obviously, however, (64) is unimodal; and it may be easily shown that the modal value à and the expected value G are given respectively by dy = # = \!"—to – co, (65) 77 and (**) g = W. O = C20. (66) (**) We shall have many occasions to make use of the factors ci and c2 occurring in these two equations. Hence they are * Biometrika, Vol. VI, 1908, pp. 1–25; Vol. XI, 1917, pp. 416–417; Metrom, Vol. V, No. 3, 1925, pp. 18–21. * Biometrika, Vol. X, 1915, pp. 522–529. *Ibid., pp. 507–521; Proc. Cambridge Phil. Soc., Vol. XXI, 1923, pp. 655–658; Metron, Vol. V, No. 3, 1925, pp. 3–17 and 22–32. FLUCTUATIONS-UNDER STATISTICAL CONTROL 185 tabulated in Table 29 for sample sizes most likely to be of interest. TABLE 29.-CoRRECTION FACTORS c1 AND co 7. C 1 C2 77 Cl C2 3 O. 57735 o. 72360 22 o. 95.346 o. 965.45 4. o.707II o. 79788 2 O. 95553 O. 96697 5 o. 77.460 o. 84069 24 O. 95.743 O. 96837 6 o. 81650 o. 86863 2. O. 95917 o. 96965 7 O. 845I 5 O. 8882O 3O o. 96609 O. 97.475 8 o. 86603 o. 9027O 35 O. 971 OI O. 97839 9 o. 88.192 o. 91388 4O O. 97468 O. 981 II IO O. 894.43 O. 92275 45 O. 97753 O. 98322 II O. 90453 o. 92996 5O o. 9798o O. 9849 I I Q o. 91.287 O. 93594 55 o. 9816.5 o. 986.29 I3 o. 91987 O. 940.98 6o O. 983.I.9 o. 98744 I4 O. 92582 O. 94.529 65 O. 9845o O. 9884I I 5 O. 93O94 O. 9490I 70 o. 9856I o. 98924 I6 O. 9354 I O. 95225 75 o. 986.58 O. 98.996 17 O. 93934 O. 955 II 8O O. 98742 O. 99059 I 8 O. 9428 I o. 9576.5 85 o.98817 O. 99 II 5 I9 O. 94,591 O. 9599 I 90 o. 98883 o. 99164 2O O. 94.868 o. 96.194 95 O. 98.942 O. 992O8 2. I O. 95 IIQ o. 96378 IOO o. 98.995 O. 992.48 For sample sizes greater than five, the difference between modal and expected values of standard deviation is so small that in most practical problems we may assume that the Camp-Meidell inequality applies, where the standard deviation O' Von Here again it is not feasible to duplicate theoretical con- ditions in practice. It is therefore interesting to see how closely the 1,000 standard deviations in samples of four drawn from the experimentally normal distribution previously de- scribed can be approximated by (64). The results of such a comparison are shown in Fig. 60. The closeness of fit between the observed and theoretical distributions certainly appears to warrant our acceptance of the theory as a guide to practice in such a case. It is also of interest to note how closely the of the distribution of a is taken to be I86 ECONOMIC CONTROL OF QUALITY ſ&O r- O | 6 O H. O OBSERVED DISTRIBUTION STUDENT DISTRIBUTION |4 O H. |2O H. |OO H. 6O H. 2O H. | | | | | | 1.575 |.875 | l I l O.675 O.975 |.275 ſ O.375 - STANDARD DEVIATION Oſ | O.O75 FIG. 6o.—DISTRIBUTION OF STANDARD DEVIATIONS IN SAMPLEs of Four DRAwN FROM NorMAL UNIVERSE theoretical and observed values of modal and average standard deviation agree as indicated in Table 30. TABLE 30.-AGREEMENT OF THEORETICAL AND OBSERVED VALUES of MoDAL AND Aver AGE VALUES OF STANDARD DEVIATION Observed in Theoretical | 1,000 Samples of Four Modal Standard Deviation in Samples of Four. . . . . o. 7071 o,7168 Expected or Average Standard Deviation in Samples of Four. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O. 7979 o.8007 6. Standard Deviation—Non-Normal Universe Theoretically, we know nothing about the distribution function of the standard deviation of samples from a non-normal universe—not even the values of the moments. If, then, we are to be able to establish ranges of variability within which the observed values of standard deviation may be expected to fall for samples drawn from other than a normal universe, we must rely at the present time upon empirically determined results. FLUCTUATIONS-UNDER STATISTICAL CONTROL 187 To indicate the nature of the results to be expected, it is of interest therefore to consider the observed distributions of standard deviations of samples of four drawn from rectangular and right triangular universes. These are shown in Fig. 61. ºf RECTANGULAR UNIVERSE O e ‘’ e ‘’ º: 9 O H. O > O OC pºss # O § 60 H ū- O O * | § 3o H § O O 2 e ‘’ O O L- f_\ | | I | | | | | | | | | | I I | 10 1 O. O.75 O. 675 |. 275 | . 875 2.475 4 STANDARD DEVIATION O’ 16O % }* RIGHT TRIANGULAR UNIVERSE O O E 12O H. O O § É - O (/) © e O CD O 80 H. O 5 L O OC uj 4 O H. O § * O Ž e O ° - -*————————————l-l-l-e-1– O.O.5. O.45 O.85 t. 25 |, 65 STANDARD DEVIATION OF FIG. 61.—DISTRIBUTIons of STANDARD DEviATIONS of SAMPLEs of Four DRAwN FROM RECTANGULAR AND RIGHT TRIANGULAR UNIVERSEs. As is to be expected, the modal and average values of the observed distributions are less than the standard deviations of the respective universes, Table 31. These results show that since the modal and expected values are approximately equal, it would be possible to apply the Camp-Meidell in- equality except for the fact that the standard deviation is not known. In other words, we are not in a place to set sampling limits on the standard deviation of samples drawn I88 ECONOMIC CONTROL OF QUALITY TABLE 31.-ExPECTED AND MoDAL VALUES OF STANDARD DEVIATION Rectangular Right Triangular Universe Universe Modal Standard Deviation in Samples of Four. I.4639 o. 7761 Average Standard Deviation in Samples of Four. I .432.5 o.786; Standard Deviation of Universe. . . . . . . . . . . . . . 1.7607 O. 9539 from other than a normal universe, unless the divergence from normality is so small as to warrant our belief that the distribution function (64) is a reasonable approximation. In cases where this assumption is not justified, we may make use of the square of the standard deviation or the variance as it is termed. 7. //ariance For variance, as for standard deviation, we know the distribution function when sampling from a normal universe. It is m — 3 mor? dy = C(22) #Te 37*.d(G2), (67) where C is a constant. In fact, “Student” first found this, distribution function empirically, and from it derived the distribution of 0. When the sampled universe is not normal, we know merely the moments of 0° expressed in terms of those of the universe.” The expected variance and the standard deviation of variance alſe • or? - 71 — toº 77 and > (68) o” In – I O.2 = — (n-of-n +3)) 7? 77 * Loc. cit. The distribution was later found rigorously by R. A. Fisher, loc. cit. * See, for example, A. E. R. Church, “On the Means and Squared Standard Devia- tions of Small Samples from any Population,” Biometrika, Vol. XVIII, Nov., 1926, Pp. 32 I-394. FLUCTUATIONS-UNDER STATISTICAL CONTROL 189 in terms of the standard deviation o and the flatness 32 of the universe. Obviously, without further investigation based upon the use of higher moments of the distribution function of variance than those given in (68), we cannot establish sampling limits in general with an assurance much greater than that afforded by the application of the Tchebycheff relationship. — X Or 8. Ratio z = — Normal Universe Thus far we have considered the distribution functions of some of the simple statistics taken one at a time. We shall find that another very helpful way of looking at this problem is to consider the ratio z of the deviation in the average to the standard deviation of the sample. “Student” was the first to derive the distribution of z for samples drawn from a normal universe. His results are given by (69): (**) F. v:(* ) (I + 2) id. (69) T a dy It is useful to know that the standard deviation oz is always I Vn = 3 the expected value Z = o, and the table of the integral of this function originally given by “Student” has now been extended by “Student” and Fisher." Fig. 62 shows how the distribution function of z differs from the normal law for the case n = 4. The broken curve is the normal law with the same standard deviation as the observed distribution of z derived from the thousand samples of four drawn from a normal universe. Two things should be noted. First, although the two distribution functions are symmetrical, they differ widely for small sample sizes. Second, we should equal to The distribution of z is symmetrical about * Loc. cit. I90 ECONOMIC CONTROL OF QUALITY note how closely “Student’s” theoretical distribution fits the observed points in Fig. 62. If the samples are drawn from other than a normal universe, very little of importance in the theory of control is known about the distribution of z other than that derived from an 400 r- C OBSERVED DISTRIBUTION * STUDENT's" DISTRIBUTION — — — NORMAL DISTRIBUTION 35 O H. 3 OO H. 25 O H. 2OO H. | 5 O - - 3. O - 2.4 - 1.8 - 1.2 - O.6 O O.6 1.2 1.8 2.4 3.O FIG. 62.-CoMPARIson of “STUDENT’s ” DISTRIBUTION witH THE Normal LAw AND witH THE OBSERVED VALUES OF z For SAMPLES OF Four. empirical study of the sampling results given in Appendix II. The success of “Student’s” theory in predicting the distribution of Z for samples of size four drawn from rectangular and right triangular universes is indicated in Fig. 63. There can be little doubt that “Student’s” distribution is a closer approx- imation to the observed distribution than is the normal law. Analysis of these results indicates that for most ranges – z to + z (when z < 3) the associated probability given by “Student’s” distribution must be considered as an upper bound! *See for example, “Small Samples—New Experimental Results,” by W. A. Shew- hart and F. W. Winters, journal of American Statistical Association, Vol. 23, pp. I44-I53, 1928. FLUCTUATIONS-UNDER STATISTICAL CONTROL 191 when sampling from a universe with values of 31 and 32 lying in the 3132 plane above the line 62 – 31 – 3 = O. 9. Distribution of Average and Standard Deviation We shall now briefly outline another way in which sampling limits may be set on statistics. Instead of considering the distribution of each statistic separately, we may consider the distribution of pairs of simultaneously observed values of two statistics. As an example, Fig. 64 shows such distributions for averages and standard deviations of the samples from the 4. O - - - O OBSERVED RATIO NORMAL UNIVERSE • A O OBSERVED RATIO RECTANGULAR UNIVERSE % 3. O H. O OBSERVED RATIO TRIANGULAR UNIVERSE £-> CUSTOMARY THEORY º ––––"STUDENT's" THEoRY O 2.0 - O)’ ( 9/ N Jº O |.. O 2. 2 H. 2.2% < • OC ** O O ** ul • cº É |2+2 (/) 142 90 - 1,0 e O Pé92 •. 2. / -2, O C 22* Jºſgo / •/ O -3, O > y y 90 / • 4... O / g O.O. O. O.2 || 2 5 || O 20 50 8O 90 95 98 99.8 99.9 PROBABILITY AssociateD witH RANGE (-CO To z) FIG. 63.-PROBABILITY Associated witH RANGE (– 20 to z). normal, rectangular, and right triangular universes. It is apparent that the distributions for the rectangular and right triangular universes differ materially from that of the thousand samples drawn from the normal universe. For control purposes NORMAL UNIVERSE RECTANGULAR UNIVERSE (a) (8) RIGHT TRANGULAR UNIVERSE (c) Fig. 64.—Observed FREQUEncy SURFAce Histograms Showing CoRRELATIon between IN SAMples of Size n = 4. the Average AND STANdARD Deviation Š FLUCTUATIONS-UNDER STATISTICAL CONTROL 193 we may make use of the form of presentation given in Fig. 65 showing the curves of regression of average on standard deviation. For the samples drawn from a normal universe, we see that the regression curve is a horizontal line. In the other cases, however, the regression is non-linear. For the rectangular universe the curve of regression is a parabola symmetrical about the ordinate through the mean of the dis- tribution; for the right triangular universe the curve of re- gression cannot be so simply described. Recent work of Neyman gives the equation of the curve of regression of the average on variance in terms of the moments of the universe. Neyman also gives the standard deviation of the distribution from this curve of regression. These results of Neyman were used in constructing the theoretical curve of regression and the dotted limits corre- sponding to three times the standard deviation of the dis- tribution about the line of regression for the data presented in Fig. 65. Of course we are not justified in using Neyman's work in this particular way, except to get an approximation. Therefore, it is interesting to note that the results so established . include approximately 99 per cent of the observed values as they should if the distribution about the curve of regression were normal and the theoretical value of the standard deviation used in constructing the limits were not subject to compu- tational error. So far as we are concerned at the present moment, emphasis is to be laid upon the importance of these results as indicating the wide variety of possible ways in which we may establish limits within which observed statistics may be expected to fall. In such a case the theoretical determination of the regression curve together with the standard deviation of an array about such a curve gives us a basis for establishing limits which we may interpret at least upon the basis of Tchebycheff's relationship. A review of the theoretical work that has already been done in this connection, however, indicates certain * “On the Correlation of the Mean and the Variance in Samples Drawn from an ‘Infinite” Population,” Biometrika, Vol. XVIII, pp. 401–413, 1926. § N OR MAL UN | V E R S E R ECT A N GULAR UNIVER SE RIGHT T R A N GULAR UN | V E R SE W. & 75 - - 2.9 15.5 1.725 2.7 14.5 on 1.575 2.5 13.5 -4 : 1.425 wn 2.3 .* § 1.27s X 2. I ; 11.5 3 1. 25 S 1.9 Ž Io.s g > * * g § O. 975 §-1.7 IJ 9.5 O s O. 825 O I. 5 O 8.5 Do n ſn = o.67s S 1.3 s: 7.5 O Jº > 2. o.52s | 9 = 1 . I – 6.5 CA O O O. 375 } | 2 O.9 Z 5.5 O. 225 | Q 0.7 Q 4.5 O O75 | O.5 3.5 -1.5 * I.O. -O 5 O _0.5 I.O 1.5 O.3 2.5 AVERA GE X O | | |.5 e OBSERVED POINTS OF REGRESSION OF -STA NOARD - 2.4 - 1.6 – O.8 O O.8 1.6 2.4 O.5 DEVIATION ON AVE RAGE Ave RAGE X ---- THEORETICAL LIMITS - 12 – 8 - 4 O 4 8 | 2 - Av ERAGE X -- THEOR ETICAL REGRESSION CURVE OF STAN DARO DEVIATION ON A V E RAGE FIG. 65.-ContRol CHARTS TAKING INTo Account THE CorrelATION BEtween STANDARD Deviation AND Average For SAMPLES OF 4. FLUCTUATIONS-UNDER STATISTICAL CONTROL 195 inherent difficulties in attaining a high degree of precision in the derivation of the necessary regression curve and the standard deviation from such a curve. Io. A Word of Caution Before passing on to a consideration of the distribution functions of other statistics, it is well to sound a further word of caution about accepting theoretical results in the form of distribution functions of statistics derived upon assumptions of continuity of universe where for one reason or another the measurements cannot be made under the ideal conditions assumed. As an illustration, it is interesting to examine the effect of grouping in any universe, such for example as the rectangular one, upon the regression of the variance on the average in small samples. We find that the apparent close- ness of fit of a second order parabola to the means of variances depends upon the number of cells. The approximation in many cases is not very good as is illustrated by Fig. 66 corresponding to the scatter diagram of the 256 pairs of values of variance and standard deviation based upon the data of Table 25. Obvi- ously the mean values of variance corresponding to a given average and represented by the solid dots do not lie on a second order parabola. It follows that the precision of the estimate of the number of points to be expected outside the limits derived after the manner of those shown in Fig. 65 is quite uncertain. In fact, we cannot use Tchebycheff's theorem in connection with the parabola of regression to estimate even the upper bound to this number. The reader may appreciate now the significance of the experimental results previously cited to show that the effect of grouping into a finite number of cells and the effect of the finite range of the experimental universe were not sufficient to invalidate the application of the distribution functions for averages, standard deviations, and ratios of deviations in averages to observed standard deviations derived upon the assumption of a continuous universe of infinite range. As a result of these considerations, we see that in the derivation of a I96 ECONOMIC CONTROL OF QUALITY distribution function for a given statistic in a sample of size n drawn from a given universe, we must realize that in practice we can never attain the condition of continuous universe. 2.4 }* O 2.2 e MEAN VARIANCE FOR A GIVEN AVERAGE o MEAN AVERAGE FOR A GIVEN VARIANCE 2 - O H. |. 8 lº O |. 6 He O (NJ b 1.4 H. UA) o: 8 § O Q |.. O H. O O O 8 H- O O O O .6 H. O O O .4 hº O 2 H. O O O 1 —º- O - 40. | © wºr | 2 3. 4. AVERAGE X FIG. 66.—ScATTER DIAGRAM For Aver AGES AND WARIANCES OF ALL PossIBLE SAMPLES OF FOUR FROM UNIVERSE I, 2, 3, 4. Of course, it must be kept in mind that so far as theory is concerned, it is quite possible that the curve of regression even for a continuous universe is not rigorously a second order parab- ola. In other words, the theory involved above rests upon the assumption that a second order parabola is simply a good first approximation to the actual curve. This fact, however, does not invalidate the argument of the previous paragraphs to the effect that the form of the best fitted curve of regression depends to a certain extent at least upon the number of cells into which the universe is divided. FLUCTUATIONS-UNDER STATISTICAL CONTROL 197 I i. Skewness k and Flatness 82 Very little is known about the characteristics of the dis- tribution function of either k or 32 except for large samples drawn from a normal universe under which conditions these distribution functions approach normality. It has long been known, however, that the standard deviations of these two statistics in samples of n drawn from a normal universe are O'k = Nº. (70) 24 06, n (71) and If the sample size n is of the order of magnitude of 500 or more, we may assume that the distribution functions of these statistics about k and B2 respectively of the universe are such that the normal law integral may be assumed to give approx- imate values for the probabilities associated with symmetrical ranges about the expected values." 12. Other Measures of Central Tendency In our discussion of quality control methods, we shall have occasion to use two measures of central tendency other than the arithmetic mean. These are the median and the Max. + Min. r) * The distribution function for the median of samples of n drawn from a normal universe is known to approach normality as the sample size becomes indefinitely large. Little is known, however, about the distribution of medians in samples drawn from other than a normal universe or in small samples drawn from any universe. Also the dis- 1 Isserlis, L., “On the Conditions under which the ‘Probable Errors’ of Frequency Distributions have a Real Significance,” Proceedings of the Royal Society, Series A, Vol. XCII, 1915, pp. 23–4I. I98 ECONOMIC CONTROL OF QUALITY tribution function of the Max. + Min. . o is apparently not known except for samples of n drawn from a rectangular universe." For both these measures of central tendency, we can say that their distribution functions for symmetrical universes are symmetrical so that the expected value for both distribution functions is the average X of the universe. Although, in general, we do not know the standard deviation of either measurement for small samples from even a normal universe, we do know that the standard deviation of the median in large I.2530. samples from a normal universe is *—, where o is the - 77 standard deviation of the universe. 2OO r A R T H M ET IC MEAN O O M E D | A N OBSERVE D o MAXIMUM + MINIMUM | DISTRIBUTION 2 – NORMAL DISTRIBUTION | 8 O H. | 6 O He }4O H. |2O H. | OO H. 80 H. 60 H. 4 O H. 2O H. - [.5 - I.O -ö.5 O O.5 1.O |.5 2.0 FIG. 67.-DISTRIBUTIONs of THREE MEASUREs of CENTRAL TENDENCY IN SAMPLES of SIZE Four DRAwN FROM A NoFMAL UNIVERSE. Wherever it has been found necessary to make use of the distribution functions of these two measures for small samples, they have been determined empirically. For example, Fig. 67 shows the experimentally determined distributions of these two measures for the 1,000 samples of four from the normal universe previously mentioned. For purposes of comparison we have included the theoretical and observed distributions of 1 Rider, P. R., “On the Distribution of the Ratio of Mean to Standard Deviation in Small Samples from Non-Normal Universes,” Biometrika, Vol. XXI, pp. 124–141, I929. FLUCTUATIONS-UNDER STATISTICAL CONTROL I 99 arithmetic means of these samples. We see that all of these are approximately normal. Obviously they would be identical one with another for samples of size two. The observed standard deviations shown in Table 32 are, however, significantly dif- TABLE 32-CHARACTERISTICs of DISTRIBUTIONs of THREE MEASUREs of CENTRAL TENDENCY Efficiency as Compared to that of Mean as Measure of Standard Ioo Per Cent Central Average | Devia- Skewness | Flatness Tendency 6) tion ke 826 Observed Theoretical O G for Samples for Large of 4 Samples Arithmetic Mean. o. or 4 O. 502 || –o.og8 || 2.985 IOO ... O IOO ... O Median . . . . . . . . . o. oo 6 o. 559 —O.O28 2.92I 8o. 6 63.8 Max, T Min.# in o.O36 o. 547 | –o. of 5 || 2.986 84.2 ferent one from another, indicating that the measures differ in efficiency as defined in Paragraph 6 of the previous chapter." Max. + Min. The theoretical efficiency for the measure eVen 2. for large samples is not known, although it is known that it will be less than that of the median. The interesting thing to note is that the efficiency of a measure depends upon the sample size. For example, that of the median starts with IOO per cent for sample size n = 2 and drops off to 63.8 per cent for large samples. 13. Other Measures of Dispersion One of the competing measures of dispersion, particularly in engineering work, is the mean deviation. In general our state of knowledge in respect to the theoretical distribution 1 See discussion in Chapter XIX of Part VI for special interpretation of efficiency for the case of small samples. Just as in Paragraph 6 of the previous chapter, effi- ciency here applies to the estimate of the mean X of the universe obtained from the Max. + Min. mean of m medians or values of — in samples of four. 2. 2OO ECONOMIC CONTROL OF QUALITY of the mean deviation even for samples from the normal universe is in a far less satisfactory state than is that of the standard deviation under similar conditions. For large samples it is true that the distribution function of the mean deviation is sufficiently near normal for us to use the normal integral in establishing sampling limits in control theory. Under these conditions, however, the efficiency of the mean deviation is only about 88 per cent. | 80 r- O e ROOT MEAN SOUARE DEVIATION O M EAN DEVIATION MULTIPLIED BY V; O 16O H. —"STUDENT'S” Distri BUTION FOR STANDARD O DEVIATION 14 O. H. g O 9 120 – O H. < > É # Ioo H ſo O u 80 H. O OC O # 6O H. O > O D 2. 4 O H. Q O 2O H. O O O O O ſ | | I I | | I | O I O O.4 O.8 1.2 |.6 2.O 2.4 FIG. 68.-EMPIRICALLY DETERMINED DISTRIBUTION OF Root MEAN Square AND MEAN DEVIATIONS IN SAMPLEs of Four. The empirically determined distribution of the thousand mean deviations (multiplied by V/t/2) in samples of four is presented in Fig. 68. We see that it is distinctly different in functional form from that of either the theoretical or the observed distribution of standard deviations of this same group of 1,000 samples of four. We also see from Fig. 68 and Table 33 that the mean and modal values of the distribution of mean deviations differ from those of the corresponding distribution of standard deviations. Hence, until further theoretical work has been done, the use of the mean deviation =# FLUCTUATIONS-UNDER STATISTICAL CONTROL 201 for small samples offers comparatively serious limitations as compared with the use of the standard deviation. Furthermore, we shall see that under these conditions the standard deviation is the more efficient measure.' Hence we should not expect to find many cases in quality control work where the mean deviation is to be preferred to the standard deviation as a measure of dispersion. TABLE 33.-CHARACTERISTICs of DISTRIBUTION of THREE MEASURES of DIS PERSION Standard Basis of Estimate of Average | Mode || Devia- |Skewness | Flatness Standard Deviation €) ë tion ke 82e 0 0 Root Mean Square Deviation. . . . o.8007 o. 7161 o. 34o o. 486 2.952 W: (Mean Deviation). . . . . . . . . o. 8612 o. 7353 o. 379 o. 622 3.261 X. — Xi. . . . . . . . . . . . . . . . . . . . . 2. oogo | 1.7564 o.875 o. 548 3. O3O Sometimes we need to use a measure of dispersion which can be readily obtained on the job. For this purpose we may make use of the absolute value of the range between the maximum and minimum observed values in samples of size n. The observed distribution of ranges in samples of four drawn from a normal universe is given in Fig. 69. The average of the thousand observed ranges is 2.Oo30 where q is the standard deviation of the universe. Upon the basis of these experimental results, we could take times the range as # 2.OO3 an approximate value of the standard deviation of the universe; or looked at in another way, knowing the standard deviation of the normal universe, we may set limits within which the observed range in the sample size n may be expected to fall with a given probability P if, as in the previous examples, we can find the distribution function of this range. 1 Chapter XIX of Part VI. 2O2 ECONOMIC CONTROL OF QUALITY Considerable theoretical work has been done within recent years in an attempt to find this distribution function. For example, Tippett gives the expected value and standard deviation of the distribution of ranges in samples of size n drawn from a normal universe. From his results we get Fig. 70. He also gives the theoretical values B1 and 32 of the distribution of the range. In this way, he shows that the distribution of this statistic diverges more and more from normality as the size n of the sample is increased. Obviously, therefore, the 2OO | 75 | 5 O |25 | O O 75 5 O 25 O | | | | l | l | | O | 2 3 4. 5 RANGE BETWEEN MAXIMUM AND MINIMUM VALUES FIG. 69.-OBSERVED DISTRIBUTION of 1,000 RANGES IN SAMPLEs of Four DRAwN FROM A NoFMAL UNIVERSE. best that we can hope to do in the present state of our theoretical knowledge, in using the range for control purposes, is to establish symmetrical limits about the expected value of the range given in Fig. 70 for a specified sample size by making use of theoretical standard deviations also given in this figure. Since we do not know the distribution function, all that we can say is that Tchebycheff’s theorem applies to the limits thus established. In this same connection, it is interesting to compare the * “On the Extreme Individuals and the Range of Samples taken from a Normal Population,” Biomeirika, Vol. XVII, pp. 364–387, December, 1925. FLUCTUATIONS-UNDER STATISTICAL CONTROL 263 observed distribution functions of estimates of the standard deviation g of the universe derived from the root mean Square deviations, mean deviations, and ranges for the thousand samples of size four drawn from the normal universe. These distributions are shown in Fig. 71. The root mean square and mean deviation estimates of the standard deviation a are those usually employed in error theory although they are not consistent as we shall see in Part VI. We shall have occasion later, in discussing the efficiency of measurements, 6.5 O.95 6.O O.9 O ul 5.5 o.85 § Z 5.O o.80; uj 4.5 O.75 O U) 3 2. - 3. 4.0 o:70 = 0. < > 2 3-5 0.65 . < C) u! 3.0 O.60 o X CC 2.5 o.55 ; Z < O.50 - 2.O (ſ) 1.5 O.45 I.O O.4O 2 3 5 | O 2O 3O 5 O |OO 2OO 3OO 50 O IOOO SAM PLE SIZE n FIG. 79-THEORETICAL VALUES of MEAN RANGE AND STANDARD Deviation of RANGE FOR SAMPLEs of SIZE m. to emphasize the significance of the differences in these three distributions. Sometimes in commercial work we may have occasion to use a range other than the extreme range because often the available data represent the quality of product after a previous inspection has excluded the extremes. We shall enter into this discussion only far enough to indicate the nature of the problems involved. At the present time we must rely almost entirely upon the results of empirical studies to indicate the nature of the 2O4. ECONOMIC CONTROL OF QUALITY distribution functions that we may expect to get under such conditions, and also to determine how these functions depend upon the functional form of the universe from which samples are drawn. Fig. 72-a shows the observed distributions of four ranges in samples of four drawn from a normal universe. To obtain these distributions, the four values in each of the thousand samples of four were arranged in ascending order of magnitude. Thus, if we let X1, X2, X3, X4, represent the |OOO r– 9 OO }- 800 H. 7OO H. OBSERVED O MEAN DEVIATION MULTIPLIED BY 2 ſ DISTRIBUTION O RANGE DIVIDED BY EXPECTED RANGE w 5 O O -e O ROOT MEAN SOUARE DEVIATION } STUDENT’s.” Distribution FOR STANDARD DEVIATION FIG. 71.- DISTRIBUTIONs of THREE Estimates of STANDARD DEVIATION of UNIVERSE. values in a sample thus arranged, the four ranges are: the extreme range X4 – X1, the range between the first and second X2 – Xi, the range between the second and third X3 – X2, and the range between the third and fourth X. — X3. The striking thing to be observed is that the distribution functions of the last three ranges are less symmetrical than that of the extreme range. Furthermore, the standard de- viation of the extreme range is larger than that of any one of the other three distributions in absolute magnitude, although FLUCTUATIONS-UNDER STATISTICAL CONTROL 2.05 when expressed as a coefficient of variation, the variation in the extreme range is less than that in any other. For purposes of comparison, the distribution function of observed differences between successive pairs of observed values is also reproduced in this figure. Table 34 shows the observed expected value, standard deviation, skewness, and flatness for these five dis- tributions. TABLE 34.—CHARACTERISTIC OF DISTRIBUTION of RANGEs Average Standard Skewness Flatness Range - Deviation k 6) O 9 6 826 A 2 – X1 o.7863 o. 6087 I. 2 I 33 4. 5604 X3 – X2 o. 6338 O. 494 I I. 245 I 4. 5974 A 4 – X3 o. 7752 O. 5953 1. 1672 4. 3608 X4 – X1 2. OO44 O. 87.59 o. 5627 3. O3I 2 Successive Drawings I. 2136 o. 8661 O. 914O 3. 5884 Turning our attention to Figs. 72-6 and 72-c, we see the marked influence of the functional form of the universe upon the distribution functions of the ranges. This is significant in connection with our present study in that it shows that the interpretation of control limits set upon some statistic such as a range depends much more upon the nature of the func- tional form of the universe than does the interpretation of similar limits placed upon standard deviations and, particu- larly, limits placed upon arithmetic means. I4. Chi Square The statistic X* is a measure of the resultant effect of sampling fluctuations in the cell frequencies. Thus, if the universe of possible effects be divided into m cells such that in a sample of size n the expected frequencies in these cells are respectively y1, y2, ... , yì, . . . , ym; and if the observed 2O6 ECONOMIC CONTROL OF QUALITY frequencies for a given sample in these same cells are y1, y2, . . . , yi, . . . , ym, x* is defined by the relationship 778 X (yi — yº 1 = 1 yi 2 In 1900, Pearson gave the distribution function of the statistic x”, which may be written m — 3 -* (x”) 2 d(x2), (72) : f,2(x”, m) = Ce" where C is a constant. Similarly it may be shown that the expected value X2, the IOOO |OOO > Z - g 800 u 8 OO ID ; 7OO § 700 OC u- 6OO N OR MAL u- 6CO luſ UN | VERSE [L] > 500 > 500 H. H- $ 400 $ 400 2 300 # 300 RIGHT TRIANGULAR ; ; U N | V E R SE O 200 U 200 |OO |OO O O O O.5. I.O 1.5 2.O 2.5 3.O 3.5 4.O 4.5 5,O 5.5 DIFFERENCES IN TERMS OF OF |OOO x- G 900 Z uj 8 OO 3 7OO § ----- DIFFERENCE x2 – xi. Ul- 6OO ––– DIFFERENCE x3 - x2 $ 500 RECT ANGULAR - – D F FERENCE X F U N IV E R SE 4 - X3 $ 400 DIFFERENCE x4 – x1 . 3 300 ---- SUCCESSIVE DIFFERENCES ...) O 200 |OO O O O.5 LO [.5 2.O 2.5 3.0 3.5 4.O 4.5 5.0 5.5 DiFFERENCES IN TERMS OF Or * FIG. 72.- How DISTRIBUTIONs of ORDERED DIFFERENCEs DEPEND Upon THE UNIVERSE. * Karl Pearson “On the Criterion that a Given System of Deviations from the Probable in the Case of a Correlated System of Variables is such that it can be Reason- ably Supposed to have Arisen from Random Sampling,” Philosophical Magazine, 5th Series, Vol. L, 1900, page I 57. FLUCTUATIONS-UNDER STATISTICAL CONTROL 207 modal value X3, and the standard deviation ox of x2 are given by X* = m — I J% (73) X* = m – 3 o,” = \/2(m – 1). (74) Tables of values of the integral of the x” function for the range of values of number m of cells of most importance were O.9999 NUMBER OF D E GREES OF FREEDOM O.999 O.99 O.9 O.8 O.7 O. 6 O.4 O.3 O.2 O.] J. Ol O.OO! O.OOO! | 1.5 2 3 4 5 6 7 8 [O | 5 20 3 O 4 O 50 60 o BSERVED x2 Fig. 73–PROBABILITY Associated witH A GIVEN VALUE OF x*. originally given by Elderton and are reproduced in useful form in Pearson’s Tables." Tables in slightly different form are given by Fisher.” Making use of these tables, we can read off the probability P associated with almost any pair of limits in which we may happen to be interested. Fig. 73 indicates the way in which the probability associated with a given value of x* varies with the number of degrees of freedom.” 1 Tables for Statisticians and Biometricians, Table XII. 2 Statistical Methods for Research Workers. 3 The number of degrees of freedom is equal to one less than the number of cells if, as we have assumed above, the universe frequencies are known a priori. 2O8 ECONOMIC CONTROL OF QUALITY The distribution function of X* is unimodal; and since the mean and the mode differ by only two, the Camp-Meidell inequality applies quite accurately to symmetrical ranges about the expected value. Furthermore, it is of interest to note that, for a comparatively large number m of cells, the O.1 ! T- • x2 Distribution m = 30 ~ SECOND APPROX IMATION DISTRIBUTION O.I.O H. O.O9 H. O.O8 H O.O7 H. O.O6 H. i # ; O.O4 H. O.O3 H O.O2 H O.O | H. O I | | l I | | ſ I I L l | O 4. 8 | 2 | 6 2O 24, 28 32 36 4O 44 48 52 56 6O x2 FIG. 74.—Distribution of x* For m = 30. distribution of x” can be quite accurately obtained by the second approximation function (23). For example, Fig. 74 shows the second approximation fitted to the theoretical dis- tribution of x” for m = 30 cells. Hence, for a number of cells of the order of magnitude of thirty or more, the normal probability function can be used to give a close approximation to the probability associated with a symmetric range about the expected value. & It is of interest to note that the distribution of X* is not explicitly limited by the functional form f of the universe or by the number n in a sample. A limitation, however, does enter in that the functional form of the distribution depends upon the assumption that the variable xi is distributed FLUCTUATIONS-UNDER STATISTICAL CONTROL 209 normally, where xi = y; — yi. From our study of the point binomial distribution function, we see that this assumption requires that the probability pi associated with the ith cell must be such that the probability distribution (qi + p.)” is approximately normal. This condition cannot be rigorously fulfilled, nor do we have any available analytical method for determining its significance. We may, however, again make use of the experimental results presented in Appendix II, this time to give information of an empirical nature which indicates the magnitude of the effect of grouping upon the distribution of X*. We shall make use here of only the four samples of one thousand drawings each from the normal UIn 1WerS62. TABLE 35.-CALCULATIONS INvolved IN DETERMINING x*. True Observed (y – y)* Distribution Distribution y – y (y – y)* *mºs-tº- y y y 3 5 2 4. I. 333 9 9 O O O . OOO 28 36 8 64 2.286 65 55 IO IOO I. 538 I 2 I I Q. T]. 4. O. O33 I 74 165 9 8I o. 466 198 2O3 5 25 o. 126 I 74 172 2. 4. o.O23 I 2 I I23 Q 4. O. O.33 65 68 3 9 o, I 38 28 3 I 3 9 O.32 I 9 8 I I O . I I I 3 2. I I O. 333 x* = 6.74I P = o 873 For purposes of reference, Table 35 shows the calculations involved in determining the value of X* for the first sample of one thousand, grouped into thirteen cells. We see that the probability p associated with the end cells is only gis, which is exceedingly small. We may, therefore, consider the ; TABLE 36.-EFFECT of GROUPING ON x* Observed Distributions º Sample No. 1 Sample No. 2 Sample No. 3 Sample No. 4 tion y I 3 I I 9 7 I 3 I I 9 7 I 3 I I 9 7 I 3 I I 9 7 Cells | Cells | Cells | Cells | Cells Cells | Cells | Cells | Cells | Cells | Cells Cells | Cells | Cells | Cells | Cells 3 5 I 2 2. 9 9 I 4 I 4 I 5 I O I 2 9 I I 28 36 36 5O 24 || 24 || 39 2 29 || 4 I 25 2 36 65 55 || 55 55 IOS 5 I | 5 I | 5 I 90 2. 2 2 II 3 || 49 || 49 || 49 || 85 I 2. I I 2 I 2 I 2 I 23 II 3 II 3 || II 3 || II 3 || I24 I24 I24 I24 || II 2 | II 2 II 2 | I I 2 I 74 165 I65 I65 165 187 | 187 | 187 | 187 || 181 | 181 | 181 | 181 || 191 I9I I9 I I9 I I98 2O3 2O3 2O3 2O3 I95 || IQ5 I95 || IQ5 | I 8o | I 8o I 80 | 180 || 204 || 204 || 204 || 204 I 74 172 172 172 172 176 176 176 176 || 169 | 169 | 169 169 || 182 182 | 182 182 I 2 I I 23 I 23 I 23 I 23 125 | 125 | 125 | 125 || I2O | I2O | I2O | I2O || 123 | 123 | 123 123 65 68 68 68 Io9 71 71 71 II.4. 67 67 67 || 1 13 64 64 64 | Io9 28 3 I 3I + I 3 I 3 I 43 2. 2. 46 25 2 39 9 8 IO 8 I 2. I I I4 I 2 I 4 3 2. 4. 3 2. X? 6.74I 5.630 || 4.882 o. 833 |Io. 7 IC| 6.91 I| 5.519 4.614|| 4.455| 3.900 3.885| 3.364|9. I74| 7.924, 7.291| 6.761 x2 I 2 IO 8 6 I 2 IO 8 6 I 2 I O 8 6 I 2 IO 8 6 FLUCTUATIONS UNDER STATISTICAL CONTROL 2 II advisability of grouping the tails of the distribution after the manner often suggested in the literature. Table 36 shows the effect of grouping the tails of each of the four experimental distributions. In all but one case the observed value of x” is less than the theoretical expected value, although the average difference between the two decreases as we increase the probability associated with the last cell by decreasing the number m of cells. These experimental results indicate that the effect of the limitation as to the normality of the distribution of the variable x, may be much more serious from an experimental viewpoint than one might be led to believe by reading the literature on the subject. In any case the use of x” in control work must be subjected to careful scrutiny to eliminate the obvious effects of grouping even under con- ditions where, as in the present case, we should expect the x” test to be applicable. I5. Summary Broadly speaking, distribution functions of statistics are basic tools with which the engineer interested in quality control must work. In this chapter we have sketched briefly the present state of our knowledge of the distribution functions of some of the more important statistics. A summary of these results is given in Table 37. From this we see how little is really known about the distribution functions of even the simple statistics, particularly when the universe is not normal, with the two exceptions, viz., the fraction defective p and the average X. Subject to limitations set forth in this chapter, we can make use of the average and standard deviation of a statistic, even when the distribution function is not known. When theoretical information about the distribution of a statistic is not available either in the form of the function or certain moments of the function, and we have reason to believe that the universe is not normal, we may make use of the empirical laws presented herein to indicate the extent to which the normal law theory may be applied. We see that there is much É TABLE 37—SUMMARY of AvAILABLE INForMATION IN RESPECT to SoME of the MoRE IMPORTANT STATISTIcs . Expected Value 6 in Standard Deviation or Modal Value Ö Statistic fo (€), n) Samples of Size n 69 €) f Normal | f not Normal | f Normal | f not Normal | f Normal | f not Normal | f Normal | f not Normal p (q –H p)” (q -- p.)” p p Vº Vº p p 77 71 - Approximate- * {- or _of_ tº-º-º: * A Normal ly normal X X vºn vºn X Xf Table 29 or * Table 29 Or (64) (66) Van (65) 2 * T 4.2 * T *-2 * T 3-2 O’ (67) == 77 (68) (68) —-g I 2. (69) O V n — 3 O A: of 6 77 £32 3f Wº: 77 X” (72) (72) (73) (73) (74) (74) (73) (73) e * I. 2530. f tº-e Median X ~7- X Range Fig. 70 Fig. 70 Max. + Min. X X 2. Distribution Function * Correction for sample size of little commercial significance. t If n is large. f Subject to limitations stated in the text. FLUCTUATIONS-UNDER STATISTICAL CONTROL 213 room for future development in distribution theory, all of which will have a direct bearing on the theory of control. However, we shall soon see that in many cases the gain in precision through possible developments of this nature may not be of so great practical importance as might at first be expected. CHAPTER XV SAMPLING FLUCTUATIONS IN SIMPLE STATISTICs— CORRELATION COEFFICIENT 1. Correlation Coefficient Having considered in the previous chapter the distribution functions for statistics of a single variable, we now turn our attention to the distribution function of simultaneously observed quality characteristics correlated one with another. Since, as is to be expected, the problem of deriving the distri- bution functions for correlation statistics is in general much more difficult than those previously considered, we shall confine our attention to the use of the correlation coefficient as a measure of relationship. In Part II we saw how this simple function may be used to present the information con- tained in a single set of n data. There, however, we did not consider how much an observed value of r tells us about what we may expect to get in the future under the same esseytial conditions or, in other words, under the same constant system of chance causes. What was said there about the corrélation coefficient as an expression of observed relationship is true for a given sample. Naturally, however, even under con- trolled conditions this statistic is subject, as are those previously studied, to sampling fluctuations. As an illustration Fig. 75 shows the observed scatter dia- grams and corresponding values of correlation coefficient for eight samples of five simultaneous pairs of values produced by the same constant system of causes wherein there was no correlation or commonness of causation between the two variables. In other words, the correlation r in the universe was zero; yet we find in one sample an observed correlation of — o.82. 2I4. FLUCTUATIONS-CORRELATION COEFFICIENT 21; The method of obtaining these eight samples was as follows: Eighty consecutive values were taken from Table A, Appendix II, and these were grouped into forty pairs by taking the first and second, the third and fourth, and so on. The first five pairs were taken as the first sample, the second five pairs as the second sample; and in this way eight samples of five pairs each were obtained from a non-correlated universe. The result of this experiment is sufficient to show the impor- tance of knowing the distribution function of the correlation 3 ſ = -O.2| T = O.30 ſ = - O.82 ſ = O.23 - - O . - O O O >3' o © "e C O © . © O O O O O O O – 3 t—l—l—l-l—l— —l——l—l—1– (——l—1–1—1– (–1–1–1–1–1–1 3 r = O.O4 n = O.76 ſ = O.42 T = -O. 6 O O eº sº O O O O * O O O O O O e Go O O -3 O l—l—l—l—l—l— |-l—l—1–1–1– |-l—l—1–1—1– \—l—l-l—l-l— – 3 O 3 -3 O 3 -3 O 3. –3 O 3 X X X X FIG. 75.-EIGHT ScATTER DIAGRAMs REPRESENTING SAMPLING FluctuATIONs of THE OBSERVED CoRRELATION IN SAMPLES OF FIVE DRAwN FROM A UNIVERSE IN which THERE was No CoRRELATION. - coefficient as a basis for interpreting the significance of an observed value of the correlation coefficient r in a sample. As might be expected, the distribution function of the observed correlation r in samples of n drawn from a universe in which the correlation is r involves both r and the sample size n. Table 38 presents experimental evidence. Thus Column 2 of this table shows the observed distribution of correlation coefficient r in one hundred samples of four drawn from a universe in which the correlation r was o. It will be seen that the mean value r is o.o.3OO and that the standard deviation or is o.562o. The distribution itself is approximately rectangular. In a similar way, Column 4 shows the observed 3. TABLE 38.-OBSERVED DISTRIBUTIONs of CoRRELATION CoEFFICIENTs n = 4 n = 25 I = O r = o. 5 r = o.895 r = O r = or 5 r = o. 9 r = o. 98 Cell Cell Cell Cell Cell Cell Cell Mā | *..., | Mid- Fre- || M. | Fre- M. ..., | Mid- ..., | Mid- ..., | Mid- Fre- point quency point quency point quency point Q y point quency point Q y point quency —o. 9o I 2 —o. 9o 4. O. I 5 I —o. 6o I — O. I 2 I o. 8oo 3 o. 950 I —o. 6o II —o. 6o 3 O. 3O O —o. 45 2. O ... O O o. 825 3 o. 956 I —o. 30 I6 —o. 30 7 O. 45 2. —o. 30 6 O. I 2 I o. 850 6 o. 962 4. O. O I6 O. O I Q. o, 6o 7 —o. I 5 I 8 O. 24 6 o. 875 22 o. 968 5 O. 3O 17 O. 3O I6 O. 75 II O. O 25 o. 36 I4. O. 9oo 22 O. 974 17 o. 6o 17 o. 6o 37 O. 9o 57 o. I 5 23 o. 48 25 O. 925 I8 o. 980 30 O. 9o I I O. 9o 46 I. O5 22 O. 3O II o. 6o 29 O. 950 IO o. 986 I9 o. 72 IO O. 975 2. O. 992 9 772 = IOO 772 = I2 772 = IOO m = 86 m = 86 m = 86 m = 86 f = o.o.3oo r = o. 4872 r = o.8790 F = ooo37 r = or 5009 ? = o.8968 r = o. 9792 or = or 562O or = o. 4673 or = o. I 5I 5 or = O. I932 or = o I 522 or = o.og77 or = or of 16 FLUCTUATIONS-CORRELATION COEFFICIENT 217 distribution of r for one hundred and twenty-five samples drawn from a universe in which r was o.5. The differences between columns 2, 4, and 6 are attributable to the fact that r is not the same in the three cases. Columns 8, Io, I2, and I4 give the distributions of observed values of the correlation coefficient in samples of twenty-five for different values of r. A comparison of these results with those in the other part of the table indicates the influence of the size of sample. 2. Distribution Function of Correlation Coefficient From experimental results, “Student” derived in 1908 an empirical distribution function of correlation coefficient r in samples of n drawn from a normal universe in which r = O. In 1913 Soper” obtained the mean and the standard deviation of the distribution of correlation coefficient to second approx- imations for samples of n drawn from a normal universe with correlation coefficient r. In 1915 R. A. Fisher 8 showed that the distribution function of r is n — I (1 – rº) * oº:: * d" (* * º) ºmº-º: ==5, G-Fº v/1 rº,2 / (75) d(rr)"-" This function is so complicated as to require a table of values giving the distributions for different values of universe cor- relation r and sample size n. Such tables were provided in 1917 by Soper* and others, and the reader is referred to these for a comprehensive and detailed picture of the dis- tribution of the correlation coefficient. It will be of interest, however, to note the way it varies with the size of sample and the correlation in the universe as shown in Fig. 76. * “On the Probable Error of a Correlation Coefficient,” Biometrika, Vol. VI, p. 302 et seq. * “On the Probable Error of the Correlation Coefficient to a Second Approxi- mation,” Biometrika, Vol. IX, 1913, page 91, et seq. * “Frequency Distribution of the Values of the Correlation Coefficient in Samples from an Indefinitely Large Population,” Biometrika, Vol. X, 1915, page 507, et seq. * H. E. Soper, A. W. Young, B. M. Cave, A. Lee, K. Pearson, “On the Distribution of the Correlation Coefficient in Small Samples,” Biometrika, Vol. XI, 1917, pp. 328– 413. 2I8 ECONOMIC CONTROL OF QUALITY T = O r = O.9 | | -I.O O 1.O - I.O O I.O -I.O O !.O SAMPLE SIZE n = 4 M n = O r = O.5 r = O.9 l 1 1 1 I | | - I.O O I.O -I.O Ö !.O - I.O O 1.O SAMPLE S1 Z E n = 1 O' n = O r = O.9 —1– 1 | I - | I 1 - -I.O O I.O -1.O O I.O - [..O O I.O SAMPLE SIZE n = 25 FIG. 76.-TYPICAL DISTRIBUTIONs of CoRRELATION CoEFFICIENT. 3. Standard Deviation or of Correlation Coefficient The article by Soper and others shows that the standard deviation or of the correlation coefficient in samples of n is given approximately by the simple formula . . - I – r2 FLUCTUATIONS-CORRELATION COEFFICIENT 219 The degree of approximation is indicated by the curves in Fig. 77. In general, it will be seen that, except when the sample size n is small and the universe correlation r is large, formula (76) gives a two-place accuracy. For greater precision the reader must refer to the tables. O.5 // // * || 0:4 // i. %// à O.3 V % / # %/ i //// ... 0.2 Z ºf i=EE: ** TRUE STANDARD DEVIATION OF CORRELATION coEFFICIENT or FIG. 77.-STANDARD DEVIATION OF CoRRELATION CoEFFICIENT IN RELATION TO THE SIMPLE APPROxIMATION (76). 4. Modal and Expected Values of Correlation Coefficient Except for the case of samples from a normal universe with correlation coefficient r = o, the modal value r and the expected or mean value r of correlation coefficient do not coincide with the universe value r. Fig. 78 shows the rela- tionship between these three values for several sample sizes. 22O ECONOMIC CONTROL OF QUALITY We see that for samples of less than twenty-five the absolute differences | r – f | and |r – F | are quite large. Even for n > 25, we often have occasion to make corrections for the fact that these two differences are not zero. !... O O.9 n = 5 s º n = 6.< 22 cal Mobº E sisſ 21.2% As º sº 2% % O.7 *— 2. % lſ. / O. 6 sº Sºsiº //J^2 غ 4% i 7 / % ; | W// / % n = 5 A n = 7 ſal O.2 H. / ń – 25 MEAN n > |OO O. I O O O. I O.2 O.3 O.4 O.5 O.6 O.7 O.8 O.9 l.O TRUE CORRELATION COEFFICIENT " FIG. 78.-RELATIONSHIP of MoDAL VALUE r AND ExPEctED VALUE r IN SAMPLEs of SIZE m FROM A NoFMAL UNIVERSE witH CoRRELATION CoEFFICIENT r. 5. Transformed Distribution of Correlation Coefficient Let us consider the problem of establishing sampling limits on the observed value of correlation coefficient in samples of n drawn from a normally correlated universe for which the correlation coefficient is r. The tables of r and or previously referred to, make it possible to write down limits r + to r, FLUCTUATIONS-CORRELATION COEFFICIENT 221. and applying Tchebycheff's inequality, we can say that the probability of observing a value of r within these limits is I ~. tº º § - F-vº º greater than I – 2’ Since, as is illustrated in Fig. 76, the shape of the distribution function changes so much with size of sample and the correlation of the universe, the actual probability associated with such a pair of limits will vary materially for different sample sizes and different values of r. Under these conditions, some of the recent work of Fisher! can be used to good advantage. He has shown that the dis- tribution of z where 2. = #|loge (I + r.) – loge (I — r)] (77) is approximately normal independent of the sample size and the correlation coefficient r in the normally correlated universe. Furthermore, he has shown that I -v- where oz is the standard deviation of the distribution of the transformed variable z. Fisher has also shown that the expected value Z is greater O2 5 (78) numerically than z by an amount sº where z is the value of z given by (77) for r = r. Making use of these results we can establish sampling limits z + to 2 such that to a high degree of approximation the probability that an observed value of z in samples of size n drawn from a normally correlated universe with correlation coefficient r will fall within the range fixed by these limits is given by the normal law integral. 6. Conditions under which Distribution of r has Significance What has been said about the sampling fluctuations of r has significance only when all samples are drawn from the * Statistical Methods for Research Workers, Second Edition, 1928. 2.2.2 ECONOMIC CONTROL OF QUALITY same constant system of chance causes, so that the probability p that the point (X, Y), corresponding to an observed pair of values of X and Y, will fall within a given area X to X + d X and Y to Y + dy is constant for each observed pair of values. Correlation between variables coming from non-constant cause systems is termed spurious correlation. A correlation coefficient calculated from n observed pairs of values arising from a non-constant system of chance causes is a spurious cor- relation coefficient for which the sampling distribution function (75) does not apply. Such a coefficient is not subject to the usual interpretation as a measure of relationship discussed more in detail in the following section. If then we do not take great care to eliminate lack of constancy in the cause system giving rise to a set of n pairs of values of two variables, we may obtain a false conception of the relationship between these variables. This is very important as we shall now show by a simple illustration. Let us assume that we are using Rockwell hardness Y as a measure of tensile strength X for nickel silver sheet and that for this kind of material of given thickness the relationship is statistical in that the probability of an observed pair of values (X, Y) falling within the rectangle X to X + dx and Y to Y + d'Y is constant. It can easily be shown under these conditions that the correlation coefficient R between X and Y for two universes considered as one, or for the total number of observations is ab roaqi + -- 4. R = “) ov 2 (79) 9 &” •) /2 NG- + º (ºr *) 4. 4. where the difference between expected values of tensile strength and that between expected values of Rockwell hardness are a and b respectively. This equation shows that, under these assumptions, the spurious correlation R may be either greater or less than r. Fig. 79 gives a simple illustration. The two sets of dots rep- FLUCTUATIONS-CORRELATION COEFFICIENT 223 resent two sets of 12 observed pairs of values of tensile strength and hardness for nickel silver sheets of two thicknesses. The observed correlations of the two groups taken separately are r1 = O.59 and r2 = 0.54; considered together the correlation R is O.90. Lines of regression (I, 2, and 3) of hardness on tensile strength are shown for correlations ri, r2, and R respectively. Obviously R is a spurious coefficient. To use it as an indication of the statistical relationship between hardness and tensile ſ | O O 9 O 7 O 6 O8 O }* 9 – 1– | | - ſ | | O 5 O 6 O 7 O 8 O 9 O IOO ROC KW E L L H A RDN ESS B FIG. 79.-EFFECT OF SPURIOUS CoRRELATION. strength would obviously be misleading. Furthermore, as already stated, the distribution function (75) does not apply to this case. 7. Commonness of Causation Measured by r Let us assume that we have any two physical quantities XI and X2, and that variations in the first are produced by (1 + 3) independent causes, which we shall designate by U1, U2, . . . , Ul, /1, W2, . . . , P’s, 224 ECONOMIC CONTROL OF QUALITY whereas variations in the second are produced by (1 + m) inde- pendent causes U1, U2, . . . , Ul, W1, W2, . . . , Wm, so that l of the causes are common to the two variables. Let us consider first the following simple hypotheses con- cerning the causes: (1) Each cause produces a single effect, and this effect is unity for all of the causes. (2) The probability that any one of the causes produces its effect is constant and equal to p. (3) The resultant effect X1 or X2 is made up of the sum of the effects of the individual causes. These conditions, of course, lead to a binomial distribution of effects for each of the variables X1 and X2. Denote by z the contribution to X1 and X2 of the l common causes, by x the contribution of the V's, and by y that of the W’s. Then, for any particular operation of the cause systems, Xi = x + z, and A 2 = y + z. It may easily be shown that under these conditions / V(7I)(7.Tºm) TX1X, (8o) If s = m so that there are the same number (1 + m) of causes for each of the variables X1 and X2, then / r = 1-Hºmº (81) or the ratio of the number of common causes to the total number of causes in either variable. Let us consider now the more general case in which XI and FLUCTUATIONS-CORRELATION COEFFICIENT 225 A 2 are related to their respective causes by some unknown functional relationship. Thus X1 = F (U1, U2, • * - 5 Ul, V1, V2, • * * 3 Vs), and A 2 = F2(U1, U2, • * - 5 Ul, W1, W2, e ſº º , Wm). Now we shall think of the U's, P's, and W’s as symbols for groups of causes, each group producing a discontinuous dis- tribution of effects. Assuming that Xi and X2 can each be expanded in a Taylor's series, that terms beyond the first powers in the expansions can be neglected, that equal deviations in the U's, P's, and W’s produce deviations in Xi and X2 proportional to the cor- responding number of causes, and that the standard deviation of effects of one of the l-H s + m causes is the same as that of any other, it may be shown that ryaz, is again given by (80). 8. Simple Example Showing How correlation Coefficient Meas- ures Commonness of Causation Let us take eight chips experimentally identical—three red, three green, and two white. On each chip let us mark one side with zero and the other with unity. Now let these chips be tossed; let z be the sum of the numbers turned up on the two white ones, and x and y be the corresponding sums on the green and red ones, see Fig. 80. We may think of the turning up of a chip as a cause and the number on a chip as the effect of the cause. If we let Xi be the sum of the numbers on the three green and two white ones, and similarly let X2 be the corresponding sum on the three red and the same two white ones, then X1 and X2 may be thought of as two variables having two out of a total of five causes of variation common to both. - * * In general, the resultant effect of the first system is A1 = x + z, and that of the second system is X2 = y + 2. Inasmuch as each observed value of X1 and X2 has a common 226 ECONOMIC CONTROL OF QUALITY component, i.e., the effect of the two common causes, we would naturally expect a certain relationship between the values of Aï and X2 in successive operations of the two systems. Now the correlation coefficient rºw, between Xi and X2 is a measure of this relationship; and since these two systems of causes obey all the laws laid down for the general case in Paragraph 7, we have merely to set l = 2, m = 3, and we have Q. — = O.A.O.O. 2 + 3 4. r xix, * The observed correlation coefficient between Xi and X2 in one observed set of 500 pairs of values was o.422, giving a rather close check on the expected value O.4OO. (3) G) g, 9 @ FIG. 8o.—Two SystEMS HAVING Two CAUSEs IN COMMON. Fig. 81 gives the scatter diagram and lines of regression for these 5oo observed values of X1 and X2. - Practical Significance.—In Chapter IV attention was directed to the fact that the quality of material must be expressed in terms of physical characteristics which are, in general, not independent one of another because we do not know the independent ultimate quality characteristics or properties of a thing which make it what it is. In this con- nection the importance of considering not only the quality characteristics that are used in expressing quality but also the relationships between these was emphasized. We are now in a position to see more clearly the reason for so doing. FLUCTUATIONS-CORRELATION COEFFICIENT 227 Let us consider first the simplest kind of a case in which we have a product with two quality characteristics, Xi and 22. It is apparent that simply to specify that the two quality characteristics should be controlled about the averages XI and X2 with standard deviations on and go does not place the ; * 2 21 L^ |5 | O XI FIG. 81.-SCATTER DIAGRAM AND LINEs of REGRESSION For 500 OBSERVED VALUES. same requirement on the constancy of the inherent quality of the material as to state that these two properties shall be controlled in the way just indicated and in addition that the correlation between them shall be, let us say, homoscedastic and linear with a coefficient of correlation r. In the second form of specification in which the nature of the correlation between the two characteristics is specified, we have intro- 228 ECONOMIC CONTROL OF QUALITY duced certain restrictions on the quality of the material in that the two characteristics must have a common causal source of an amount consistent with the causal interpretation of r outlined above. Passing to the more complicated case where the quality of the material is specified in terms of m quality characteristics Å1, X2, . . . , Xi, . . . , Xm, there is a corresponding interpre- tation of the correlations which becomes of importance in the consideration of ways and means of specifying the quality of materials. It is beyond the scope of our present discussion to do more than call attention to some of the recent develop- ments in statistical theory indicating possible causal inter- pretations of certain inter-relations between all the pairs of the m variables measured in terms of the correlation coeffi- cients. For example, it has been known for several years that four variables may be thought of as due to one general causal factor plus four specific non-correlated factors when T12I34 = T131ſ24 = T14ſ.23. T. L. Kelley + has recently given an interesting discussion of the causal significance of inter-relationships of this character. Such work suggests an avenue of approach to the difficult problem of specifying quality in terms of those attributes which make it what it is. 9. Interpretation of r in General The correlation coefficient is often used as a measure of relationship when the condition of constancy of cause system is not satisfied. This is particularly true of time series. We shall consider one simple example in sufficient detail to show that the sampling distribution for such a coefficient of correlation is not necessarily the same as that discussed above, and that the above interpretation of r as a measure of commonness of caus- ation does not apply. * Crossroads in the Mind of Man, Chapter III, Stanford University Press, 1928. FLUCTUATIONS-CORRELATION COEFFICIENT 229 For this purpose we shall use an example given by Yule 1 in his presidential address before the Royal Statistical Society in November, 1925. The data are given in Fig. 82 and show the apparent relationship between the number of marriages in the Church of England and the decrease in the standard mortality rate over the same period. In this case the observed value of r is o.95. Needless to say this value of r may be thought of, as in Part II, as a summary presentation of the observed pairs of © STANDARDIZED MORTALITY PER THOUSAND PERSONS 78 O CHURCH OF ENGLAND MARRIAGES - PER THOUSAND MARRIAGES — 7.4 O — 7 OO – 66 O —l 62O | 8 Hºme * – ſ 1 | I | ſ –1– | 8 7 O |8 8 O | 8.9 O |9 OO | 9 |O * YEARS FIG. 82.-ExAMPLE of A NoNSENSE CoRRELATION. values. For example, an assumed line of regression would involve the statistic r. However, one is led to agree with Yule that there is no causal relationship between the two quantities shown in Fig. 82. Even if there were, the interpretation of r as a measure of commonness of causation in the sense of the previous two paragraphs would not hold. 1 “Nonsense Correlations between Time Series,” journal of Royal Statistical Society, Vol. LXXXIX, pp. I-64. CHAPTER XVI SAMPLING FLUCTUATIONS IN SIMPLE STATISTICs— GENERAL REMARKS 1. Two Phases of Distribution Theory Starting with the simple problem discussed in detail in Chapter XIII of Part IV, we have noted that there are two phases to the theory of distribution. A. Mathematical Distribution.—Given a discrete universe, it is theoretically possible to set down all of the ways in which one may draw therefrom a sample of size n just as we did in the case of the simple example discussed in Paragraph 5 of Chap- ter XIII. It is then possible to calculate any given statistic for any one of the N possible samples. The fundamental nature of the problem of determining the mathematical distribution of a given statistic may then be represented schematically as in Table 39. The first column of this table is supposed to stand for the N possible different samples. Obviously, 0; stands for the value of the ith statistic for the jth sample, the permuted column of values corresponding to any statistic 0, representing the distribution of possible values of that statistic. The problem of determining the mathematical distribution of a given statistic 0, is that of finding the distribution cor- responding to the N possible different samples. This part of the work, it should be noted, is purely formal or mathematical. From a logical viewpoint, this table has nothing to do with the universe in which we live until we have connected it up in some way or other with reality. . This we shall now do. B. Objective Distribution.—We may think of the equation of control (58) as defining the universe of possible values of X from which we may select all possible different sets of samples of 23O IFLUCTUATIONS-GENERAL REMARKS 23 I size n just as we have done above. Strictly speaking this is true only when (58) is discrete. If it is continuous we can, of course, calculate the relative frequency of occurrence of a sta- tistic within a given interval. • TABLE 39.-SchEMATIC of DISTRIBUTION of STATIstics Sample Statistic Statistic Statistic Statistic 61 62 €); €s I 611 6321 63;1 €s1 2 €12 622 6;2 6s2 j 01; O2; 0; 6s; N 63 1N €2AW €);N. €sN In a similar way, it should be possible to calculate mathe- matically the distribution of any statistic for a sample of given size drawn from such a universe of possible effects. Up to this stage, the procedure is, as before, purely mathematical. At this point we make use of the postulate of control previously discussed in which we assume that there exist constant systems of chance causes such that the observed distribution of effects approaches in the statistical sense the mathematical distribution function. It does not appear feasible to justify this assumption other than in an empirical way as we have tried to do in Parts I 232 ECONOMIC CONTROL OF QUALITY and III. The comments of Dodd again become relevant. Whether one chooses to call a mathematical distribution a prob- ability distribution or not would seem to be a matter of choice. The mathematical distribution itself, as any mathematical formula, merely becomes a tool in the hands of an experi- mentalist. It is essential therefore that in all that follows we carefully keep in mind the difference between the mathematical theory of distribution and the physical theory of distribution which it would appear must rest upon the assumption that the law of large numbers is a law of nature. 2. Importance of Distribution Theory Again let us return to the simple problem discussed in Chapter XIII of Part IV. I think that most people would agree that if they were to draw samples of four from an experimental universe such as described in that chapter, they would get as statistical limits the distributions shown in Fig. 57. I doubt, however, that many of us would have much of an idea how the distributions of standard deviation, mean deviation, skewness, and flatness, would look in such a case until we had gone through the mathematics of distribution as was done there. This is just the kind of situation that the engineer of control faces when he considers the problem of predicting what he may expect to get in the future based upon an assumed equation of control of the type (58). It is obvious that the reasonable way of predicting under such conditions, assuming the existence of the law of large numbers, is to make use of mathematical distribution theory such as that briefly discussed in the previous chapters. Our excursion into the field of mathematical distribution theory, however, has been a sort of pleasure trip in which we stopped to look at a few things which in the present state of our knowl- edge appear to be of immediate practical interest. It is well, therefore, that we take another look at this field for the purpose * Loc. cit. FLUCTUATIONS-GENERAL REMARKS 233 of getting a little better picture of the theory of mathematical distribution as a useful tool. 3. Mathematical Distribution Theory—Method of Attack Given the problem of determining the distribution function of a given statistic 0 for samples of size n drawn from a given universe, there are, in general, two methods of attack depending to a certain extent upon whether the universe is discrete or continuous. One of these methods consists in finding the exact mathematical distribution function through the use of integral calculus. The other, already illustrated in the previous chap- ters, consists in finding merely certain moments of the dis- tribution function. As a simple example of the exact method, let us consider the problem of determining the distribution function of x where x = x1 + x2. Furthermore, let us assume that values of x1 and x2 are normally distributed about zero. One method of finding the distribution of x is to fix on a definite range, say & to x + dx, and then to find the total probability of the occurrence of all possible combinations of xi and x2 which will yield a value of x within the prescribed interval. The distribution function of x thus obtained will be the one desired. # The probability that xi lies within the interval x1 to x1 + dx1 at the same time that x2 lies within the interval x2 to x2 + dx2 is given by the expression 20-22 dx2. I __** I 20-12 p = — 6 l dx1 —— e o, V2+ gov/or Having fixed on a value of x1, and x being initially fixed, the value of x2 is of necessity x − x1. Hence we may write (a — ac1)? 20-22 dx2, __*. 20-12 dx1 I I p *e gºvº" gºvº." which for a proper choice of dx1 and dx2 is the probability, to within infinitesimals of higher order, of pairs xi and x2 which yield a value of x within the interval x to x + dx. When x; 234 ECONOMIC CONTROL OF QUALITY is allowed to take all values between – CO and + CO and dx, is made to approach zero, we see that the sum of terms like p approaches the total probability that x1 + x2 lies within the prescribed interval. Hence the total probability P(x)dx that the sum x1 + x2 lies within the interval x to x + dx is by definition * = +& 1 #####) I ——º P(x)dx = e 2\oti” ' a 2* /dx1dz = −7–e 2(q13-Ho-22) dx, gig 12t, ac1 = – OO V (or *-Hors”) 2T - since dx2 → dx as dx1 – O. Thus we are led to the well-known result that the dis- tribution of a sum of two variables, each of which is normally distributed, is normal with a variance equal to the sum of the variances of the given normal distributions. This method may be extended to a linear sum of any number of variables. Whit- taker and Robinson 1 show how through the use of Fourier's Integral Theorem it is possible to obtain the distribution function of a linear function of deviations in a more elegant Iſla 111] C1". If one can obtain in some such way an exact distribution function, it is theoretically possible to obtain the integral of this function over any given range, either exactly or by quadrature methods. As an illustration of the modern mathematical tools avail- able for finding the moments of the distribution of a statistic, let us consider one * method of finding the moments of the mean of a sample of n drawn from any discrete universe. Assume that the universe is defined by s different values X1, X2, . . . , Xi, . . . , Xs, the relative frequencies of which are P1 > P2, . . . ; Pi, . . . ; Ps respectively. * Loc. cit., p. 168. See also the very interesting papers, “Application of Thiele's Semi-Invariants to the Sampling Problem,” C. C. Craig, Metrom, Vol. 7, No. 4, Dec. 31, 1928, pp. 51–107, and “Sampling When the Parent Population is of Pearson's Type III,” C. C. Craig, Biometrika, Vol. XXI, Parts I to 4, Dec. 1929, pp. 287–293. * W. Romanovsky, loc. cit. : FLUCTUATIONS-GENERAL REMARKS 235 Let the frequency of X1, X2, . . . , Xs in a sample of m independent trials be Ji, f2, tº º ſº ſs, where, of course, some of the fºs may be zero. Then our problem is to investigate the distribution of 2 fix; M = 1 = 1 7? in possible sets of n trials. Denote the average and higher moments of the distribution of the universe by X, H2, P3, . . . , Hi, . . . . and of the distribution of the mean by Xz, H.2x, P3x, . . . , Pix, . . . . where, in each case, the moments are measured about the mean. What we shall do is to express the pix's in terms of the p’s, which for a given universe are known constants. Since the mean value of X in an indefinitely large number of samples is X, we may replace Xz by X in finding expressions for the higher moments of X. Romanovsky has developed an elegant and simple way of obtaining these moments as follows: Consider the function of t defined by S t - 77, –(X. — X U = | pºº | = 1 7, - *, *, * t gº- t ==, 75 - |nº + pº + . . . pººl & By the multinomial theorem we have 77 | #(X, -X) f #(x,-X) f *(xs—X)T ſ U = X ! {..., | |ple 1 | "|pº 2 | ‘. . . |pº S | S fi! fe' . . . fs! 71. ->717. Tºp S 1ſt p3% & C & ps” eT #2 J;(Xt – X) (a) 236 ECONOMIC CONTROL OF QUALITY the summation being extended to all fºs whose sum is n. Now the factor n! Ji --/ f pr” pº . . . ps” S : fi! fe! . . . f. is the probability of getting in n trials fi Xi's, f2 X2's, ..., fs Xs's. Or, in other words, this factor is the probability of getting an X constructed in a particular way. Also for a par- ticular construction of X, the exponent of e is s f;(X; – X) = t|X – X). ; Making use of this fact, we have, on differentiating U r times with respect to t and then setting t = o, d' U 77 *-* mº r = X →pſ, p3% . . . pºſs(X = X)". /, (. |- fi! fel . . FP p2 ps's ( ) (b) This is true since each differentiation of a particular term in the sum (a) merely multiplies this term by (X. — X). By virtue of the way in which the right-hand side of (b) has been built up, it is clear that this sum is precisely the rth moment ºr of the mean about its mean value. The method of obtaining any moment of X is then a very simple one. To facilitate the work, set t S - (X; — X) w = 2 pier e {= 1 Then U = w”. Then the zeroth moment of X is (U), –0 = (p + p2 +...+ ps)” = 1. dU I , (x,-X) - Fºlx = ( )." |-- * 2 pie” (Xi — x, dt 71 i = 1 \ t = 0 2 pi(X; – X) = o. 4 = 1 FLUCTUATIONS-GENERAL REMARKS 237 22 P2; F (#). n — 1 s t S t 2 70) :-(X; –X) ==, a 71 - I -, :-(X; –X) * -: X pie" " (X;–X)2+–w” ($ pie" (x-x) | 77 4 = 1 77 1 = 1 t = 0 2 77 I -º * = −X pi(X; – X)* = P2. 774 = 1 In an exactly similar way, it can be shown that |13 — I |13+ = m2 and H4+ = **** |4 73' Denoting by Bix and Bºx the skewness and flatness re- spectively of the distribution of the averages, we have by definition |4× 3(n − 1), a , 4) nº P2 = 3 (****); -º where 31 and B2 are the skewness and flatness respectively of the universe. Of course it is possible, by the above method, to go much further than this and to find expressions for Bix of any desired order i. However, our present purpose is merely to illustrate one of the modern methods of finding the moments of the distribution of a statistic. A. Some Numerical Results.--To fix in our minds the sig- nificance of the above results, let us use them to calculate the statistics of the universe of averages, column 3, Table 25. We get + 3, Xx = X = 2.5oooooooo OT I. I 180339887 ºx-vº –––– assºciºs _ Bi – o – By = , = i-e 32 – 3 I.64 – 3 B2+ = + 3 = —” + 3 = 2.660000000. 7? 4. 238 ECONOMIC CONTROL OF QUALITY These results obtainable through the use of the first four moments of the universe without going through the details of getting the distribution in column 3, Table 25, are the same to the number of places shown as the results obtained directly from the distribution in column 3. In this same connection, it will be interesting to compare the values of mean variance o’ and o, obtained from (68) for the distribution of variance in samples of four drawn from the experimental universe of Chapter XIII with that calculated directly from column 4 of Table 25. From (68) we get a-sº 77 — I Oſ = = #(I.25) = 0.9375 o” In — I •-ºw iſ(n − 1)2 -n +3 I #ºvisa.g.) — 4 Is = O.3125 Voy;(3.92) = ( O .3125)(1.71464282O) = O.5.3582.58812. These results check to the number of places shown those obtained directly. . B. Comparison of the Two Methods.--Whenever the exact distribution of a statistic can be found by integration, we have more information than can be provided by the knowledge of any number of moments of the distribution of the same statistic. In other words, when the distribution of a statistic 0, is known as a function of 0s, the probability that the statistic will take on values lying between any given limits can be found either by direct integration or by quadrature methods. t On the other hand, if only the moments of the distribution of 0; are known, we can never be quite sure what the form of the distribution is. For example, Bix → 0 and 32x → 3 as m becomes large but even if we actually had 31z = 0 and FLUCTUATIONS-GENERAL REMARKS 239 P2x = 3, we could not infer that the distribution of X was normal; for obviously the distribution defined by A : — I O -H I f: I 4. I has 1 B1 = O, 32 = 3, 33 = O, which are identical with the first three betas for a normal universe, although this distribution is far from normal. As a matter of fact, it would be necessary in this instance to go as far as the sixth moment before we would discover any difference between it and the normal law function, so far as moments are concerned. Suppose then, that the universe we started with had a form such that the distribution of means actually was identical with the simple one given above, but we had calculated merely the moments of this distribution by the above method. We would find that the first five moments were identical with those of the normal law, and we might perhaps be tempted to infer that the distribution of means was normal, although, as we have seen, such an inference would in fact be far from the truth. 4. Mathematical Distribution Theory—Important Results Looking back over the work in the previous chapters, we see that distribution theory provides us, in certain instances, with distribution functions of a given statistic 0 of the form fo (€), m) such that the integral of this function for a given range gives us the probability of occurrence of a value of 0 within that range. Illustrations of this type are the distribution func- tions of average, standard deviation, and correlation coefficient. Similarly, we may have distribution functions of a ratio z between two statistics 0; and 0; such that fe (z, m)dz represents the probability of occurrence of a value of z within the interval z to z + dz. This kind of function has been illustrated by the distribution of the ratio of the error of the average to the observed standard deviation. - * Of course, uncorrected moments are used here. 24O ECONOMIC CONTROL OF QUALITY The other important form of distribution to be noted is that of the distribution of two statistics 0; and 0}, such that fei, e, (0, 0), n)deide; represents the probability of the occur- rence of values of 0; and 0; within the rectangle 0; to 0; + de; and 0; to 0; + dej. It is important to note also that the distribution function of a given statistic depends upon the functional form of the universe from which the sample is drawn, and that, in general, the average or expected value 6 in samples of size n is not the same as the value of this same statistic for the universe. 5. Mathematical Distribution Theory—Present Status Any summary of the status of distribution theory today will likely be out of date before the ink is dry. Here, as in the field of modern physics, progress is so rapid and along so many different lines that even those actively engaged in extending the theory find it difficult to keep abreast of all that is being done. A few brief remarks, however, may be of service to the engineer who cares to become acquainted with some of the important recent contributions. The exact distribution of means of samples from normal populations dates back at least to the time of Gauss, whereas the exact distribution of variance and standard deviation were found in 1915 by R. A. Fisher. In the same article, Fisher gives the exact distribution of the correlation coefficient in samples from an indefinitely large normal population. The same author has since given the exact distributions of the regression coefficient,” partial correlation coefficient,” and multiple correlation coefficient,” assuming a normal universe. * Loc. cit. * “The Goodness of Fit of Regression. Formulae and the Distribution of Regres- sion Coefficients,” journal of the Royal Statistical Society, Vol. LXXXV, Part IV, 1922, pp. 597–612. * & “The Distribution of the Partial Correlation Coefficient,” Metrom, Vol. III, No. 3-4, 1924, pp. 329-332. - tº 4 “The General Sampling Distribution of the Multiple Correlation Coefficient,” Proceedings of the Royal Society, A, Vol. 121, 1928, pp. 654–673. FLUCTUATIONS.–GENERAL REMARKS 24. I Pearson, Romanovsky,” and Wishart 8 have also studied these same distributions. In 1925, Hotelling 4 gave the distribution of the square of the correlation ratio subject to the conditions that the variates are not correlated, that the population is indefinitely large, and that the variates are normally distributed. Exact distributions of means for certain of the Pearson type curves other than the normal have been given by Church,” Irwin,” and Craig." Important contributions to the theory of distribution through the use of moments have been made by Pearson,8 Tchouproff,” Church, 10 Fisher," and Wishart.12 The list of references given in the last few paragraphs is by no means complete. Instead, it is selective and is intended to indicate the rapid development” that is going on in this field. * “Researches on the Mode of Distribution of the Constants of Samples Taken at Random from a Bivariate Normal Population,” Proceedings of the Royal Society, A, Vol. II 2, 1926, pp. I-I4. * “On the Distribution of the Regression Coefficient in Samples from a Normal Population,” Bulletin de l'Academie des Sciences de l’U. S. S. R., 1926, pp. 645–648. * “The Generalized Product Moment Distribution in Samples from a Normal Multivariate Population,” Biometrika, Vol. XXA, 1928, pp. 32-52. * “The Distribution of Correlation Ratios Calculated from Random Data,” Proceedings of the National Academy of Science, Vol. II, No. 10, 1925, pp. 657–662. Tables of the integral of the function given by Hotelſing have recently been given by T. L. Woo, Biometrika, Vol. XXI, 1929, pp. I-66. & * Loc. cit. * Loc. cit. * Loc. cit. * “On the Probable Errors of Frequency Constants,” Biometrika, Vol. II, 1903, pp. 273–281 and Vol. IX, 1913, pp. I–Io. “Further Contributions to the Theory of Small Samples,” Biometrika, Vol. XVII, 1925, pp. 176–179. * “On the Mathematical Expectation of the Moments of Frequency Distributions,” Biometrika, Vol. XII, pp. 185–2Io. 10 Loc. cit. * “Moments and Product Moments of Sampling Distributions,” Proceedings of London Mathematical Society, Vol. 30, 1929, pp. 199—238. * “A Problem in Combinatorial Analysis Giving the Distribution of Certain Moment Statistics,” Proceedings of London Mathematical Society, Vol. 28, 1929, pp. 309-321; Proceedings of Royal Society of Edinburgh, Vol. XLIX, 1929, pp. 78–90. 13 Rider, P. R., “A Survey of the Theory of Small Samples,” Annals of Mathematics, Vol. 31, No. 4, pp. 577–628, October, 1930. An excellent bibliography is appended to this article. - 242 ECONOMIC CONTROL OF QUALITY 6. Importance of Distribution Theory—Further Comments We are now in a position to consider a little more critically than has been done the significance of some of the recent work on the mathematical theory of distribution as it bears upon the theory of control. Assuming that an engineer is going to make use of statistical theory in helping him to do what he wants to do, it is but natural that he must sooner or later express what he wants to do in terms of some distribution function of a given quality X which he is to take as standard; that is to say, he must specify as a standard of what he wants to do some distribution function typified by the equation (58) of control dy = f(X, \l, M2, • • • 3 \i, • * * x \m)d X. (58) Assuming the existence of a constant system of causes having as its objective statistical limit this equation of control, it is necessary to set up limits on one or more different statistics of samples of size n. In many cases the control engineer may also desire to set up limits upon the allowable variation in X itself and in the fraction of the observed values of X which lie beyond some particular pre-assigned value. Let us consider first the problem so often met in practice of setting a limit X + to on the variable X such that the objective probability that an observed value of X will fall between this limit and + o is p, where X and g are the average and standard deviation of the universe (58) of control. To do this it is necessary to find the value of t from the equation p-ſ f(X, \1, A2, . . . , \, . . . , \m)dX. X--tor Expressed in this general way, the formal problem of establishing the value of t for a given value p appears to be quite simple. When, however, we consider the theory of frequency distributions, we find that this problem is not so simple as it appears when the value t corresponding to the * The same discussion obviously applies to the negative tail. FLUCTUATIONS-GENERAL REMARKS 243 W chosen value of p is greater than three, at least for most of the standard functions involving not more than four parameters. In fact, certain of these frequency functions may be found to have negative frequencies for values of X outside of a sym- metrical range something like X == 30. This is true of the second approximation (23) for certain values of k. This fact is significant because it shows that when an en- gineer attempts to set some particular limit X + to such that the objective probability of an observed value falling beyond this limit shall be p (where p is perhaps of the order of O.OOI or less), even the solution of the formal problem may be dif- ficult. Of course, he might appeal to experience, observe the value of X a large number of times under what he assumes to be a controlled condition, and in this way try to approach as a statistical limit the exact objective frequency distribution to which any of the customary theoretical distributions would simply be an approximation. One does not need to go far to see, however, that such a procedure is not, in general, feasible if for no other reason than because it would require a large number of trials in order to justify the establishment of such a limit in anything like a satisfactory manner—it being true, of course, that one could never be sure of results obtained in this way. Passing to the more general problem of establishing sampling limits on any statistic 0 in samples of n drawn from the universe (58), it is of practical importance to note that with but few exceptions the exact frequency distribution function of such a statistic is unknown even when the universe (58) is continuous. When the universe is not continuous—it never is in practice— we must be satisfied with a knowledge of the moments of the distribution function of the statistic expressed in terms of the moments of the universe (58). For example, in the previous paragraph we have spoken briefly of a method of expressing any moment of the average of a sample of m in terms of the * This point is emphasized in the writings of Edgeworth and is touched upon in various places in Bowley's summary of “Edgeworth's Contributions to Mathematical Statistics,” published by the Royal Statistical Society, 1928. 244 ECON $1IC CONTROL OF QUALITY moments of the universe. We have seen that to be able to specify the moments of the distribution of averages in samples of size n beyond the fourth moment requires a knowledge of moments of the universe higher than the fourth. This is significant from an engineering viewpoint because it shows that if we are going to try to establish sampling limits even on such a simple statistic as the arithmetic mean with a comparatively high degree of precision in respect to the objec- tive probability associated with the tail of this distribution, we must certainly be in a position to specify the moments of the accepted standard (58) of control beyond the fourth— something that it is obviously very difficult to do. What we have said in respect to the establishment of sampling limits on the average is all the more true when we attempt to establish limits on other statistics such, for example, as the variance. This follows from the work of Tchouproff and Church 1 showing that the equation relating the fourth moment of the distribution of variance in samples of m to the moments of the universe involves the eighth moment of the universe— to obtain which is certainly not feasible. There is another reason why it is difficult to attain great precision in the estimate of the probability associated with an asymmetrical range as we shall now see. Several times in the previous section we pointed out the significance of the fact that sampling from a discrete universe may give results radically different from those obtained when sampling in a similar way from a continuous universe. This is particularly important because we seldom see fit to classify measurements into more than ten to twenty cells, and it does not appear feasible to introduce moment corrections which allow us to go from the discrete to the continuous case with a known degree of precision. We have considered at some length the approach of the distribution of the average to normality with increase in sample size irrespective of the parent population (58) as characterized by the first two 3's of this distribution. The comparatively * Loc. cit. FLUCTUATIONS-GENERAL REMARKS 245 recent work of Holzinger and Church 1 shows that the dis- tribution function of averages from a U-shaped universe is not even unimodal for small samples and appears to approach uni- modality and symmetry only for samples of the order of fifty or more. In fact, they conclude that the distribution function of averages of less than fifty cannot be satisfactorily repre- sented by a continuous curve. In such a case we must rely upon the application of the Tchebycheff inequality as we have done. This kind of evidence indicates the nature of the difficulites involved in trying to establish asymmetrical limits on the sampling fluctuations of any statistic and it helps us appreciate the significance of the powerful Tchebycheff inequality in the establishment of symmetrical limits with at least a known upper bound to the error that we may make in the estimate of the probability associated with these limits provided only that we know the two simple statistics X and g of the universe. The fact that we do not, in general, know the exact distri- bution function of measures of correlation other than the correlation coefficient in terms of the specified correlation in the universe precludes the use of these statistics in that we cannot establish their control limits. For this reason, we have not discussed the mathematical distribution theory for these statistics. * “On the Means of Samples from a U-shaped Population,” Biometrika, Vol. XX-A, pp. 361-388. PART V Statistical Basis for Specification of Standard Quality The Establishment of Economic Toler- ances and Standards of Quality Involves the Use of Three Simple Statistics CHAPTER XVII DESIGN LIMITS ON VARIABILITY I. Tolerances Since all pieces of a given kind of product cannot be made identical, it is customary practice to establish allowable or tolerance ranges of variability for each of the measured quality characteristics. For example, if a shaft is to work in a bearing, we must allow for a certain clearance. In such a case the specifications usually require that a shaft have a diameter not less than some minimum nor more than some maximum value, and that the diameter of the bearing must not be less than some minimum nor more than some maximum value. An illustration taken from practice is: Diameter of Shaft Maximum limit O.75oo inch inimum limit O.7496 inch Diameter of Beari Maximum limit o.7507 inch 12 Iſlet C1. Of Jij Ca.1111 - * | Minimum limit O.7502 inch Assuming that the diameters can be measured accurately to the fourth decimal place, we see that the minimum and max- imum clearances are O.OOO2 inch and O.OOII inch respectively. The tolerance range for a given quality X is defined as the range between the maximum and minimum tolerance limits specified for this quality, Fig. 83. Sometimes these limits are called tolerances. Perhaps more often, however, these limits are given in the form X = X – AX and X2 = X + AX, and in this case AX is called a tolerance. To avoid any misunder- standing that might arise because of the apparent lack of uni- formity in the definition of tolerance we shall use the terms 249 25O ECONOMIC CONTROL OF QUALITY tolerance range and tolerance limits wherever necessary to make the meaning clear. 2. Tolerances Where zoo Per Cent Inspection Cannot Be Made Where the quality X can be inspected on every piece of apparatus by some go-no-go gauge, it is easy to separate product into two classes—that which does and that which does not fall within the tolerance range. If, however, we are testing for *–TOLERANCE RANGE-- CUALITY x M : NIMUM LIMIT MAXIMUM LIMIT XI X2. FIG. 83.—RELATIONSHIP BETween ToleRANCE RANGE AND Toler ANCE LIMITs. some quality such as tensile strength, it is obviously not possible to make Ioo per cent inspection to see that the tolerance is met. In this case our information about a lot of N pieces of product must be obtained from tests made on a sample of n pieces. The usual practice is to establish tolerance limits for the quality X and also tolerance limits for the fraction defective in the lot, or, in other words, the fraction of the total number of pieces of product in the lot having a quality X lying outside the tolerance limits for this quality, Fig. 84. Usually zero is taken as the lower limit for the fraction defec- tive in the lot. Since our information must depend upon a sample, it is also necessary to establish tolerance limits on the fraction defective found in the sample, the lower limit being zero. These two kinds may be thought of as lot and sample tolerances, and they are related one to the other through a risk associated with the given sampling plan as will be indi- cated in Part VII, thus making it necessary for the sample tolerance to depend upon the number n in the sample. 3. Importance of Control in Setting Economic Tolerance In general, a tolerance range on a quality X should be as small as possible. If it is too small, however, the rejections DESIGN LIMITS OF WARIABILITY 25 I will be excessive. In other words, the design engineer tries to balance the rate of increase in value of reducing a tolerance range against the rate of increase of cost of such a procedure because of increased reiections. TO LERANCE RANGE —- OUALITY X M lN | MUM Ll M1T MAXIMUM LIM lT XI X2 TOLERANCE RANGE _fRACTION DEFECTIVE - | | LOWER LIMIT UPPER LIMIT Pl P2 FIG. 84.—Two SETs of Toler ANCE LIMIts NEcessARY WHEN Ioo PER CENT INSPEc- TION CANNOT BE MADE. From what has previously been said, it is obvious that, if a design engineer knows that the quality X of a material or == CONTROLLED DISTRIBUTION du =f (X, A, A2, ..., \m).dx PROBABILITY OF REJECTION P I MINIMUM LIM lT MAXIMUM L |M|.T X X2 QUALITY X FIG. 85.—ToleFANCE on FRACTION DEFECTIVE FOR ContRolleD QUALITY. piece-part entering into his design is statistically controlled in accord with some probability distribution such as illustrated by the smooth curve, Fig. 85, then he knows the expected number 252 ECONOMIC CONTROL OF QUALITY pV of rejections that will occur in the production of a number N of these piece-parts for a given set of limits. Only under these conditions of control is it a comparatively simple process to find an economic tolerance range. Hence, to set an economic tolerance range it is necessary that the qualities of materials and piece-parts be controlled. 4. Tolerances where Zoo Per Cent Inspection Cannot be Made— Importance of Control When Ioo per cent inspection cannot be made, we never know that the tolerance on a quality X is being met, even though it is met in the sample. Later we shall show that any inference about what exists in the remainder of the lot from what was found in the sample depends entirely upon what we assume about the lot before the sample was taken, and that the significance of such an assumption depends upon whether or not we assume that the product is controlled. If, however, instead of trying to use the double tolerance described in Para- graph 2 above, the design engineer makes use of raw materials and piece-parts previously shown to be statistically controlled with accepted expected values and standard deviations, he need only specify that the qualities of all materials and piece- parts going into his design be controlled with accepted average values and standard deviations. Hence we see that it is very desirable to know that the quality of a product is controlled when it cannot be given Zoo per cent inspection. 5. Tolerances for Quality of Finished Product in Terms of Tolerances of Piece-parts Let us consider a very simple problem. Assume that an engineer wishes to design a circuit containing m different pieces of standard apparatus, such as relays, transformers, and so on. Suppose that he wishes to set a tolerance range on the overall resistance in the circuit and that the tolerance limits on the resistances of these m different pieces of apparatus are respectively R11 and R12; R21 and R22; . . . ; Rii and Rīz; DESIGN LIMITS ON VARIABILITY 253 . . . ; Rini and Rim2. What shall the engineer use as the tolerance range for the overall resistance? The answer to the question is obviously R12 + R22 + . . . --Riº + . . . -- Rºma – R11 — R21 — . . . – R$1 — . . . – Rimi, R11 R12 Ril Riz | I –-----sº Rm I Rm 2 FIG. 86.--ToleRANCE RANGES ON OBSERVED DISTRIBUTIONS. if we hold to the definition of a tolerance as the range between the maximum and minimum possible values of the quality. Before accepting this answer, however, let us consider the problem further. Oftentimes we find that the previously observed distri- butions in the m different resistances are somewhat as indicated by the smooth distribution curves in Fig. 86. We see that in some instances the tolerances are such as to cause rejections 254 ECONOMIC CONTROL OF QUALITY in both the upper and lower ranges of the resistance as in the case of the quality R1. At other times the condition may be such as indicated for the resistances R; and Rim. When the number m of different resistances is large, it is obvious that the number of times that we may expect to get a combination of m resistances chosen at random (one each from the m different kinds of resistances) that will add up to either the maximum or minimum limit is very small indeed. The question arises, therefore, as to whether or not it is eco- nomical to allow in design for an over-all tolerance range equal to the range between the possible maximum and minimum resistances that may occur. Let us consider this problem upon the basis of the assump- tion that each of the m kinds of apparatus is manufactured under conditions such that the resistances are controlled about average values R1, R2, tº & º , Ri, tº tº , Rm, with standard deviations O 1, O2, • * - 5 O’i, • * - 5 O m. For the sake of simplicity, let us assume that the resistances are normally controlled, or, in other words, that the distribution function for each resistance is normal. From what has previ- ously been said, it would be quite reasonable to adopt the tolerance limits Ri + 30: on the ith resistance. If we adopted such a set of m toler- ance limits, and followed the practice previously described of taking the difference between the sums of the maximum and minimum possible resistance as the tolerance for the sum of the resistances, we would have a tolerance range such as that schematically indicated in Fig. 87-a. Let us now consider why such a tolerance range may not be economical. As may be shown, the expected distribution of the sum of m resistances chosen from the m different kinds of resistances DESIGN LIMITS ON VARIABILITY 255 as indicated above would be normal with an expected or mean value equal to XR, and a standard deviation g = Voº + c2% -- . . . 4 gº +...+ gm”. Suppose that we assume, as a simple case, that each of the m standard deviations is equal to, let us say, G1. It is obvious that the standard deviation of the sum is g = V/mol. Starting with these simple assumptions, we may easily draw the frequency distribution function of the resultant resistance TOLERANCE RANGE g- * I MINIMUM LIMIT ŽR; MAXIMUM LIMIT £R; – 3×oi § zRi -- 3:0; O U A L | T Y X (a.) 6. º } | l i I Mi NlMUM LIMIT sº- XIR; gº MAXIMUM LIMIT zR; – 3xo; ŽRi –30 zR + 30 XR1 + 2×6; OUALITY X (b) FIG. 87.-Illustrating PROPER WAY to SET LIMIts. for any special case. Fig. 87-9 shows such a distribution cor- responding to nine component resistances in the circuit or to the case m = 9. For purposes of comparison, the additive tol- erance previously described is also shown for m = 9. We see at once that the practice of adding tolerance limits may be uneconomical because the chance is relatively very small that a resultant resistance would ever lie outside the limits XR3 = 30. Having considered this simple illustration, we are in a 256 ECONOMIC CONTROL OF QUALITY position to discuss the general problem of setting overall toler- ance limits in terms of tolerance limits of piece-parts. 6. The General Problem of Setting Tolerances on Controlled Product As a perfectly general case, let us assume that the quality X upon which we wish to set tolerance limits depends upon the qualities X1, X2, . . . , Xi, . . . , Xm of m different piece-parts or kinds of raw material. Interpreted from the viewpoint of control, this means that we wish to set two limits on X which will include a certain fraction P of the product in the long run. We shall show how this can be done upon the basis of the assumption that each of the m component qualities are con- trolled about expected values X1, X2, . . . , Xi, . . . , Xm, with standard deviations 01, O2, . . . , O7, . . . , 0 m, subject to certain limitations. Let us assume that we may write X = F(X1, X2, . . . , Xi, . . . , Xm). Furthermore, let us assume that the quality X may be expanded in a Taylor's series so that to a first order of approximation we may write X = F(X1,X2, & e G .xi, tº ſº gº , Xm) + a 1 x1 + q2 x2 + ... + ai &i +...+am Wom, where xi = Xij — Xi (27 = (...) 2. ox: Xi’ it being understood that Xij in this case is any one of the possible values of the quality X. and DESIGN LIMITS ON VARIABILITY 257 It may easily be shown under these conditions that the expected value X and the standard deviation ox of the distri- bution of quality X of product assembled at random are given by the following equations: X = F(X1, X2, . . . , Xi, . . . , Xm) (82) ox = Val Ol + (22 Og + . . . -- (Zº Cº. + . . . -- dºm O ºn No matter what the nature of the distribution functions fi(X), f2(X2), . . . , fi(X), . . . , fm(Xm), Equations (82) enable us to write down the expected resultant quality X and the standard deviation ox of this quality about the expected value subject to the limitations already considered. Making appli- cation of Tchebycheff's theorem, we can say that the probability Pl, that the resultant quality will lie within the interval X + /O x satisfies the inequality h For example, one can say with certainty that in the long run more than (I – #) of the product will have a quality X lying within the limits X + 30x. In the simple case con- sidered in the previous paragraph, where it is assumed that the distribution function for each of the m quality character- istics is normal, we see that the probability Pagx is equal to O.9973. It is exceedingly important from our present viewpoint to note that so long as we know nothing about the distribution function of each of the m quality characteristics, we can only make use of (82) in connection with Tchebycheff's theorem. The more we know about these functions, the more accurately we can establish the probability Play. If the distribution functions of the m quality characteristics are alike in respect to their second, third and fourth moments, it may easily be shown that the skewness kx and the flatness 2.58 ECONOMIC CONTROL OF QUALITY Box of the distribution of quality X are given by the following equations: k kx = y: y (83) B2 – 3 £2x = 777 + 3 where k and 32 are the skewness and flatness of the distribution of any one of the m quality characteristics. Thus we see that under these conditions the skewness and flatness of the resultant distribution will be approximately normal, even though the individual qualities are distributed in a way such that their skewness and flatness are appreciably different from zero and three respectively. In the more general case, where the distribution functions for the m different quality characteristics are not all alike, it may also be shown that the distribution of the resultant effect A will approach normality as m – Co. * These results are of great importance as indicating the magnitude of the advantages that accrue from specifying the distribution of any one of the m qualities other than by saying that they shall be controlled about known average values with known standard deviations. Even though the distribution func- tion of X approaches normality as m increases, it is usually true in a specific case that it would be very difficult to characterize the functions of the m component qualities with such precision as to enable the determination of the probability Pirx to within, let us say, I per cent. In other words, it appears that, from a design viewpoint, there are many advantages to be gained by specifying that the quality of raw materials and piece-parts shall be controlled about known averages and with known standard deviations, although it appears that the advantages to be gained by trying to specify the functional forms of the controlled distributions and more than these two parameters of the distributions are offset by certain disadvantages. * Compare with (63). * See Appendix I. DESIGN LIMITS ON VARIABILITY 259 Hence from a design viewpoint we conclude that the specift- cation of control should include the specification of expected value X; and standard deviation gº of any quality characteristic X. We are now in a place to consider the more general problem of designing a complicated piece of apparatus so that the quality of the product will have minimum variability. 7. Design for Minimum Variability Again let us assume that the resultant quality X is a function F of the qualities X1, X2, . . . , Xi, . . . , Xºn, or that X = F(X1, X2, . . . , Xi, . . . , Xm), and that we wish to make a product having an expected quality X with minimum standard deviation ox. We shall assume that the m quality characteristics are controlled about expected values X1, X2,..., Xi, ..., Xn with standard deviations G1, G2, . . . , Ji, . . . , Gm. Making the same kind of assumptions as in Paragraph 6 about the expansibility of the quality X by means of Taylor's theorem, we may write 9) 2 2 -2 rº- . 2 Ox = Valºgº + as O2* + . . . -- alº gº + . . . -- amº Om”, X = F(Xi, X2, . g .., Xi, . iº ..., Xm), where, as in the preceding paragraph, as is a function of the m mean values. Our problem now is one of minimizing ox subject to the restriction imposed by the last equation. This will be recognized as a problem in the theory of maxima and minima. Expressed in terms of the Lagrange indeterminate multiplier X it involves the solution of the following m + 1 equations: bºx") – 2X 2F = O OX; oXi tº (84) X = F(X1,X2,..., Xi, ..., Xm) 26o ECONOMIC CONTROL OF QUALITY It may not be feasible to solve this set of m + 1 equations for the unknowns X1, X2, . . . , Xm and X because of their com- S P E C | ES A | 1– | | * I l l | L | I | L– O 2Ts |4 :----|--|--|--|-> L ] X L2 S P E C | E S B l | | | | ſ ſ | | | I O | 2 || 3 4. 5 6 8 9 |O || | | 2 || 3 |-l X |-2 S P E C | E S C t L 1 _j | | I | | | I I O | 2 3 4 5 6T 7 8 9 |O || || | 2 || 3 |- X |-2 S P E C : E S D 1–1 l I | l | f | | O l 2 3| 4 5 6 7 8 9 io || || | 2 |3 |-| . X |-2 FIG. 88–TYPICAL RELATION BEtween Expected VALUES AND STANDARD DEVIATIONs. plexity. Again it is possible that the solution may contain zero, infinite, or imaginary values of the X's. Such solutions are DESIGN LIMITS ON VARIABILITY 261 obviously of no practical significance. We see that, in addition to knowing that the qualities of piece-parts and raw materials are controlled, it is essential only to know the averages and standard deviations of the distribution functions of the component qualities. In practice limitations are often imposed upon the possible magnitudes of the expected values of the m quality charac- teristics other than those already considered. For example, one or more of these quality characteristics may be properties of material such as density, tensile strength, resistance, coeffi- cient of expansion, and so on. Obviously, in choosing the expected values in such a case, we are limited to the expected values of the available raw materials, unless we develop some alloy having the desired expected value. Also, in practice, the choice of an expected value of a quality cannot usually be made independent of the choice of its stand- ard deviation. Thus in the case of a physical property of a material there is, in general, some relationship between the expected value of the property or quality and its standard deviation. This fact is illustrated in Fig. 88 showing the relative expected values and standard deviations of modulus of rupture of four kinds of telephone poles. We see that, broadly speaking, the standard deviation increases with increase in expected modulus of rupture. CHAPTER XVIII SPECIFICATION OF STANDARD QUALITY I. Standard Quality We often think of a standard of quality as being either a specified value X's or a value X lying within some specified tolerance limits X1 and X2. If, however, we try to produce all units of a given kind of product with a standard quality Xs, the best we can hope to do, as we have seen in Parts I and III, Xs H | | X X O UA Li T Y X XI X2 O UAL IT Y X (a) (b) FIG. 89.-CoMMON Concepts of STANDARD QUALITY. is to make a product whose quality X satisfies the equation of control - dy = f(X, M, M2, • * * x \i, • * - 5 \m'), (58) with an expected value X somewhere near the specified standard or ideal value Xs as indicated schematically in Fig. 89-a. Sim- ilarly, if one attempts to make a product all units of which will have a quality within the tolerance range XI to X2, he will usually end up, after having done everything feasible to attain constancy, by making a product whose quality will be dis- tributed as indicated schematically in Fig. 89-6. It is possible that the tolerance limits X1 and X2 will lie outside the limits of the curve (58) although this is seldom the case. 262 SPECIFICATION OF STANDARD QUALITY 263 These standards are, as it were, ideals. We may, however, gain certain advantages by looking upon standard quality in a slightly different way as being the distribution function representing what we can hope to do in our attempt to attain an ideal standard quality. This objective standard quality distribution represents what we may expect to get when we have done everything feasible to eliminate assignable causes of variability in the quality. Hence, if we are to be able to inter- pret the significance of observed variability in quality, it is necessary to adopt or specify some such distribution function to be accepted as a standard for each quality characteristic. Then, so long as the observed variability in quality of m pieces of product may be interpreted as a sampling fluctuation in the effects of the constant system of chance causes characterized by the accepted standard distribution function for this quality characteristic, there is no need to worry over the observed variation because it is likely that there is nothing that we can do about it. The question now to be considered is: What are the factors that determine how far we should try to go in specifying dis- tribution functions to be used as standards? In the previous chapter we have shown that, from a design viewpoint, it is usually satisfactory to specify only the average X and the standard deviation o of the distribution, whereas complete specification would require the functional form f and the numer- ical value of each of the m' parameters. Furthermore, it is obvious that the specification must be such as to provide a satisfactory basis for detecting lack of control in the two important design characteristics X and g of the distribution of effects of the chance cause system. It is necessary that we consider at this time the character of the specification to be required, because upon the choice of specification depends much of the treatment to follow in the discussion of the two problems: (a) Establishment of sampling limits to detect lack of control to be treated in Part VI. 264 ECONOMIC CONTROL OF QUALITY (b) Statistical estimation involved in establishing quality standards to be treated in Part VI. 2. Types of Specification Type I: The probability of the production of a defective piece of product shall be p. * This type of specification corresponds to making the toler- ance limits either – CO and some value X2, or some value X1 ſ\ QUALITY X FIG. 90.--THREE UNIVERSEs of EFFECTs SATISFYING THE SPECIFICATION THAT THE PROBABILITY p SHALL BE Constant. and + Co., and to specifying that the probability of X lying outside such a tolerance shall be p. It is obvious that this form of specification does not fix the form of the distribution func- tion (58). For example, Fig. 90 shows three distribution functions which satisfy the specification Type I, although they are distinctly different. Hence the necessary design infor- mation, viz., the average and standard deviation of the distri- bution function, is not fixed by this type of specification. It follows from what was said in Part IV that we may establish sampling limits within which the observed fraction p SPECIFICATION OF STANDARD QUALITY 265 defective in a sample of n may be expected to fall with a specified probability P. Hence this form of specification provides a basis for detection of lack of control although it fails to give requisite design information. Type II: The expected or average quality shall be X. This form of specification is sometimes considered when we would like to specify that the quality should be some ideal standard value Xs. It is apparent that there is an indefinitely large number of frequency functions satisfying this specification, but differing in respect to dispersion, skewness, and other char- acteristics as is illustrated schematically in Fig. 91. QUALITY X FIG. 91.--THREE UNIVERSES OF EFFECTS SATISFYING THE SPECIFICATION THAT THE ExPECTED VALUE SHALL BE X. It follows that specification Type II fails to give the infor- mation which makes possible the establishment of design limits on the variability of quality. Neither does it give information basic to the establishment of limits within which the observed quality may be expected to vary without indicating lack of control. Hence this form of specification is of comparatively little value from the viewpoint either of design or control. Type III: The average or expected quality shall be X and the standard deviation shall be G. This specification gives the requisite design information, and so long as quality of product satisfies this specification, we know from Tchebycheff's theorem that the probability Pig. that a piece of product will have a quality X lying within the range • * ~~ e I e between the two limits X + to is greater than 1 – ; This 266 ECONOMIC CONTROL OF QUALITY statement is true independent of whether the function f in the objective equation of control is or is not continuous." To emphasize the importance of the use of Tchebycheff’s theorem in this connection, we show in Fig. 92 four distributions | i nº | | H —4. to- | Y | QUALITY X FIG. 92.-Four UNIVERSEs of EFFECTs SATISFYING THE SPECIFICATION THAT THE AND ExPECTED VALUE SHALL BE X STANDARD DEVIATION SHALL BE or. having approximately the same average X and standard deviation G. The dotted limits are drawn at X + 30. Hence we should expect to find more than 89 per cent of the total are a for each distribution within the limits. In fact, no matter what distribution we might construct with average X and standard deviation g, we would find that more than 89 per cent of the area would fall within the dotted limits. From the viewpoint of control, we have seen in Part IV that sampling limits may be set on averages of size n if we know g and that the probability associated with any limits XI to X2 for the average X of a sample of n is given quite accurately by the normal law integral, at least when n is large. Furthermore, sampling limits can be estab- lished for observed standard deviation or variance in samples of n, and the probability asso- ciated with a given range on to a 2 can be quite accurately estimated if we can assure ourselves that the function f is approximately normal. * This is true at least for objective distributions of the type possible in practice. SPECIFICATION OF STANDARD QUALITY 267 From this discussion we conclude that the specification Type III is far superior to either of the two types previously mentioned. Type IV: The average, standard deviation, skewness, and flatness of the distribution of quality X shall be X, o, k, and B2. Let us see what the specification of k and B2 adds in the way of valuable information. In the first place, the knowledge of these two statistics of the distribution function adds nothing to our knowledge of the integral of the function over any range X, to X2 over and above that given by X and g and the use of Tchebycheff's theorem. This statement rests upon the assumption that we know nothing about the function f. Under the same conditions the knowledge of k and B2 is of little practical value from a control viewpoint, since, as we have seen in Part IV, not even the expected values and standard deviations of k and 62 for samples of size n are known for other than normal universes, so that we cannot establish sampling limits on these two statistics. Hence we come to the important conclusion that the speci- fication of standard quality in terms of X and a gives us the maximum amount of usable information, unless we specify f. 3. Importance of Specifying the Function f. From the discussion of Chapter XII, Part III, we see that there is some justification for the belief that the distribution of a controlled quality is approximately normal or at least is approximately representable by the first two terms of a Gram- Charlier series, which has previously been referred to as the second approximation (23). If then we specify that the func- tion f shall be normal with X and g as the two parameters, the specification becomes complete from the viewpoint of both design and control in that we know for such a product the probability associated with any interval Xi to X2, and we can set sampling limits on almost all of the common statistics, Table 37. Similarly, if we specify that f shall be the first two terms of a Gram-Charlier series, we can make use of most of the distribution functions of the simple statistics for a normal universe as first approximations, and the normal law integral 268 ECONOMIC CONTROL OF QUALITY gives the probability associated with any symmetrical interval X == to. In these two cases we find X and o playing an impor- tant rôle. Formally, of course, the specification of f and each of the m’ parameters makes possible the determination' of the proba- bility associated with any interval X1 to X2. We have seen, however, that little is known about sampling fluctuations in statistics of samples of n drawn from such universes with the exception of average, variance, and x*. Hence, from a control viewpoint, having specified X and o, the specification of f or of any number of parameters does not add as much as one might at first expect. However, we shall soon see that we must specify f in order to make possible the most accurate estimates of such statistics, as p, k, and 32. 4. Specification—Further Discussion. Thus far we have considered the problem of specification as though we could make the function f and parameters A1, A2, ..., \, . . . , \m', whatsoever we chose to make them. Obviously we do not have such freedom of choice. We assume that there is one and only one objective distribution function representing the state of control for each quality X, although we do not assume that these functions are necessarily even of the same form f for all qualities. This means that the distribu- tion function for any quality X must be found before it can be specified. Our previous discussion is of interest therefore in indicating the relative importance of different forms of specifica- tion, thus indicating the extent to which we should try to go in finding the distribution function of control in a specific case. In any case we need to estimate the expected value X and standard deviation o of the objective distribution representing the state of control. Whether we try to go further and specify p, k, B2, and f depends upon whether or not the kind of infor- mation given by such a specification justifies the added expense of estimating these characteristics of the objective distribution * The use of complicated quadrature methods is often necessary. SPECIFICATION OF STANDARD QUALITY 269 and the expense of the extensive inspection required to assure the producer that the quality of product does not vary beyond reasonable sampling limits in respect to these characteristics. It is of interest to point out at this stage of our discussion that the specification of p, k, and 32 introduces a problem in estimation, the solution of which requires the assumption of a particular functional form f. To illustrate this point, let us assume that we have a comparatively small sample, say five observations, in which we are to estimate p. Assuming that the objective p is of the order of O.OI as is often the case in practice, it is obvious that we cannot use the observed fraction p in a small sample as a basis of estimating p. The best we can do perhaps is to make use of our estimates of X and a derived from the sample as a basis for the estimate of p. On the other hand, the estimate of p derived in this way involves an assumption as to the functional form f. We may, by mak- ing use of Tchebycheff's relationship, state certain bounds within which it is likely that p lies. Of course, when we have a large sample representing what we assume to be the condition of control, it is possible to use the observed fraction p as a basis for an estimate of p, although even then it is reasonable to believe that we should consider the general functional form of the distribution in arriving at an estimate. For example, Column 2 of Table 4o gives a dis- tribution of observed values of a variable X. Column 3 of this table gives a theoretical distribution based upon the assumption that the distribution function is w2 \-m 1 * - t •= | * y – yo I + ...) & 1) Ga, Il a. (Z- The theoretical and observed distributions, shown in Fig. 93, indicate close agreement between theory and observation. When there is such close agreement it seems reasonable to assume that the integral of the assumed theoretical distribution between any two limits X1 and X2 should be taken into con- 1 Elderton, W. P., Frequency Curves and Correlation. 270 ECONOMIC CONTROL OF QUALITY sideration along with the observed fraction p within the same limits in estimating the objective fraction p. In other words, we see that the estimate of p required in a specification involves the assumption of a particular functional form f which in turn must be justified upon the grounds that it appears to be the objective frequency function representing the condition of control in this specific case. TABLE 4o.—IMPORTANCE OF DISTRIBUTION FUNCTIONS IN Estimating FRACTION IN TAIL of DISTRIBUTION Cell Observed Distribution |Type IV Distribution Midpoint of Variable X of Variable X 5 IO 6 IO I3 I6 I 5 4 I 49 2O II 5 I35 25 326 32I 3o 675 653 35 I, II 3 I, Io8 4O I,528 I,535 & 45 I,692 I,712 5o I,53O I,522 55 I, I22 I,O74 6o 6Io 604 65 255 274 70 86 I O2 75 26 32 8o 8 8 85 2. 2 90 I I 95 I O X 9I 54 9I 54 We have seen in the previous paragraphs that if we are to make use of information given by k and 32, we must also have a specification of f. Thus, in the example just quoted from Elderton, the observed values of k and 32 are O.O73 and 3. I70 respectively. The fact that the use of these two observed values of k and 32 in the assumed functional form f gives an apparently SPECIFICATION OF STANDARD QUALITY 271 close fit to the observed data, provides us with a certain amount of assurance that the objective values of skewness k and flat- ness B2 are, for example, different from O and 3 respectively corresponding to the normal law, or that they are somewhere in the neighborhood of the values derived from the observed data. - Enough has been said to show that the problem of estimation involved in the specification of characteristics other than X and |OO 9 O e OBSERVED DISTRIBUTION • e NORMAL D | STRIBUTION 92. 8 O z 22* — — — — TYPE IV DISTRIBUTION …” P i 5 O 4 O º 3O 26 º * 2O a’ 2^ |O 24.2 2 ^ 2 O O,OI O. O.5 | 5 |O 3O 5 O 7O 9 O 98 99.5 99.9 99.99 PROBA B | LITY AS SOCIATED WITH A G|VEN VALUE OF VARIABLE FIG. 93.-GRAPHICAL PRESENTATION OF DATA IN TABLE 4o. o and the objective fraction p of the distribution of control involves the assumption of specific forms for f. 5. Conclusion The specification of quality from the viewpoint of both design and control should provide X and g. In certain cases it is desirable that we specify p so as to provide a basis for catching erratic troubles which, as we shall see later, may not be detected 272 ECONOMIC CONTROL OF QUALITY through sampling limits established on statistics used to detect lack of control in X and o. The accurate estimate of p, how- ever, involves the introduction of some assumption as to the functional form f of the distribution (58) of control. The speci- fication of k and B2 is, in general, of less importance than that - of p, X, and 0. PART VI Allowable Variability in Quality Five Criteria for Determining When Variations in Quality Should Not Be Left to Chance CHAPTER XIX DETECTION OF LACK OF CONTROL IN RESPECT TO STANDARD QUALITY I. The Problem In Part V we saw that standard quality is characterized by the equation of control dy = f(X, \1, M2, . . . , Mi, ..., \m)dX. (58) In particular, we saw that it is desirable to maintain constancy of this distribution at least in respect to the average X and standard deviation o. Of course the qualities of samples of m pieces of product of standard quality may be expected to show sampling fluctuations. The problem to be considered in this chapter is that of establishing an efficient method for detecting the presence of a cause of variability other than one of the chance causes belong- ing to the group which gives the accepted standard distri- bution (58), or of determining when an observed sample is such that it is unlikely that it came from a constant cause system characterized by this distribution. 2. The Basis for Establishing Control Limits Knowing the distribution function (58), we saw in Part IV that it is possible, in general, to find a distribution function fe(0, n) for a given statistic 0 calculated for samples of size n such that the integral 62 gives the probability that the statistic 0 will have a value lying within the limits 61 to 62. Of course, if the function fe(6, n) 275 276 ECONOMIC CONTROL OF QUALITY is limited in both directions, we may choose 01 and 02 such that P = I; and, in this case, any observed value of 6 falling outside the limits is a positive indication that standard quality is not being maintained. If the function fe(0, n) in (85) is dis- continuous we must replace the integral sign by the symbol of summation 2 for discrete ordinates and change our discussion accordingly. The conclusions, however, remain unchanged. For the most part, however, we never know fo(0, n) in sufficient detail to set up such limits. More important yet is the fact that, even if we knew the function well enough to set up limits within which a statistic 0 must fall provided the cause system has not varied from the accepted standard, we could not say that the occurrence of an observed value of 6 within this range is sufficient to prove that the sample came from a constant system characterized by the accepted standard distribution function (58). How then shall we establish allowable limits on the vari- ability of samples? Obviously, the basis for such limits must be, in the last analysis, empirical. Under such conditions it seems reasonable to choose limits 61 and 62 on some statistic such that the associated probability P is economic in the sense now to be explained. If more than one statistic is used, then the limits on all the statistics should be chosen so that the probability of looking for trouble when any one of the chosen statistics falls outside its own limits is economic. Even when no trouble exists, we shall look for trouble (I – P)N times on the average after inspecting N samples of size n. On the other hand, the smaller the probability P the more often in the long run may we expect to catch trouble if it exists. We must try to strike a balance between the ad- vantages to be gained by increasing the value P through reduction in the cost of looking for trouble when it does not exist and the disadvantages occasioned by overlooking troubles that do exist. It is conceivable, therefore, that there is some economic value P or pair of limits 01 and 62 for each quality characteristic. It is perhaps unnecessary to say that the determination of the economic value P and the associated DETECTION OF LACK OF CONTROL 277 limits must be an approximation in any case. Furthermore, it is obviously necessary to adopt some value which will be acceptable for practically all quality characteristics, although the economic value P for one quality may not be the same as that for another. With these points in mind we shall consider a few principles to guide our choice of 61 and 02. In general, it is reasonable to believe that the objective economic values of 61 and 92 are not symmetrically spaced in respect to the expected value 6 of the statistic. It is perhaps more reasonable to assume that they are so spaced as to cut off equal tails of the function fe(0, n). Under these conditions it is reasonable to try to set limits 01 and 02 that will satisfy this condition. From the discussion in Part IV we see, however, that even when the distribution (58) is known, the distribution function fo(0, n) for a given statistic 0 is seldom known in sufficient detail to make it possible to choose 61 and 62 to cut off equal tails. Even more important is the fact that we seldom care to specify f accurately enough to make possible the setting of such limits. For these reasons we usually choose a symmetrical range characterized by limits - 6 == toe (86) symmetrically spaced in reference to 6. Tchebycheff's theorem tells us that the probability P that an observed value of 0 will lie within these limits so long as the quality standard is maintained satisfies the inequality I 2" P × I – l We are still faced with the choice of t. Experience indicates that t = 3 seems to be an acceptable economic value. Hence the method for establishing allowable limits of vari- ation in a statistic 0 depends upon theory to furnish the expected value 6 and the standard deviation go of the statistic 0 and upon empirical evidence to justify the choice of limits 6 == toe. 278 ECONOMIC CONTROL OF QUALITY 3. Choice of Statistic to Detect Change in Average 9aality Suppose, for example, that dy = f(X, \1, M2, . . . , Ni, . . . , Mn.)d X, (58) with an expected value X, is the standard of quality and that we are to detect a change in quality in which only the expected value changes from X to X + AX. What statistic of the sample should we use to detect this change in order to minimize f (x,xi,..., xm) X sºa5 x + A X the number of observations Q U A L | T Y (b) required? exeiº Wºlve To start with, let us assume that (58) is a normal distri- bution. Obviously then, we | | might use either the median / : (C) | or arithmetic mean of a (d) sample to detect a change AX Jºãº, ºr in the expected value X. To illustrate, let us assume that the standard quality is distrib- uted as in Fig. 94-a and that the shift AX in expected value is represented by Fig. 94-6. Let us assume also that the l | | | | (f) e tº tº te & D IS T R B UTIONS OF ME D : A NS distribution of arithmetic means US ED IN DETECTING SH | FT § and that of medians are nor- mal as indicated in Figs. 94-c and 94-e respectively. This situation is practically met when the sample size is large, in which case the standard deviation of the distribution of FIG. 94.—ILLUSTRATING IMPORTANCE of PROPER CHOICE of STATISTIcs. tº ſº Or O' medians is I.253 – and that of means is —. These values 53 v/n g Vn of standard deviation were used in drawing Figs. 94-c and 94-e. Limits including equal areas of Figs. 94-c and 94-e are shown. The curves of Figs. 94-d and 94-frepresent the distributions of DETECTION OF LACK OF CONTROL 279 averages and medians about the expected value X + AX. Obviously, the area of Fig. 94-d outside the dotted limits for means is greater than the area of Fig. 94-foutside the limits for medians. Hence, for a given increase AX, we may expect to have an indication of trouble more often by limits set on arith- metic means than by those set on medians. In general, if €1 and 02 are two statistics (such as median and arithmetic mean) used to detect a change in some charac- teristic 6 of the universe; if the functions fo,(01, n) and fe,(02, m) are symmetric, monotonic, and unimodal; if the standard deviations of 61 and 02 fall off in the same way with increase in sample size m; and if €1 = €2 = 0, then we may say that that statistic having the smaller standard deviation should be used in detecting the change AX. Now, if there exists a statistic 0 such that the use of any other statistic 01 does not throw any further light upon the value of the parameter to be estimated, then 0 is said to be a sufficient statistic, and is, of all statistics of this class, the one to use, provided it can be shown that it is also the most efficient. In this connection, some very useful theory has been con- tributed by R. A. Fisher. He shows that if q and 0.1, the standard deviations of 6 and 61 respectively, fall off as ––. 77 and if 0 and 01 are normally correlated with correlation coeffi- cient r, then the above criterion of sufficiency leads to the relationship O = ro1, showing that 6 is more efficient than 61 and that under the given conditions r = VE, (87) where E is the efficiency of 61 as compared to 0. If, in practice, we find that the correlation surface for two statistics, such as the median and arithmetic mean, is normal and satisfies (87), then it is reasonable to assume that the more efficient of the * “On the Mathematical Foundations of Theoretical Statistics,” Philosophical Transactions, Series A, Vol. 222, pp. 309-368, 1922. 28O ECONOMIC CONTROL OF QUALITY two is a sufficient statistic and perhaps also the most efficient statistic that can be used. It should be noted that under the given conditions the more efficient of the two statistics has the smaller standard deviation and hence is the better one to use in detecting a change of parameter. We have already seen that the distribution of medians for samples of size n = 4 from a normal universe is symmetrical and not so very different from normal, whereas the distribution of arithmetic means is normal in this case. It is interesting to see, therefore, whether or not the arithmetic mean is not only better than the median for detecting a shift AX but really the best statistic that can be used. Fig. 95 shows the observed scatter diagram of correlation between medians and means for samples of four. In this case the observed efficiency E and correlation coefficient r are E = O.80 r = O.899, and (87) is practically satisfied. Since we know of no statistic whose standard deviation falls off more rapidly than wº 7? we may conclude that the arithmetic mean is the best statistic to be used for detecting a shift AX, subject to the conditions stated above. We are not in a place to prove that the average is the best statistic when the distribution function (58) is not normal. However, since we do not know of a better statistic than the arithmetic mean to detect a shift of AX when the universe differs from normality by no more than it usually does in prac- tice, we shall always make use of the arithmetic mean for this purpose. It is of interest to note that the efficiency E of the median in respect to the arithmetic mean for samples of n drawn from a normal universe decreases asymptotically with increase in sample size from Ioo per cent for n = 2 to 63 per cent when m is large, as indicated in Fig. 96. The point for n = 4 is that DETECTION OF LACK OF CONTROL 28 I observed for the 1,000 samples of four. This curve shows that for large samples the efficiency of the median is such that it contains only about 63 per cent of the information in respect to the change AX; in other words, that the average of a sample | 4 Z | , OO 4 4 y 9 3. | | 3 || || 2 || 28 / | O | y 33 . O, 5 O 3 6 || 44 % |4 | 55 A | |O | 5 | 2 | 3 2 75 «[ o O 9 || 34 45 || 6 | 9 O Ll > | 28 || 7) 36 5 | eZV – O. 5 O | 15 % 54 7 2 28 % 8 2O 7 | | 14 — 1. OO 3. | | 4. 2 Z 6 3 ( | © MEDIAN | OBSERVED O MEAN AVERAGES - |, 5 O O 2 - LINES OF REGRESSION 3 gº y O | -1, 50 -], OO -O.5C) O O, 5 O |, OO |, 5 O A RITH M ET || C M E A N FIG. 95.-ScATTER DIAGRAM of CoRRELATION BETweEN MEDIANS AND MEANs. of size n = 63 will detect in the long run a shift AX as often as the median of a sample of n = IOO. Max. + Min. If, instead of the median, we use the alS al 2 statistic, we have seen that the efficiency is ſoo per cent for samples of two and about 88 per cent for samples of four. 282 ECONOMIC CONTROL OF QUALITY: By making use of some of the recent work of Tippett, E. S. Pearson, and N. K. Adyanthāya,” we may show that the Max. + Min. O efficiency of the falls off as indicated in Fig. 96. sº This curve is in striking contrast to that for medians. The concept of efficiency here used is different from that introduced in Part IV, and is perhaps the more usual one. It is simply the ratio of the sample sizes of two different |OO r- ARITH M ETIC MEAN 9 O H. 8 O hºmºe 7 O * M E D : A N 6 O º 4. O º 3.5 OO sºº 2O H. | O H. MAXIMUM + MINIMUM 2 O 1 I ſ L 2 4 |O 2O |OOO OO SA M P L E S Z E n MAx. + MIN. FIG. 96.-EFFICIENcy of THE MEDIAN AND — As A FUNCTION r) wº of SAMPLE SIZE m. consistent statistics required to give the same standard deviation. Consider for example the arithmetic mean X and median M of a sample of n. The standard deviation of X in samples of m drawn from a normal universe with standard deviation q is o/v/n and for medians M, the standard deviation is c(n)0, * “On the Extreme Individuals and the Range of Samples Taken from a Normal Population,” Biometrika, Vol. XVII, December, 1925. * Egon S. Pearson and N. K. Adyanthāya, “The Distribution of Frequency Con- stants in Small Samples from Symmetrical Populations,” Biometrika, Vol. XX-A, pp. 356–36O. - DETECTION OF LACK OF CONTROL 283 where c(n) is some function of n which approaches I.253/v/n as n becomes large. Choose a particular sample size nM for the median and find the sample size mx for the arithmetic mean required to give the same standard deviation as that of the median for the chosen sample size. This requires merely the solution of the equation O' = C(?? Mr.) O’ v/n: ( M) for ny. In fact I cº(ny)" and therefore by definition the efficiency of the median for the chosen value of nM is 77 57 I E = ** = 7? M. narcº(nºw) The trouble with this value of efficiency for small values of n is that it depends upon the fact that the value of nºt was chosen first. Thus if we assign to my the same value nM, and solve for the new value m'ºf we should come out with the same value of E, if the efficiency for small samples is to have the same interpretation as for large samples. However, if we solve for m'M from the equation O” — = C 77'M)0. v/n, (n'M)0, and then take the ratio E = +, it will be found to be different, 77' Mſ. in general, from the value of E computed above. In other words, this means that for small samples we get one curve of efficiency by assigning to nºt an increasing sequence mi, m2, . . . and a different curve of efficiency when my is assigned the same series of values. For this reason the curves of Fig. 96 should not be con- sidered as exact but as merely indicating, in a general way, how 284 ECONOMIC CONTROL OF QUALITY Max. + Min. falls off with in- the efficiency of the median or creasing sample size. 4. Choice of Statistic to Detect Change in Standard Deviation Suppose now that we consider the problem of determining the statistic which will detect a change only in the standard deviation of the effects of the cause system. Let us start, as in the previous paragraph, with the case where the universe of effects (58) is normal. Naturally, we may use any one of several infinite sets of estimates of G as a means for detecting a change AG. Thus, for example, 2. • _* g t & In , = ſ złe Poºdy = fºr(H) (88) 0 O V 27ſ T 2. where x = X – X, and i = (1, 2, 3,...). For a given value of i, we can write O” = &mi, where b is a constant for a particular i. Obviously, the ith moment mi of the absolute values of the deviations in a sample from the observed average X of a sample can be used as an estimate of q in samples of size n = Co. In other words, the statistic l 6 = (bmd)? may be used as an estimate of q if the sample size is sufficiently large. In general, the distribution function fe(0, n) of any statistic 0 is not symmetrical; hence the expected value 0 is not q. This situation is represented schematically in Fig. 97. For samples of a given size n, there is some constant c by which to divide 6 so that the expected value of º becomes equal to 6. 6) * & & Hence z may be used as an estimate of 6 or in this case of G: it is called a consistent estimate. DETECTION OF LACK OF CONTROL 285 In a similar way we may make use of either a symmetrical or an asymmetrical range as an estimate of G. For example, we have already considered the distribution of 1,000 observed ranges in samples of four drawn from a normal universe. The statistics for these distributions were given in Table 34. Since these ranges are measured in terms of the standard deviation of the universe, the empirical factors for estimating o are those N ~ fe (e, n) § 6 e STATISTIC 9 FIG. 97.-SchEMATIC ASYMMETRICAL DISTRIBUTION of A STATISTIc. given in Table 41. Now, as in the discussion of Fig. 97, if € represents the expected value of the distribution of any range 9, the expected value of the distribution of e is 0 or the sta- C tistic 0 of an infinite sample or of the universe. Of course, TABLE 4I.-EMPIRICAL FActors For Estimati NG or Range s & e s e e º e º e º e º e e s tº º e º 'º e º e s tº e X4 - X1 X2 * A 1 A 3 *> A 2 X4 - A 3 Empirical Factor for Estimation. . . . . 2.oO44 o.7863 o. 6338 o. 7752 this statement rests on the assumption that 0 is measured in units of 6 as in Table 41. The second row of this table gives the empirically determined factors with which to transform the observed ranges into consistent estimates of G. It will be noted that we use 6 as a statistic of an infinite universe. If 6 is also a parameter in the equation of control (58), as it usually is, then there is some parameter A numerically equal to 6. 286 ECONOMIC CONTROL OF QUALITY Enough has been said to show that there is an indefinitely large number of ways in which to estimate q. Which one shall we choose as being the most likely to detect a change Ag? Let us start with a comparison of the standard and the mean deviation as a basis for estimating o. In Part IV we saw that the expected value for small samples is not equal to 0 for either of these statistics, the situation being that characterized by Fig. 97. Hence, before we can use either statistic as an estimate of G, we must know the correction factor for transforming the statistic into one for which the expected value will be q. Such correction factors are given in Table 29 for the standard devi- ation a of the sample and a similar table could be given for the mean deviation. Of course these factors approach unity as the sample size becomes large. If we also assume that the distributions of these two statistics approach normality as the sample size n becomes large, we can make use of the same reasoning as that given in Paragraph 3 to show that a is the better estimate since the mean deviation estimate is only 88 per cent efficient. When the sample size is small, these two estimates have more nearly the same efficiency. This situation is shown in Fig. 98. The question arises as to whether or not the standard deviation a is the most efficient statistic for estimating q from a small sample, assuming that it is the most efficient for a large sample. The only available method for doing this is to apply the test of (87) which is strictly applicable only when the correlation between the two estimates is normal, which condi- tion is, as we know, not fulfilled in this case. The experimental results for the 1,000 samples of four are shown in Fig. 99. The correlation coefficient r in this case is O.895, whereas the effi- ciency of the estimate I. I547 ml as compared with the estimate I.2533a is practically IOO per cent. We are, therefore, uncertain from this test whether or not the standard deviation is the most efficient estimate although we see from Fig. 98 that even for small samples it is more efficient than the mean deviation. The difference is negligible, of course, for comparatively small Samples. DETECTION OF LACK OF CONTROL 287 It will be of interest now to consider the efficiency of the range between the maximum and minimum values of a sample as an estimate of G. Again making use of the work of Tippett, E. S. Pearson, and N. K. Adyanthāya,” we get the range effi- ciency curve shown also in Fig. 98. The very rapid decrease in efficiency of the estimate derived from the range is striking. The same concept of efficiency is used here as was used in Paragraph 3. We have here an added difficulty in that the | OO r- Ts ROOT MEAN SOUARE DEVIATION 9 O H. * *R*-- TI MEAN DEVIATION (? 8 O hº i: 234.567 OOOOOO ºº-}*lºss}*gº O }* RANGE O I | | 2 4. | O 2O |OOO OO SA M P L E SIZE ſh T FIG, 98.-EFFICIENCY of N; TIMES MEAN DEVIATION AND RANGE AS ESTIMATES of or CoMPARED witH THAT of THE STANDARD DEVIATION. root mean square deviation, VT/2 mean deviation, and the range are not even consistent estimates of G. For this reason the curves of Fig. 98 are supposed merely to indicate, in a general way, how the efficiencies of the above two statistics fall off with increasing n. It should be noted that, in our discussion of the importance of choosing the most efficient statistic for detecting a change AX or AG, we tacitly assumed that the distribution functions of the statistics compared were symmetrical and of the * Loc. cit. * Loc. cit. 288 ECONOMIC CONTROL OF QUALITY same functional form. This is a very important requirement for, in general, the most efficient statistic in the sense of being the one with the smallest standard deviation need not be the statistic most likely to catch a given change in X or g. For 1,875 / 2^ 2%|2'ſ }^ 36 I, 275 º 8 9 // O. 975 |9 2. | 6 2 z 9 y^ 4 O o,675 2 º 2 O. 375 2% 76 e STANDARD DEVIATION UOBSERVED * A o MEAN DEVIATIONYITſAVERAGEs 2 - 2 2% - L| NES OF RE GRESSION 34 vºy y Z O. O.75 y/ l O. O. 94 O,47 O O, 846 1.222 I, 59.8 l, 974 Yg M E A N D E V | A T | O N T FIG. 99.-CoRRELATION BETweeN STANDARD DEVIATION AND V: TIMEs MEAN DEVIATION FOR SAMPLEs of Four FROM A NoFMAL UNIVERSE. example, the comparison of the four ranges of Table 41 for detecting a given change AG involves the algebraic magnitude of AG, and the knowledge of the functional forms of the distri- bution of the different ranges. The same could be said of the comparison of the statistics based upon the moments mi of the DETECTION OF LACK OF CONTROL 289 absolute values of the deviation. To make such a comparison is certainly not practicable at the present time. It appears, therefore, that there is good reason to choose the standard deviation o of the sample as a basis for the es- timate of the standard deviation o of the universe to detect a change AG. 5. Additional Reason for Choosing the Average X and Standard Deviation a We are now in a place to consider an additional and very important reason for choosing the average X of a sample to detect a change AX and the standard deviation a to detect a change AG. The previous discussion has been limited to the assumption that the universe or distribution (58) of standard quality is normal. º In Part IV, however, we saw that, no matter what the nature of the distribution function (58) of the quality is, the distri- bution function of the arithmetic mean approaches normality rapidly with increase in n, and in all cases the expected value of means of samples of n is the same as the expected value X of the universe. Hence the arithmetic mean is usable for de- tecting a change AX almost equally well for any universe of effects which we are likely to meet in practice. It appears that the same cannot be said of any other known statistic. We also saw in Part IV that, although the distribution function f,(0, n) of the standard deviation a of samples of n is not known for other than the normal universe, nevertheless the moments of the distribution of variance oº are known in terms of the moments of the universe. Hence we can always establish limits O2 + to 22 within which the observed variance in samples of size n should I * fall more than ico( ºmºs ...) per cent of the total number of times f a sample of n is chosen, so long as the quality of product is controlled in accord with the accepted standard. 290 ECONOMIC CONTROL OF QUALITY This generality of usefulness is not shared by any other known estimate of q or, more specifically, of gº. 6. Choice of Statistic to Detect Change Ar in the Correlation Coefficient r In the present state of our knowledge of the distribution of product moments, the only available basis for detecting a change Ar is the distribution function (75) of the correlation coefficient in samples of size n. 7. Choice of Method of Using Statistics Having chosen statistics with which to detect variability from standard quality, it remains for us to choose the way of 6 | * * *- " " - - - - - - - - - - - - - SUCCESSIVE SAMPLES OF n FIG. Ioo.—SIMPLE ForM of ContROL CHART. using them. We shall illustrate this point by a discussion of the ways of using the average X and standard deviation a of samples of size n. Making use of the control limits Ö == 30 0, we may construct a control chart such as shown in Fig. IOO. The occurrence of a value of 6 outside these limits is taken as an indication of a significant variation from standard quality or as an indication of trouble. DETECTION OF LACK OF CONTROL 29I Instead of using this simple form of chart for each of several statistics, we may use a chart based upon the probability of the simultaneous occurrence of the different statistics. Two possible forms of such charts for two statistics 01 and 02 are shown in Fig. IOI. In Fig. IOI-a the occurrence of a sample for which the point (61, 62) falls outside the shaded area is taken as an indication of trouble, the boundary of this area having been chosen so that the probability P of falling within the boundary is economic. Similarly, in Fig. IOI-b, the prob- ability P of falling inside the dotted limits on either side of the e el e2 62 (a) (b) FIG. IoI.—Two TYPICAL ForMs of ContRoL CHART. curve of regression represented by the solid curve is economic. Such a test is often referred to as the doublet test. To construct a chart of the type of Fig. IOI-a requires the knowledge of the distribution function fel, e,(01, €2, m) of the two statistics 01 and 02. For the averages and standard deviations of samples from a normal universe this function rapidly approaches normality as we see from a study of the distribution functions of X and g of Part IV. Hence we can set up correlation ellipses corresponding to a desired probability P. In general, however, little is known about the distribution function of pairs of statistics, even for the arithmetic mean and standard deviation, for samples from other than a normal universe. 292 ECONOMIC CONTROL OF QUALITY The work of Neyman already referred to in Chapter XIV of Part IV makes possible the construction of a chart of the form of Fig. IoI-A for averages and variances of samples from any known universe. This theory also makes it possible to establish approximate limits for pairs of averages and standard devia- 2.8 REGRESSION CURVE – STANDARD DEVIATION ON AVERAGE - - - - LIMITS - STANDARD DEVIATION ON AVERAGE ſ 2.6 º --------> 2.4 H. 2.2 }* O.1.!,l.!.!.2. 8O24.68O }*ººº*}*}* O. 6 }== Av ERA GE X FIG. IO2.—EFFECT of SAMPLE SIZE on LIMITs. tions. Fig. IO2, for example, shows such sets of limits for samples of n = 4, n = IOO, and m = IOOO. This figure is of particular interest in that it indicates that such a test may be more sensitive to a change in the functional form f of the uni- verse when the sample is small than when it is large. In other DETECTION OF LACK OF CONTROL 293 words, such a chart can be made sensitive to changes in the function representing standard quality, even though the average X and standard deviation o of the universe remain constant. 8. Choice of Method of Using Statistics—Simple Example Table 42 gives forty observed values of tensile strength of steel strand in pounds per square inch (psi). Let us assume TABLE 42.-TENSILE STRENGTH OF STEEL STRAND Company No. I Company No. 2 I2,600 I3,800 I4,3OO I4,550 I3,750 I4,250 I3,900 I4,25o I 3,440 I3,370 I4,460 I3,390 I3,960 I 3, 5 IO I4,480 I4, I 3O I 3,570 I3, I IO * I4, I 7O I 3,910 I3,550 I3,4OO I3,610 I 3, 180 I3,570 I 3,860 I 3,990 I 3,790 13,430 I 3,44C I4, I4O I3,810 I3,250 I3,9CO I3,4OO I3,260 I 3,32O I3,910 I4,29O I4355O that the accepted standard quality for the tensile strength of this particular product is normally distributed with X I3,540 psi, and O' = 440 psi. Is there any indication that the quality of product of either supplier is significantly different from standard quality in the sense that the observed samples may not be considered as random samples from standard quality? In what follows, we shall describe three different ways of using the statistics X and a to answer this question. A. One way is to construct control charts for averages and standard deviations of samples of twenty with the following 294 ECONOMIC CONTROL OF QUALITY limits. Of course, G is 440C2 where the value of c2 is that given in Table 29 for n = 20: — O' 449 (13,245 X == 3–7– = I.3,540 + 3–5– = } *V, 33.54 *V. |. and This is done in Fig. Iog. Using this method, we assume that there is an indication of the existence of significant deviations “I 900 r- }* 80 OH- 13, 8 OOH b 7ooH 2 mºms º ºsmºs ºs º ºgº tºº ºmº mº O EX }*s = 600H <ſ Lil > § 13,600 - º, SooH CC Lal © O P- OC ſº < !-m < 4OO H. O Z < H- 13,400 H. on 30O H- H. 200 E--— — — — — — 13, 2 OO L ! O O L. I | J L | J O | 2 O | 2 C O M P A N Y C O M P A N Y FIG. Iog.—ONE ForM of ContRol CHART Test. from standard if the observed values of either average or stand- ard deviation or both for a given sample fall outside of the control chart limits. The observed values of average and standard deviation for the two samples of twenty are represented by the black dots. We take the fact that one of the averages falls outside its limits as an indication of lack of control in respect to standard quality. DETECTION OF LACK OF CONTROL 295 B. Another way of testing whether or not the two samples of twenty came from standard quality is to construct a control chart of the type shown in Fig. IoI-a. Since for samples of twenty from a normal universe the correlation surface of X and o is approximately normal, we may construct the ellipse which should include, let us say, P = 99.73 per cent of the observed pairs of values of X and g. Doing this for the case in hand, we get the results shown schematically in Fig. IO4. 8 OO | 7OO - 6OO H. 5OO - 4OOH. 3 O O H. 2OOH- | O O H. | 1 | | | Y ſ 13,600 13,800 14,000 A V E R A G E X ſ f 13,200 13,400 FIG. Ioa.—ANOTHER ContRol CHART TEST. The fact that one point is outside this ellipse is taken as an indication of trouble. C. A third way of testing whether or not the two samples came from standard quality is to test whether or not the dif- ferences |X, * X2 | - 428.50, and |al – go = 71.28 are likely to have occurred if both samples came from standard Cf. Chapter XV, Part IV, 296 ECONOMIC CONTROL OF QUALITY quality. Obviously a test of this nature comparable with the previous two is to consider the occurrence of an absolute dif- ference in averages greater than OT 44O –F– = Q –– = A.I’7.42 'N. 3Vis 4. 7.4 X 2 or in standard deviations greater than - OT 44O —= = q += = 20 K.I6 3V, 3.Vas 95 as indicative of trouble. Again we get a positive indication. 9. Choice of Method of Using Statistics–Continued Let us look at the results obtained by the three different tests just described. It will be seen that the first test indicates 8 OO ſ 7O6P- 6OO H. 5OOH- 4OOH- 3OOH- * *-* * *m, ame sºme 2OOH- |OOH- | | | | | | | + | | | | | | | | | | | | | | l- | | | | | | I | l l — 13, 600 13,800 14,000 AVERA GE X I l | | 13,200 13,400 FIG. Io;.—CoMPARIson of THREE TESTs. trouble when a point (X, g) falls outside the dotted rectangle in Fig. IoS, whereas the second test indicates trouble when a point falls outside the ellipse. It is easy to see that the two DETECTION OF LACK OF CONTROL 297 tests are inherently different. In the first place the probabilities associated with the areas of the rectangle and ellipse are o.9946 and O.9973 respectively. More important, however, is the fact that the two tests could not be made to exclude the same region even if the areas were equal. Now the third test is basically different from the other two in that it indicates trouble when either the distance a or b ex- ceeds certain limits. Since, as in the simple illustration of the previous paragraph, experience indicates that the three tests so often give consistent results, since the third test is obviously very difficult to apply when we have many samples of size n, and since the second test is more difficult to apply than the first although it gives approximately the same results, the first test appears to be the practical choice. Io. Choice of Statistic for Detecting Change in Universe of Effects Let us consider next the problem of detecting a variation from standard quality represented by a change of cause system from one which gives standard quality, say dy = f(X, \1, M2, . . . , Mi, . . . , \m')d X, (58) to one which gives something different from standard and represented by some unknown distribution of the form dy = f(X, \'i, M2, . . . , Mi, . . . , Aſm”)dX. Perhaps the single statistic most sensitive to a change of this type is the x” function. Subject to the limitations set forth in Part IV, we may divide the original distribution into any number of cells and calculate x” for samples of size n grouped into the chosen cells. A control chart for x” may then be con- structed by making use of the known values of x2 and ox”. In general, it is desirable to use a grouping which gives as nearly as possible equal probabilities for all cells. One difficulty is that the x” control chart can only be used for comparatively large samples. - 298 ECONOMIC CONTROL OF QUALITY II. Detection of Failure to Maintain Standard Quality Thus far we have considered the comparatively simple problem of detecting a change of a given kind and amount in the effects of a constant cause system, such as a change AX in expected value or a change AG in the standard deviation of the effects of the cause system, everything else remaining fixed. In practice, however, we never know that the quality has changed from standard in a specific way. What we do is to take a sample of m to determine whether or not the product has changed. It may or may not have changed one or many times within the period in which the sample of n is being taken. Our success in detecting trouble in such a case depends among other things upon the way in which the sample is taken, or, more specifically, upon whether or not the sample of n comes from one or more constant systems of causes. For example, in testing whether or not the tensile strength of strand, Table 42, had been controlled in accord with standard quality, we divided the data into two groups of twenty obser- vations, one group from each of the two suppliers. Of course we could have tested in a similar way the hypothesis that the forty observations came from a standard production process. Thus, the control limits in pounds per square inch (psi) on average X and standard deviation a of samples of forty from product of standard quality are respectively: 44O | 13,331 3&4 svº *E. ſº 440 |284 432 =3 vº- |. The fact that the observed average of the forty values of tensile strength falls outside the control limits would be taken as evidence of lack of control. Hence, no matter which test had been applied in this case, the result would have been the same. It may easily be shown, however, that the results of two such tests may not be the same. That is to say, if trouble does exist in that the product as tested by a sample of n comes from two DETECTION OF LACK OF CONTROL 299 constant systems of causes in the sense that ni pieces come from a cause system with constants X1 and O 1, and m2 pieces come from another system with constants X2 and O2, it is possible that a test for trouble using the total sample m may or may not give an indication of trouble. The same is true of the test based upon the use of the samples m1 and m2. Furthermore, one test may be positive and the other negative. Therefore it might appear that it makes little difference how a set of n data representing lack of standard control is grouped before applying the test for detecting trouble of this kind. In other words, this would mean that an inspector trying to detect variation from standard quality would be able to do so equally well irrespective of whether or not he was able to divide the data in a sample of size n into subgroups corre- sponding to different constant systems of causes. To draw such a conclusion would be utterly misleading and against what is perhaps the most generally accepted step in the scientific method, that is, classification. Assuming for the moment, how- ever, that in the long run a test using the whole group of n data as a unit is just as likely to detect trouble as one using the subgroups of data obtained by accurate classification, there still would be a definite advantage in classifying the data before applying the test. Obviously, the ultimate object is not only to detect trouble but also to find it, and such discovery natu- rally involves classification. The engineer who is successful in dividing his data initially into rational subgroups based upon rational hypotheses is therefore inherently better off in the long run than the one who is not thus successful. For such an engineer the statistical tests described in this chapter constitute a powerful tool in testing his hypotheses and in determining the extent to which an investigation must be carried in order to check beyond reasonable doubt whether or not a given hypothesis is justified. 3OO ECONOMIC CONTROL OF QUALITY Suppose, for example, that an engineer wishes to determine how large a sample is required to detect variation from standard quality by an amount AX in the expected value, where it is assumed that the functional form f and all other parameters remain the same. It is a simple matter to show that the required sample size n is given by the solution of the equation 0 8 v/m’ ( 9-a) where t is generally taken as three for reasons already set forth. In a similar way one finds that the number required to detect a change only in standard deviation and of an amount AG is given by the solution of AX = 24 Or AOT = 2f v/2n' (89-b) For example, the size of sample determined from (89-a) is such that the probability of detecting trouble of the nature of a change only in X and of an amount AX is approximately o.99 if t = 3. We can go even further and say that with this sample size the probability of detecting trouble in the form of a change only in X is greater than o.99 if the shift is greater than AX used in (89-a). A similar interpretation may be given to the value of n derived from (89–b). Thus we see how statistical theory becomes a useful tool after we have taken the scientific step of classification of data into rational subgroups. Moreover we see that, even though classification is not as it should be, statistical tests often indicate the presence of trouble. Of course, these advantages are attained with a knowledge that we shall not look for trouble when it does not exist more than a certain known fraction (I – P) of the total number of times that a sample of size n is observed. CHAPTER XX DETECTION OF LACK OF ContRol I. The Problem In the previous chapter we considered the comparatively simple problem of detecting lack of control in respect to an accepted standard distribution. Now we shall consider the problem of detecting lack of control in the sense of lack of constancy in the unknown cause system. To make clear the inherent difference in these two problems, let us consider once more the data on tensile strength of strand as given in Table 42. The three tests of the previous chapter merely served to in- dicate whether or not it is likely that the data came from a specified constant cause system. The corresponding question to be considered now is whether or not they come from some constant cause system of unknown functional form f, unknown average X, and unknown standard deviation o. The tests of the previous chapter made use of assumed known values of X and q. The corresponding tests which we can use in this chapter must involve estimates X and g, say, of the unknown average X and standard deviation o of the objec- tive but unknown distribution representing the condition of control, if it be controlled. Two criteria to guide usin making the estimates X and g are: 4. The estimates X and a used as a basis for detecting lack of control must be such that, if the quality from which the sample of size n is drawn is controlled with an average X and a standard deviation o, then the following two statistical limits should be fulfilled: Ls X = X m. – CO (90) Ls or = OT 9 77 – CO 3OI 3O2 ECONOMIC CONTROL OF QUALITY B. Insofar as possible, the estimates should be chosen so that, if the quality is not controlled, the estimates X and a actually used shall be those which will be most likely to indicate the presence of trouble or, in this case, lack of constancy in the cause system. 2. Choice of Method of Estimating X and G Let us start by considering estimates X and g in psi derived from the data of Table 42 in two different ways as follows: (a) Let 40 X. A. X = − = 13,763.75, 4O and 40 — -\ }% 2 (XI - X)* a = \ = | = 442.2O. 4O (b) Let 40 2 X: X = = 1 F I3,763.75, 4O and I 0 1 + 0.2 O = — = 4OO.45, C2 2. where a1 and 0.2 are the standard deviations of the first and second groups of twenty observed values and where cº is the factor given in Column 3 of Table 29. Obviously the condition (90) is satisfied by the estimates (a) and (b). It may easily be shown, however, that if the sub- groups are rational, then the estimate a of type (b) is on the average less than the corresponding estimate of type (a). Therefore, under these conditions criteria involving the use of estimates (b) will in the long run detect trouble more often than similar criteria involving estimates (a). Hence it is reasonable to choose method (b) for estimating X and g. DETECTION OF LACK OF CONTROL 3O3 3. Choice of Test Criterion for Detecting Lack of Control Having chosen a pair of estimates X and a, we may use them in any criterion in which we may use X and o. As an illus- tration let us apply the three criteria of the previous chapter, making use of X and a calculated as in (%). The results of the application of the first two criteria are shown graphically in Fig. Ioff. Obviously both of these criteria give a negative indication of lack of control. Comparing Fig. Io; with Fig. IO6 7OO r- 6OO H. 500 H. 400 H. 3OO H. 200F — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — | OOH. | 13,400 13,600 13,800 14,000 14,200 AVERA GE X FIG. Ioff.—TESTs For ContRol. we see that, whereas one point is out of limits in Fig. IOS, neither point is out in Fig. IO6. This is interpreted as meaning that, although the observed data are consistent with the assumption of the existence of a controlled state upon the basis of the criteria used, the equation of control is likely not the accepted standard used in the previous chapter. Now since the difference 428.50 psi in averages exceeds Oſ 3V 77/2 lack of control. As previously explained, this is the kind of situation which = 379.9 psi, the third test criterion gives indication of 3O4. ECONOMIC CONTROL OF QUALITY often arises in which the indications of two criteria are not the same. Our decision in such a case involves the use of judgment. In this particular instance and for reasons outlined in the previous chapter, we choose the first type of control chart test corresponding to the rectangular limits of Fig. Ioff. With the above discussion as an introduction, we shall now describe criteria which have been found to work successfully in the detection of lack of control. 4. Criterion ſ—General Given a set of n data to determine whether or not they came from a constant system of causes, we take the following steps: A. Divide the n data into m rational subgroups" of m1, m2, ..., ni, . . . , nº values each. B. For each statistic to be used, use estimates 6 and go satisfying as nearly as possible conditions A and B of Para- graph I. C. Construct control charts with limits 6-- 30 e for each statistic. D. If an observed point falls outside the limits of this chart, take this fact as an indication of trouble or lack of control. 5. Criterion I–Attributes In this case we make use of a control chart with limits p = 30p where p is the fraction defective in the total set of n observations and - - op = AP 4, 77 where 7 is the average sample size. The lower limit is taken as zero if p – 30 p < O. * Note in Fig. 55 the difficulties encountered if the data are not divided into rational subgroups. DETECTION OF LACK OF CONTROL 305 Example: Carrying out these computations for the Type A data of Table I, we get the following results: Month p = * Month p = * Month p = * 77 71 77 January. . . . . . . o.OO76 || May. . . . . . . . . O. O3OI September. . . . . o. oO48 February. . . . . . o.OO82 || June. . . . . . . . . o.OO60 || October. . . . . . o. oºz80 March . . . . . . . . o, or 17 | July. . . . . . . . . o.oo76 | November. . . . . o.o.112 April. . . . . . . . . . o.Ooso | August. . . . . . . o.OO5I December. . . . . o. oOSg * 2n, p = X, = O.OIO9, P g – p + 30 p = O.O25o, 7 –30 p = – O.OO32 (hence taken to be O.OOOO). With this information we get the control chart of Fig. 4-a. The fact that points fall outside the limits was taken as in- dicating the presence of assignable causes of variability, at least some of which were later discovered, thus justifying the indi- cation of trouble given by the test. 6. Criterion I–Variables—Large Samples Given a series of n observed values X1, X2, . . . , Xi, . . . Xn divisible into m rational subgroups of ni, n2, ..., ni, . . mm values each, we make use of control charts with limits X+ 30x and g + 30, where X is the average of the n observed values and _ |ni a 1° + m2 a.2° -- . . . -- nº dº + . . . -- nºm am” O = 5 (91). 711 –H 712 –H . . . —H 717 –H . . . -- 71m 3. TABLE 43.−TYPICAL APPLICATION OF CRITE RION I Frequency Cell Midpoints - July Aug. Sept Oct Nov. Dec Jan. Feb. Mar. Apr. May June T 5.5 ! . . . . . . . . . I 2 I I | . . . . . . . . . I 2 4 I - 5 - O | . . . . . . . . . . . . . . . . . . . I | . . . . . . . . . . . . . . . . . . . I I 3 2 I T 4.5 ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I 7 T 4. O | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I 2 | . . . . . . . . . . . . . . . . . . . 5 I 2 7 5 9 T 3.5 ! . . . . . . . . . . . . . . . . . . . 3 I I | . . . . . . . . . 5 IO 24 I9 I 5 24 –3. O 55 IO I I 2 59 49 48 52 I67 I 30 I IG II 6 — 2.5 I4I 90 I IQ 99 I 57 I 52 I 37 I 25 22 I I68 I46 2O6 – 2 ... O I68 238 238 I7 I I79 249 I 77 I95 239 I57 I 7 I 2 IS – I 5 249 265 2 I 3 3I 2 3O2 359 320 330 28I 24.I 237 322 — I . O 305 335 332 366 327 4I4 285 309 254 2 I 5 243 318 — O. 5 23 I 3 I 3 238 234 IóI II 7 I62 I40 I 34 I 32 ISO I53 O. O 64 46 3 3 II 9 I 3 27 49 I OO IO6 79 O. 5 26 2 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I 2 4. 24 IO 8 I. O 9 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I I 4 I. S 2 X I,250 I,300 I, I 50 I,2OO I,2OO I,350 I, ISO I,200 I,400 I,200 I,2OO I,450 Frequency Distributions for Data of Twelve Polygons of Fig. IQ July Aug. Sept. Oct. Nov. Dec. Jan. Feb. Mar. Apr. May June Total Average X. . . . . . . . . . . – I. 298 || – I. 250 | – I. 368 || – I. 325 | – I. 504 || – I. 512 ||— I. 490 || – I. 505 || – I. 765 |- I. 550 |- I. 501 || – I. 577 |- I. 475 Std. Dev. a . . . . . . . . . . . O. 829 o. 672 O. 673 O. 623 O. 7 I 3 o. 638 O. 7 IO O. 754 O. 923 O. 985 O. 92 I O. 862 o. 786 . Skewness k . . . . . . . . . . . – o. oog | – o .439 – o. 785 |—o. 770 ||—o. 541 ||—o. 490 —o. 573 – O. 717 – O. 353 |- O. 993 |- O. I91 – O. I 92 |- o. 424 Flatness 82. . . . . . . . . . . 2. 729 3. 287 4.854 4. 208 3. I43 3. O25 3. 673 4. 566 3. 33 I 2. 637 2. 390 2.5 IQ 3. 40 I Sample Size n . . . . . . . . . I, 25 O I, 3OO I, ISO I,2OO I,2OO I,350 I, ISO I,2OO I,400 I, 2 OO I,2OO I,450 Statistics for above Frequency Distributions * tº e tº rz = - I. 408 * c = = − = O. O.22204 X control limits {: + 30; I. 4 v/i X – 30 x = -1.541 Gr ſº e or + 3a or = O. 833 o or = —— = O. or 569 a control limits * 'V2; a – 30 or = 0. 739 • E. 6 = * * { k + 30 k = –0.217 Oſk. 75 O. O69 I 7 I k control limits k – 30 k = – 9.632 = 3.8 I6 := 24 = & vº {; + 306, - 3. ° 32 # O. I 38343 62 control limits B2 – 3082 = 2.986 * 7 = 1,254 or average sample size for I 2 months. DETECTION OF LACK OF CONTROL 3O7 as being the standard deviation of the ith rational subgroup. If the sizes of the subgroups are practically equal, we have O’ a F = − and 0. = * X Vºn Oſ ^/ 277 If the sizes of the subgroups are not equal, the limits for a given subgroup i must be made to depend upon the sample size n: for that group. Obviously, the condition that the statistical limit Zs O = OT 71—200 is not satisfied when mi is small. It seems reasonable to believe in the light of our previous discussion of the distribution func- tion of the standard deviation that, so long as the minimum size of a subgroup does not fall below, let us say, twenty-five, the estimate a given by (91) approximately satisfies this limit con- dition. If the rational subgroups contain a large number of observa- tions, we may also make use of control charts for the skewness k and flatness 82. Example 1: Table 43 gives the observed frequency distri- butions and the control limits for the twelve monthly records of quality shown previously in Fig. 19. Fig. IO7 shows the results in graphical form. The fact that some of the points fell outside control limits was taken as an indication of lack of control for which the assignable causes were later discovered. Example 2: Let us apply Criterion I to the data of Fig. 21 to determine whether or not there is any indication that the depth of penetration for the seven treating plants is controlled. The requisite computations are given in Table 44. In this case the sample sizes are too small to justify the use of k and 32 and the sizes differ so much among themselves that it is necessary to use variable limits as shown in Fig. Io9. Lack of control, the causes of which were later discovered, is indicated by both the averages and correlation coefficients. 3. TABLE 44.—TYPICAL APPLICATION of CRITERION I To CoRRELATION CoEFFICIENT X = Depth of Sapwood in inches Y = Depth of Penetration in inches Number of Number of - – Company Poles Borings A 0 x Y a y 7 xy I 48 35O 3.56II o. 6060 I .8966 o. 6326 O. 2597 2. 5O 239 3. I 552 o. 6922 2. O795 O. 709.I O. 44O3 3 5O 316 2.8959 o. 6667 I. 7016 O. 5925 O. 49 IG 4. 47 323 3.3963 O. 7O93 2.0653 O. 7I 53 o. I 584 5 48 346 3.6Lo7 O. 5935 I. 9642 o. 6865 —o. 1815 6 5O 24. I 3.4OI 2 o. 5987 2. O32O o. 75.46 o. 4181 7 5O 346 3. I 85o o. 6385 I.6832 o. 6563 o. 3855 Total 343 216I 3.3242 o. 6863 I. 9053 o. 691 I o. 3926 7x = o .6436 ày = o. 6781 | FX = o 2817 * 5x = o. 6422 *ā-y = o 6736 | *rzy = 6. 2656 Statistics for Data of Fig. 2I Calculation of Correlation Control Chart—Company ſt: or r = * Weighted Average. Hº - H (o. 2597)* vºn – 1 V 349 Control me O. O.5o r −F 32 r = o. 415 r — 32 r = O. II.7 f Limits for other six companies calculated as for Company I. DETECTION OF LACK OF CONTROL 3O9 Example 3: As a third example, let us apply Criterion I to a set of data which may reasonably be assumed to be controlled and see if the result of the test is consistent. For this purpose, we may make use of the four observed distributions of 1,000 given in Table 23. Since these data were obtained under con- ditions as nearly controlled as we may reasonably hope to attain, all observed points should fall within the limits. Fig. Io9 - 1,2O F- 1. of O b -130F O © 3 | ,OO H. O |>< O F O Q tº "“F ––––– ... 0.90 F O º- - mºsº ; w-–––––––––––. < 3.2 - T ~~~~ T ~~~~~ ul O.65 H. 0. \e O V Ull O > º: < O © 3. O H. ' z O.60 H. - © < O H. : C (ſ) N. .” e’ 2.8 L o.55 L * N.” DEPTH of PENETRATION IN INCHES - 2 \, 2 S < 2 O N. 2.4 r- O.75 T- 2^ --~~~~ * ... O bº O 2.2 H. Z O.7O H. IX- O O H. < 2, O H. — — —- TY" - U O.65 H. Dr. O O ul —a- O > ºr CC O < < % | 8 H T ~ -- - * - * = ** ~ J.-- ~~ 3 o'cof N. 9–– — - ~ 2^ (ſ) NS-T ^,” O O - !. 6 L O.5.5 L l l | t t ! l f t—l I I 1 | | _l O | 2 3 4 5 6 7 O | 2 4 5 6 7 N C O M P A N | E S - C O M P A N | E S PENETRATION VS. SAP WOOD O. 6 I- (- O 2 -e `-- _- T- iſ o.4 H T - — – T e > Q O u- ti- Lul wºr O O O. 2 H Z © 3 ~~~~---~~~~~ * Li J O H. Cº. O. O O O u - O.2 L 1 I l I | —1 —l O | 2 3 4 6 7 C O M P A N E S FIG. Io9.—CoNTRol CHARTs for DATA of TABLE 44 AND FIG. 21. DETECTION OF LACK OF CONTROL 3II constitute what we shall term the Criterion I test for small samples, where - - - — ` x – 21 + x2+. ... + Xi + . . . -- Xm 777 O 1 + q2 + . . . –H 0.7 -H . . . -- 0 m 772 == Oſ – C2-OF In these expressions co is the factor given in Table 29, Xi is the average, and g; the standard deviation of the ith subgroup. O.2O 1.20 T- ſ \O ºmº ºmnº mºm ºmºmºsºme - Z O. I O – tº-º ºm- wº- O . I O H O < tal - O * : O () OC |-mm- ºmº sº-ºº ºmºs º-e ºsmºs º ºsmºsºms - <ſ *- *-* * *mm ammº ºmm amºr smºs-me < i - O || O H. C, O.9 O H. 2. . o.20 F §3.5°E - * * * * co, *ms sºmº sºme (ſ) Uſ) O (ſ) u] (ſ) O U > : Lil O < yme sºme sº sºme ºme sº sºme ºne ºmº —º— —- E ºsmºs smºº sºme tºº smºº gº tº dº º -. º 'LLJ < O U) > $ 30,000 E # 2,000 H. É Ç O 3. O # U O }* = ** = * * = * sºme sºme sm-m. 9 1,000 | © «[ % t 25,000 L O O mº sºme mºm sºme me mºme me mºmº ºmºmº em me I | | f I I ſ I | C D G W S C S COMPAN } E S COMPANIES FIG. I. Io.—CoNTRol CHART For SMALL SAMPLEs SHowING LAck of ContROL. averages shown in Fig. 7-a and that for standard deviations shown in Fig. III indicate lack of control. As was pointed out in Part I the causes for lack of control were found and removed. The reader may question why the original data were grouped into subsamples of four instead of some other number. A little 800 r- © \o O à Soo -- - - - - – – — — — — — — — — — — — — — — — — — — — — º O O O > O § 400 He « » O O © 2 © —& O © Aſºº • , * jº. O g 200 © * e • ". O C e * e e O : © e º e O O O e * * * . (ſ) ° -- – - – – – — — — — — — — — — — — — — — — — — — — 1–1 | | | | | | | | i | O 5 |O |5 2O 25 3O 35 4.O 45 5C) 55 SA M P L E NUM BER FIG. III.-CoNTROL CHART FOR STANDARD DEVIATIONS OF SAMPLEs of Four— DATA of TABLE 2. consideration will show that there is nothing sacred about the number four although there are several reasons why it may be the most satisfactory when there is no a priori knowledge to justify any other sample size. 3I4 ECONOMIC CONTROL OF QUALITY Obviously, if the cause system is changing, the sample size should be as small as possible so that the averages of samples do not mask the changes. In fact single observations would be the most sensitive to such changes. Why then do we not use a sample size of unity? The answer is that if we do, we are faced with the difficulty of choosing the standard deviation to be used in the control charts. Of course, we might use the standard deviation a of the entire group of observations but, in doing so, we would find that a = 465.21, a value distinctly larger than that of º = 328.26. A little consideration will sº show that, in general, this condition will occur in the long run whenever the cause system is not constant in respect to the expected value X, although the expected values of g and a C2 are equal when there is no change in the cause system. Thus the test in which we would use the standard deviation a of the whole group of n observations is not so sensitive, in general, as the one proposed in which we divide the data into small sub- groups in the order in which they were taken. In fact, the sensitivity of the test will increase, in general, with decrease in subsample size until the size of the sample is such that the data in any given subgroup come from a constant system of chance causes. In the absence of any a priori information making it possible to divide the data into rational subgroups, there would be some advantage therefore in reducing the sub- sample size to unity. To do so, however, would obviously defeat our purpose since we could not then obtain an estimate a to use in the control charts. Hence we must choose some sub- sample size greater than unity. Sizes 2 and 3 offer some dif- ficulties in the way of computation of a and so we go to a sample of four. m Now we are in a position to see how important it is to record the data in the order in which they were taken when we have no a priori basis for dividing the data into rational subgroups. If this is not done, there would obviously be no sense in trying to apply Criterion I. DETECTION OF LACK OF CONTROL 3I 5 8. Use of Criterion ſ—Some Comments In the practical application of Criterion I, particularly in the case of small samples, certain questions arise. One of these is: How many subgroups of four must we have before we are justified in using Criterion I? That this question is important is at once apparent because the expected probability of a statistic falling within the ranges established by Criterion I approaches the economic limiting value only as the total number m of observations approaches infinity. This difference in expected probability, however, even for two subsamples of four is likely less than O.O2 and certainly less than o.os. Hence, the effect in the long run of using Criterion I when the total number of observations is small is to indicate lack of control falsely on an average of perhaps five times in Ioo trials instead of three times in, let us say, I,000 trials which it would do when the total number n is large. In almost every instance we can well afford to take this added precaution against Overlooking trouble when the total number of observations is small. It appears reasonable, therefore, that the criterion may be used even when we have only two subsamples of size not less than four. In this case, of course, we may wish to apply additional tests although, as we have already seen in the earlier part of this chapter, such tests will perhaps in the majority of cases give consistent results." The principal thing to be kept in mind is, however, that the main purpose of such a criterion is to detect lack of control in a continuous production process where we have a whole series of samples so that the question as to the minimum number of subsamples becomes of minor importance. We may also ask how the indications of Criterion I depend 1 In work not yet published, F. W. Winters has investigated the efficiency of this criterion for the case of small samples from two normal subgroups, assuming that the data have been divided objectively. In other words, he has determined the probability that the use of Criterion I with a given sample size will detect a difference of a given amount in the averages of two objective subgroups. For example, he has shown that the efficiency varies all the way from 4 per cent for a sample of four and an objective difference of or (the common standard deviation of the objective subgroups) to 97 per cent for a sample of twenty and an objective difference of 20. On the other hand the probability that this Criterion will lead us to look for trouble needlessly is, under the first condition, .oO85, and under the second, .oOOI4. 316 ECONOMIC CONTROL OF QUALITY upon the universe from which the sample is drawn, especially in the case of small samples. It will have been observed that the factor co used in setting limits for standard deviation is based upon the assumption that the samples are drawn from a normal universe whereas, in general, we know that this con- dition is not rigorously fulfilled. Furthermore, we have seen that the distribution function of both the average X and stand- ard deviation a of samples of a given size depends upon the nature of the universe. Hence, the probability associated with the limits in the control charts for the average X and standard deviation a depends upon the universes from which the samples were drawn. Of course, the distribution of averages, even for samples of four, is approximately normal independent of the universe so that the probabilities associated with control charts for averages are closely comparable irrespective of the nature of the uni- verses. This is not true, however, in respect to the distribution of standard deviations. We may get around this difficulty partly by using the control chart for the expected variance of the universe since, as we have seen, the expected value is related to the variance of the universe in a known manner. This makes it possible to establish the base line of the control chart for variance—something which cannot be done for the standard deviation unless the functional form of the universe is known. On the other hand, the stand- ard deviation of the variance involves the flatness 32 of the universe and hence cannot be estimated with great accuracy in most practical cases. Under these conditions, it seems reasonable to believe that comparatively little can be gained in most cases by making use of the variance instead of the standard deviation. In this connection, it is of interest to cite a typical instance of the way in which the control chart method, making use of averages and standard deviations for small samples, gives indications con- sistent with facts when we apply the test to samples of four drawn from either of the three types of universes previously described. For example, Fig. I 12 shows the results of the test DETECTION OF LACK OF CONTROL 317 NOR MA L UNIVERSE I.O H. O Q i O O WJ iſ ºl * ... O - O 1x f. OH O U O O § O O O O O O ſº —ſh. O * e a Or. J CŞ © O © e O ul O > O O “l O O O 1x I.O H. O O Liſ O O —ur- U O O • e " o TU O > O O -- of O H — — — — — — — — — — — — — — — — — — — — \O N OR MAL UN | V E R S E O 2.O = -me mºm amº memese amms smºs- amme - = ** - - - - - sºme º sº º-º or Z < O O O O O 9 F 1.0 O O O O e e #3: wº z-e- O ſh - O H > O O e ‘’ O • * 69 (ſ) uſ O O *- -ºº º- E=- --> * * *ms em- -mm mm-- * *-* = * * *m- ºmmiss me \O 2.O RECT ANG U LA R UNIVERSE O *m- -ºmºs º- ºmmemº -ºs º- *m' - smº- -ºmº - mºm -m-, *-* *mme -ºms amº *- Or Z < O 9 F 1 - O O © O O O © 2 < º Q O 4- A O ſh A * L O e ‘’ O O (ſ) ul O O tº- - - - - - - * = *me mºnº amus mmma emme me am- -ms sºme \O 2.O R! G HT TRIA N G U L A R U NIVERSE O * * *-* * * *-* * * * * * *m, memº ºme sºme mºsºme mºm mºme oz Z O O $9 O O O O C O O © O (ſ) uſ O O Cl *- == -- - - - * = - * = * * = * *mm sm. 1–1 I I f | 1 ! s 1 I ! 1 I I i t ſ 1–1 f I I I f ! O 5 | O | 5 2 O 25 S A M P L E FIG. I.12.-CRITERION I APPLIED to 25 SAMPLEs of 4 FROM EACH of 3 UNIVERSEs— VALUES IN TERMS OF or of THEIR RESPECTIVE UNIVERSE. 3.18 ECONOMIC CONTROL OF QUALITY applied to twenty-five samples of four from each of the three experimental universes. In each case all of the points are within the limits as we should expect them to be under the controlled conditions supposed to exist in drawing these samples. The results of the test are obviously consistent with the facts assumed a priori to be true in this particular instance. 9. Criterion II We shall close this chapter with a description of another criterion and illustrate its use by application to the 204 data of Table 2. Having the data divided into m subgroups of size n, |d Taº Table 45. If the ratio is greater than three, this fact is taken to indicate lack of control. We shall call this test Criterion II. This test provides a means of judging the nature of the conditions under which the sampling has been done. Thus, if all samples are produced by the same constant system of causes, or, in other words, if the sampling has been done in what we term Bernoulli fashion, then the expected value of d is zero. If, however, conditions change between each observa– tion of a subgroup but the same set of changes occur in the process of obtaining each subgroup of observations, then the expected value of d is greater than zero, and in such cases the sampling is said to be done in Poisson fashion. Or again, if conditions remain constant for any subgroup of observations but change in any one of a finite number of ways from subgroup to subgroup, then the expected value of d is less than zero and the sampling is said to be done in Lexian fashion. However, even though the sampling is actually done in Bernoulli fashion, the observed value of d may be positive, zero, or negative due to sampling fluctuations. Hence, we must have some way of judging when the deviations of d from zero are sufficiently great to indicate either a Poisson or a Lexian selection of samples. The standard deviation of d based upon Bernoulli sampling we calculate the ratio as indicated in the data sheet of DETECTION OF LACK OF CONTROL 3I9 TABLE 45.-DATA SHEET For CRITERION II—DATA of TABLE 2 e d Calculation of H- Od Number of observations N = 204 Size of subgroup n = 4 Number of subgroups m = 51 Sample Average X; X;2 Variance of” Number of Sample l of Sample I 4,43O. OOOO 19,624,900. Oooo I49,512.5oCO 4,372.5oCO 19,118,756.25oo 7,606. 25oo 3,827. 5000 I4,649,756.25oo 17,656. 2500 5 I 5, IOO.OOOO 26,OIO,OOO.oOOO II,25o. oOOO X 229,4O7. OOOO I,038, II 3,072. O700 4,832,876. Io;o Av. 4,498. I765 2O,355,275. 9229 94,762. 2766 777, X X;2 i = 1 R20 ax” = — X* = 20,355,275. 9229 – (4,498. I765)* 7)? ox” = I21,684.3586 772. 2 o';2 G = − = 04762.2766 O' = 94,762. 27 77 --> 777 d = O'” — max” = - 370,122.48 ro 77 — I 777 — I ºf -*- O'd = N 2(mm – I) *—Gº || = 29, Io'7.6083 m(m – 1)(n − 1) \n — I -*- |al 379,122.48to I2.71 57 O'd 29, Io'7.6083 - provides such a measure of significance. The formula for od was obtained upon the assumption that the samples had been drawn from a normal universe, in which case gº and ox.” are uncorrelated. If the universe is not normal, this formula for ad will not necessarily give the correct result, although from 32O ECONOMIC CONTROL OF OUALITY the viewpoint of detecting lack of control this simply means tº sº tº d . - that the probability that lº! will exceed 3 differs somewhat d from 99 per cent, or, in other words, we may on the average look for trouble a little more often or a little less often than one time in a hundred when it actually does not exist. CHAPTER XXI DETECTION OF LACK OF ContROL–CoNTINUED I. Introductory Statement In the previous chapter we considered the problem of detecting lack of constancy of a cause system or the presence of an assignable cause of Type I. In this chapter we shall consider the problem of detecting the presence of a predomi- nating cause or group of causes forming a part of a constant system. Such a cause will be referred to as an assignable cause of Type II. In the latter part of this chapter we shall consider what is perhaps the only available method for detecting the presence of assignable causes when the data are such that they cannot be grouped into rational subgroups and when no information is available other than the observed distribution. Assuming that the variable X satisfies the equation (58) of control, how can we detect the presence of a predominating cause or group of causes? As a basis for our consideration of this question, let us return to the picture of a constant system of chance causes presented in Part III. There we assume that such a system is composed of, let us say, multimate independent Call ISCS C1, C2, . . . , Ci, . . . . Cm, producing effects which compound linearly. It will be recalled that we do not presume to be able to describe any one of these m causes. The most that we can usually hope to do is to put our fingers on some secondary cause made up of several of the independent contributing causes. To make this point clear, let us think of the unknown group 32 I 322 ECONOMIC CONTROL OF QUALITY of m causes of error in some physical measurement such as that of the coefficient of expansion of a steel rod. Some of the secondary or macroscopic causes of error would be temper- ature fluctuations, non-homogenous heating of the rod, etc. Such a cause obviously includes a group of the elementary causes. We may represent this situation schematically as follows: C1, C2, . . . , Ci, Ci +1, Ci +2, . . . , Ci +j Macroscopic Cause Y . . Cm. } With this picture in mind, two methods of detecting the presence of an assignable group of causes suggest themselves. They are the well-known Method of Concomitant Variation and the Method of Differences of elementary logic. The first method is to vary the cause Y and see if we get an accompanying change in the resultant effect of the cause system. The other method is to remove the cause Y and observe whether or not the resultant effect is modified. In the general case where X is a chance or statistical variable subject to sampling fluctuation, the effect either of varying the cause Y or of removing this cause must be shown to be sig- nificant in the sense of being greater than can reasonably be attributed to sampling fluctuations in the variable X. It should be noted that both of these methods require that the analyst be successful in choosing the macroscopic cause which is findable in the objective sense. Hence, in the appli- cation of such a test, one must make full use of his powers of imagination, supposition, idealization, comparison and analogy in the utilization of all available data. 2. Criterion III Let us assume that we are to discover whether or not there is an assignable or predominating cause of variability in a variable X satisfying the equation of control, namely, dy = f(X, \1, M2, . . . , Mi, ..., \m')dX. (58) DETECTION OF LACK OF CONTROL 323 The application of Criterion III involves three steps: (A) Pick out some controlled variable Y which may or may not be an assignable cause of Type II. (B) Obtain n simultaneously observed pairs of values Xi Yi, X,Y2, . . . , X;Y., . . . , Xn Yn and determine the cor- relation coefficient r. (C) If r lies outside the limits V 77 — I take this fact as an indication that Y is an assignable Call ISC. If the correlation between Y and X is normal, we see that Criterion III indicates that there is a significant degree of commonness of causation or, in other words, that the observed correlation coefficient r is greater than can reasonably be attributed to sampling fluctuations where, as before, we choose sampling limits corresponding to three times the standard deviation of the statistic used in measuring the fluctuations. Since, as we have seen in Part III, there is reason to believe that the correlation between two controlled variables is at least approximately normal, we may assume that the positive in- dication of Criterion III is indicative of a significant degree of commonness of causation between the two variables, and to this extent Y may be considered to be in most cases an assign- able cause. From what has been said about the sampling fluctuations of the correlation coefficient, it is obvious that, if small samples are to be used, it is preferable to state the test in terms of the variable z given by (77). If z, as given by this equation, lies outside the range 3 –7–, V 77 – 3 the criterion is said to give a positive indication that Y is an assignable cause in the sense of our present discussion. So long as the sample size n does not exceed twenty-five, it is perhaps better to use the z transformation. O –– 324 ECONOMIC CONTROL OF QUALITY Example 1: We shall consider a case in which it is very desirable to control the hardness of a particular kind of appa- ratus. In this instance, each piece of apparatus consisted of two parts welded together, the materials for the two parts coming from different sources. Table 46 gives the hardness measurements on each of the two parts for fifty-nine pieces of this apparatus. Is there any evidence of the existence of an assignable group of causes of variability in hardness? TABLE 46.-HARDNEss MEASUREMENTs on WELDED PARTs Hardness Hardness Hardness Sample Sample Sample Number Part I | Part 2 Number Part I | Part 2 Number Part I | Part 2 I 5O. 9 || 44.3 2 I 48.7 || 36.8 4. I 47.9 || 36.7 2. 44.8 25.7 22 44.9 || 36.7 42 45.8 || 35.3 3 51.6 || 39.5 23 46.8 || 37. I 43 47.9 || 35. 5 4. 43.8 I9. 3 24 49.6 || 37.8 44 45.8 || 35. I 5 49. O || 43.2 25 5 I. 4 || 33.5 45 49. I 33.2 6 45.4 || 26.9 26 45.8 || 37.5 46 5o. O || 36. I 7 44.9 34.5 27 48.5 | 38.3 47 47. 3 || 35.9 8 49. O || 37, 4 28 46.2 || 30.7 48 46.9 || 35.2 9 53.4 38. I 29 49.5 33.9 49 49. I 38. I IO 48.5 | 33. O 3O 5o. 9 || 39.6 5O 48.2 | 35.9 I I 46. O 2.6 3 I 47.5 || 36.9 5 I 46.9 || 33.8 I 2 49. O || 35.4 32 45. O 37.5 2 49. O || 37.6 I3 43.4 || 36.2 33 46.6 32.4 53 44. 7 || 35. 5 I4. 44.4 || 32.5 34 48. O || 39.8 54 51. 7 || 36.2 I 5 46.6 || 31.5 35 44 - 5 || 35.3 55 45. 2 | 34.4 I6 5o. 4 || 38. I 36 48.5 | 38.3 56 44.8 || 27.5 17 45.9 || 35.2 37 46. O || 38. I 57 42. 4 || 3I. I I8 47. 3 || 33.4 38 48.9 || 35. O 58 48.5 || 36.8 I9 46.6 || 3o. 7 39 46.3 || 34.9 59 5O. I | 34.4 2O 47.3 || 36.8 4O 46. I 2.9 Now the only common source of causation was the heat treatment given the apparatus after the two parts were welded together. Hence the variability in the heat treatment might be an assignable cause. If it is, we should expect to find that the correlation coefficient r between the hardness measurements in Table 46 is significant in terms of Criterion III. DETECTION OF LACK OF CONTROL 325 Applying the test we find that the observed correlation r = O.513 lies outside the limits –– v/59 – 1 Hence we conclude that the heat treatment constitutes an assignable cause of variability in the hardness of the finished product. This conclusion has since been justified by further studies. O –– 3. Criterion IV Let us assume, as before, that the variable X satisfies the equation (58) of control. The application of Criterion IV involves the following steps: (A) Obtain n observations X1, X2, . . . , Xi, . . . , Xn of the variable X and calculate some statistic 6:1 for this set of n observed values. (B) Choose some variable Y which may or may not be an assignable cause and obtain n values of the variable X under a condition where it is known that the variable Y can in no way influence the variability in X. Making use of this new series of n observed values, determine the value of the statistic 6; and let us call this value 0:2. (C) If | 0.1 – 0.2 | * 37ea-ee, we take this fact as an indication that Y was an assignable Call ISC. For reasons which we have already considered, it is usually sufficient to make use of the two statistics, average and stand- ard deviation, in terms of which we say that Y is an assignable cause if either of the following inequalities is satisfied: |XI – X2 |> w; (a1% + o-2°) I |al -: go X. Nº. a 2%). 326 ECONOMIC CONTROL OF QUALITY Example: Fig. 113 shows the cross-sectional view of a com- mon type carbon transmitter. It is but natural to expect that the physical properties, such as resistance, efficiency, etc., of VARIATIONS INTRODUCED VARIATIONS INTRODUCED BY PIECEPARTS BY CARBON THREE CAUSES, OR GROUPS OF CAUSES, OF VARIATION 3 = \ſcº TG 3T G3 THIS DISTRIBUTION OF PRODUCT TRANSMITTERS IS, AS WE WOULD ExPECT, A SIMPLE COMBINATION OF THE THREE CAUSES OF WARIATION Fig. 113. this kind of instrument should be sensitive to slight variations in such factors as granular carbon, the elasticity, density, etc., of the piece-parts, and the details of assembly, such as tightness, DETECTION OF LACK OF CONTROL 327 with which the screws are set and the care with which the respective parts are centered. It is of interest to see how much influence each one of these three factors exerts upon the general variability of the qualities of the completed instrument. The method for investigating the influence of each factor immediately suggests itself—it is the use of Criterion IV. To apply this method we must eliminate the influence of all but one of the factors and study the resulting distribution of quality attributable to the remaining factor or constant system of chance causes. The results of such a study on one quality characteristic gave the three distributions shown in Fig. I 13, the standard deviations of which were 01 for piece- parts, 02 for assembly, and g8 for carbon. If a represents the standard deviation in quality of the completed instrument in a sample of n and a 1.23 = standard deviation in samples of n when piece-part variations are eliminated, a 2.13 = standard deviation in samples of n when assembly vari- ations are eliminated, and a 3.12 = standard deviation in samples of n when carbon varia- tions are eliminated, then the application of Criterion IV to standard deviations states that piece-parts, assembly and carbon represent assign- able groups of causes if I * | g – on 23 |> A. (0^+ or 1.23), I |a = 0.2.13 |> v. (a” + a”2.13), and I | O' – O'3.12 | > w; (a” + o”3.12). In this case it was found that each of these three inequalities was satisfied and hence we conclude that all three factors actually represent assignable cause groups of variation. 328 ECONOMIC CONTROL OF QUALITY Furthermore, since the value of q is approximately given by Var* -- gº + agº, we conclude that these three groups operate independently and contribute practically the entire amount of variability observed in the completed instrument. 4. Criterion / Oftentimes the observed data are given in a form such that no one of the four previously described criteria can be used. 3,OOO O OBSERVED DISTRIBUTION — SEcond APPROxiMATION $2 2,500|- Z LJ X D 2,000 H. OC H. (ſ) # 1,500- Li- O o: 1,000H Ull CD 3. # 5ooH O l- | 7 8 9 |O | | 12 |3 14 15 E F F | CIENCY FIG. I 14.—Is THERE ANY INDICATION OF LAck of ContRol? CRITERION V ANswers: “YES.” As a specific illustration we may consider the observed fre- quency distribution of efficiency of 7,686 pieces of one kind of apparatus represented by the black dots in Fig. I 14. Is there any indication of lack of control? The instruments in this group had come to the central testing laboratory from eight different shops. The measure- ments when submitted for analysis, however, had been grouped together, giving the frequency distribution of Fig. I I4 and the fourth column of the upper half of Table 47. * The method of detecting lack of control in this case is as follows: DETECTION OF LACK OF CONTROL 329 A. Calculate the average X, the standard deviation a, and the skewness k from the n observations and use these in the expression 1 QC2 al 3 ac2 I k/w I & sºme sº- - -(-; e 27°dx. ri O V 27" 2. \O. 3 o' to calculate the theoretical frequencies yol, ye2, . . . , yem for the m cell intervals into which the original data have been grouped, it being understood that x = X – X. B. Calculate 777, x2 = X (yi — ye)”. * = 1 Wei C. Read from the curves 2 of Fig. 73 the probability P of obtaining a value of x” as large as or larger than that observed, where the number of degrees of freedom is taken as four less than the number m of cells. - D. If the probability P is less than o.oO1, take this fact as an indication of lack of control. Example Z. The details of the application of this criterion to the data of Fig. I I4 are shown above in the data sheet of Table 47. It will be noted that Sheppard’s corrections are used in this case. The smooth solid curve of Fig. I I4 appears to fit the observed points very well indeed. However, Criterion V detects what the eye does not see. In accordance with the con- ditions of Criterion V, we conclude upon the basis of its appli- cation that the quality of this product was not controlled. Although the observations originally presented were grouped together without reference to the shops from which they came, it later became possible to subdivide the data upon this basis. Definite evidence of lack of constancy of the cause system was thus revealed by the control chart of Fig. I 15, and the assignable causes of variability were found. In other words, the indication of Criterion V was correct. * This is the second approximation already referred to in Parts II and III. The theoretical frequencies may be calculated with the aid of Tables A and B. * More extensive tables of P(x”) are given by K. Pearson in his Tables for Stat- isticians and Biometricians. 33O ECONOMIC CONTROL OF QUALITY TABLE 47.-DATA SHEET For CRITERION V. INSPECTION ENGINEERING ANALYSIS SHEET MBC Checked MSH 6 2 y2K yr” yx" y-ye ty-ye.” (7-ya) , m = units per cell 32O2] - e Moments about sº 7686 —4.166146. origin o' 14ló59 = 18. 450785 The origin ºf {: the . - —4%—- 86,047439 —-31918? tº: point value 7686 - ,986.050 of cell no. oy —9248299– 2l.9 9 3.142229 4. = —sº- - 421.974889. - — ;-- P = Uncorrected 142 191 tnoments about 3 arithmotio |15 l!"3 '919.1 192 4. *111 E. mean X u4 * 194 - 411 118 ° 91.11%lue - Slui" " Corrected u2(oor.) = u2 - 0.083ss3 = —l,0740ll = .0833.33 = .990678 moments about X (sheppard's 14(oor.) u.4 - 0.5 us • 0.0291.87 Corrections) : r...'Y' (z/9) º X 2. Boun F(z) f(z) kf(z) F(z) + kf(z)| Diff. Freq. DETECTION OF LACK OF CONTROL 33 I Example 2: In the development of methods of preserving telephone poles, it is of interest to know the distribution of thickness of sapwood to be expected for poles of a given kind and to know whether or not this quality of poles is controlled. Early in this study a set of 1,528 measurements of depth of sapwood on as many chestnut poles became available, although at that time it was not possible to divide this set of data into rational subgroups. 3F- E-se m-m-, sºme aumne ammammº ºm-m *=sse sº-sº sº-ºs º-se mammº sº-º ºss ºne º smº- ºr- FIG. I 15.-FURTHER Evide NCE OF LACK OF CONTROL For DATA of FIG. II.4. The observed and theoretical distributions of depth of sapwood are shown by the black dots and the smooth curve of Fig. I 16. The probability P of obtaining a value of x” as large as or larger than that observed is much less than O.OOI. Hence a search for assignable causes was begun and the following three were found: (a) The men who made the measurements favored even numbers. (b) The thickness of sapwood was determined from borings, and no allowance was made for shrinkage of these during the time that the measurements were being taken. (c) The expected thickness of sapwood was found to depend upon whether the poles had come from one or the other of the 332 ECONOMIC CONTROL OF QUALITY slopes of a mountain range. In the sample of 1,528 poles some had come from one slope and some from another. 360 F © OBSERVED DISTRIBUTION O - SECOND APPROX|MATION 3OO }- (ſ) º a 240 0. u- O or 180 F Lul CD > ID 120 H. Z 6O - O l I ſt L l l l l | ( ) 1– 5 L ſ O 2. 4 6 8 |O | 2 14 I6 DEPTH OF SAP WOOD IN #INCH FIG. I. 16.-CRITERION V INDICATES THE PRESENCE OF Assign ABLE CAUSEs. Example 3: In the initial stages of the production of a kind of equipment for which electrical resistance was an important quality characteristic, the observed frequency distribution was that given by the dots in Fig. I 17. The application of Cri- 25 O 2OO 15O |OO 5 O O OBSERVED DISTR) BUTION O – SECOND APPROX IMATION Q * I * ſ l | l | I | - I º 31.25 38.75 46.25 53,75 61,25 68.75 RES STANCE FIG. I.17.—CRITERION V GIves Positive TEST, INDICATING TRouBLE. DETECTION OF LACK OF CONTROL 333 terion V indicated the presence of assignable causes in that the probability of occurrence of a value of x” as large as or larger than that observed was much less than o.oO1. Further investigation revealed that assignable causes had entered the production process and affected the resistance of a small group of the instruments in the original lot. After the measurements for this small group had been separated from the others, the resultant distribution was found to be that given in Fig. I 18. Criterion V applied to this resultant dis- 25 Or e OBSERVED DISTRIBUTION — SECOND APPROX1 MATION 2OO H. (ſ) }- Z ; =5 |50 H. OC }- (ſ) 2 u- O | OO H. OC L] CD > T) 2 5O H. O L | | 31.25 38.75 4.6.25 53.75 61.25 R E S J S T A N C E FIG. I 18.-TRouBLE REMoved—CRITERION V GIVEs NEGATIVE TEst. tribution gave a negative test, indicating that the trouble had been removed. 5. Criticism of Criterion / In the first place the test is based upon the use of a particular frequency function, viz., the second approximation. Are we justified in assuming that quality free from assignable causes is always distributed in accord with this statistical law or frequency distribution? Is it necessary and sufficient to show that the quality of a product differs no more from a second § TABLE 48.-CHI SquaRE For EACH of THE Four DISTRIBUTIONs of SAMPLEs of 1,000 EACH FROM THE NoFMAL UNIVERSE Using OBSERVED ESTIMATES For PARAMETERs Sample No. 1 Sample No. 2 Sample No. 3 Sample No. 4 Theoretical Observed (2 Tºe)" Theoreticall Observed (2 Tºe)" Theoreticall Observedl (2 T2e). Theoretical lobserved ('Tºe)" J'e y J'0 J'e y J'e J'0 y J'0 J'e y J'e 3 5 I .333 2 I O. 500 4. 2. I . OOO 2. 2. O. OOO IO 9 O. IOO 8 I4 4.5oo IO IO O . OOO 7 9 o. 57 I 29 36 I.690 25 2. O. O4O 3O 29 O. O33 23 25 O. I 74 66 55 I. 833 61 5 I I.639 68 72 O. 235 59 49 I.695 I 2. I I23 O. O33 II6 II3 o.O78 I 22, I 24 O. O33 II6 II 2. o. 138 I73 I65 o. 370 172 187 I .308 172 I8I O. 47 I 176 I9 I 1.278 I95 2O3 o. 328 I99 I95 o. obo I93 I8o o. 876 2O5 2O4. o.OO5 I73 172 o.oob I79 176 o.O5o 172 I69 O. OS2 183 I82 o.OO5 I2 I I23 O. O33 126 I25 o.oob I 2C) I 2C) O. OOO I26 I23 o.O7I 67 68 o.OI 5 69 71 o.o.58 67 67 O. OOO 66 64 o.O61 29 3I o. I 38 3O 3I O. O33 29 32 o. 3 Io 2 25 o. I48 IO 8 O. 4OO IO 8 O. 4OO IO I I O. IOO 8 I 2. 2 . OOO 3 2. O. 333 4. 4. O . OOO 4. 3 O.25o 2. 2. O. OOO X” 6.612 8.694 3.360 6. I46 DETECTION OF LACK OF CONTROL 335 approximation curve than may be attributed to sampling fluctuations? In Part III it was shown that there is no such known neces- sary and sufficient condition for control. However, it was shown that, for a very wide range of constant systems of chance causes, the second approximation is approached as we approach the theoretical conditions of maximum control although no fre- quency function is a sufficient, even though it be a necessary, condition for maximum control unless it be known a priori that the chance cause system is constant. Now let us consider the use made of the Chi Square test in this criterion. Let us assume for the sake of argument that it is necessary and sufficient to show that the distribution function is the second approximation in order to show that the cause system is free from assignable causes. In this case can we rely upon the Chi Square test to detect the presence of assignable causes when the theoretical distribution is calculated from the second approximation using estimates of the three parameters derived from the observed data? We have seen how the Chi Square test works when the dis- tribution function is known a priori, both as to functional form and the values of the parameters. The question now to be considered is: How will it work, if we know a priori the func- tional form but not the parameters? To make the problem specific, let us consider the four dis- tributions of 1,000 observations each from the normal universe previously used to illustrate the use of the Chi Square test when the true distribution y1, y2, . . . , yi, . . . , ym into m cells is known a priori. Now, however, let us calculate theoretical distributions for each of the four samples of 1,000 by using the observed values of the averages and standard deviations in the normal function. Table 48 gives the four distributions derived in this way together with the calculated values of x*, using a thirteen-cell grouping. It is of interest to compare the observed values of x” in Table 48 with those previously calculated for the same four samples of 1,000 making use of the a priori known distribution, 336 ECONOMIC CONTROL OF QUALITY Table 36. These two sets of values are shown in columns 2 and 3 of Table 49. The average x” in the third column is TABLE 49.-OBSERVED VALUES of x* Chi Square Probability Sample Theoretical Distribution Number A Priori Theoretical A Priori Known Distribution Known * Distribution Distribution | 12 Degrees | Io Degrees of Freedom | of Freedom I 6.74I 6.612 o. 873 o. 88o o. 760 2 Io. 716 8.694 O. 554 o. 728 o. 562 3 4. 455 3.360 O. 972 O. 99 I o. 969 4. 9. I 74 6. I46 o. 688 o. 908 o. 8o0. Average. . . . . 7.772 6.2O3 o. 772 o,877 o. 773 definitely less than that in the second, and the average prob- ability calculated for the values of x” from the theoretical frequencies is o.877 as compared with the average of O.772 cor- responding to the chi squares computed from the known a priori frequencies. A little consideration shows that in the calculation of x2 from theoretical frequencies, we must make allowance for the fact that estimates of parameters are used instead of true values. We see that, when the a priori cell frequencies yi, y2, . . . , yi, . . . , ym are known, the only restriction on the observable cell frequencies y1, y2, . . . , yi, . . . , yn is that y1 + y2 + . . . -- yi + . . . -- yn = n. In other words, the set of m variables (yi — yi), (i = 1, 2, ..., m), has m — I degrees of freedom. Obviously, however, the set of m variables (yi — yet) has m – 3 degrees of freedom because we have three conditions imposed upon the possible cell frequen- cies, viz., 2y: = n 77 F. ) 2 yixi = n X, Xy;(X; – X)* = na”. DETECTION OF LACK OF CONTROL 337 When we make allowance for the loss of three degrees of freedom instead of one, we get the probabilities in the sixth column of Table 49, the average of which is O.773 as compared with the corresponding average o.772 for the values of x” calculated from the a priori known cell frequencies. This close check should strengthen our faith in the usefulness of the x” test when the functional form is known a priori and the parameters are estimated from the data. We must consider briefly certain other characteristics of the Chi Square test. Obviously the total number of observations must be large before we can apply the test, particularly when the parameters in the frequency function must be estimated from the observed data. In quality control work we seldom try to use Criterion V unless the sample size n is at least 1,000. When the sample size is very large, it becomes important that the method of estimating the parameters in the theoretical frequency distribution is such that the statistical limit I Ls ...(y. - ye;) = O CO is satisfied. Otherwise the observed value of x” as n is increased indefinitely will always approach infinity even though the quality is controlled in accord with the assumed functional dis- tribution. Enough has been said to indicate the nature of some of the limitations to be placed upon the use of the Chi Square test involved in Criterion V. Thus we see that Criterion V is a far less satisfactory test than Criterion I where the latter can be applied. We see that Criterion V in practice will usually give indication of the presence of assignable causes even though the product is con- trolled, unless the objective distribution is rigorously given by the second approximation. The criterion likely errs on the side of indicating trouble when it does not exist, although this is not a serious handicap in most industries until the state of control has been practically reached. By such time a producer will generally have set up his inspection practices so that his 338 ECONOMIC CONTROL OF QUALITY data are divided into rational subgroups and Criterion I may be applied. 6. Rôle of judgment in Choice of Criteria Even though, in general, an engineer need not go beyond the use of the five criteria previously described, certain excep- tions may arise. Such a case is shown in Fig. I 19 in which we have a control chart for averages of samples of four supposed to have been drawn from a normal universe in the order plotted. Would you conclude that the cause system is constant because Criterion I is satisfied? Almost anyone will answer this question in the negative. The probability of getting from a controlled system twenty-five samples with averages decreasing from sample to sample is so exceedingly small compared with the probability of getting twenty-five samples not so ordered 2 r- — — — — — — — — — — — — — — — — — — — — — OC # | * O $2 O Q O § * e o e e e e O © ſº Aº. tº o Tu Te O • e <{ O CC © e Ll O § – I H O <ſ – 2 L L I l | 1–1– l | | ſ H l | | | l | | | | | l I ! I | O 5 |O 15 - 2O 25 D A YS FIG. I.19.-A CASE where JUDGMENT IS REQUIRED. as to suggest the presence of an assignable cause or trend. Here is a case then where common sense suggests the use of a criterion other than one of the five. As another example of a situation requiring Judgment in the use of criteria, let us consider again the distribution of successes in 4,096 throws of twelve dice where the throw of a 1, 2, or 3 is to be considered a success. A manufacturer of these dice might reasonably have wished to produce dice which are not biased. In such a case the distribution of successes, Column 2 DETECTION OF LACK OF CONTROL 339 of Table 50, should not differ from that given by the successive terms of the point binomial, 4,096 (3 + #)” by more than may be attributed to sampling fluctuations. Would he conclude that the discrepancy between the theoretical and observed dis- tributions indicates bias? To answer this question he might TABLE 50.-Does THE DISCREPANCY BEtween THEORETICAL AND OBSERVED DISTRIBUTION INDICATE BIAs? Number Theoretical Number Theoretical Observed Observed of Frequenc Frequency of Frequenc Frequency Successes Q y 4096 (# +})*|| Successes Q y 4096 (# +})* O O I 7 847 792 I 7 I Q. 8 536 495 2 6o 66 9 257 22O 3 I98 22O IO 71 66 4. 43O 495 I I I I I 2 5 731 792 I 2 O I 6 948 924 apply Criterion V. Doing so, he would get a probability of fit of o.oO15. Since this probability exceeds the value o.ool set as a limit in the statement of Criterion V, he would be supposed to conclude that the product was controlled in the sense that it did not show a significant bias from the a priori standard. If, however, we compare the graph of the smooth curve through the frequencies determined from the binomial expan- sion, Fig. I2O, with the observed values, we see that the smooth curve appears to be shifted to the left. Instead of using Criterion V, we might have compared the observed fraction p = 0.512 of success with the expected value o,500 upon the basis of the assumption of no bias. We might take the occurrence of a value of p outside the range o.5oo +30p, where op is the standard deviation of p in samples of size n, as being significant. In this case 34O ECONOMIC CONTROL OF QUALITY Hence this test indicates control as did Criterion V, because the observed value p = 0.512 is well within the limits of o,500 + 3(o.or 58). Thus both tests indicate control. It is left as an exercise for the reader to calculate the theo- retical distribution upon the assumption that the dice were biased so that the probability of success is the observed value O.512. He will find that the probability of fit is thus remarkably | OOOf- O OBSERVED DISTRIBUTION 9 OO H. - 4 O 96 (4 +4)* Bog 7OO 6OO i NUMBER OF SUCCESSES FIG. I2O.--THE FACT THAT THE SMooth THEORETICAL CURVE APPEARs to BE SHIFTED TO THE LEFT SUGGESTs LAck of ContRol Even THOUGH CRITERIon V GIVEs NEGATIVE TEST. improved, and that the differences between observed and cor- responding theoretical cell frequencies show a mixture of signs as they should. In this case he will find that the observed results are more likely on the assumption of bias than on that of no bias. Most likely his judgment will lead him to accept the hypothesis that the dice are biased. DETECTION OF LACK OF CONTROL 34 I 7. Sampling Inspection in Relation to Contro/–Attributes We are now in a position to consider the significance of control in relation to sampling inspection designed to give the consumer certain assurance in respect to the quality of product which he receives. t The consumer, in general ignorant of the production process, naturally wants some protection against accepting a bad lot of product. Of course, the ideal situation would be to inspect the entire lot, and thus make absolutely certain of its quality. This, however, is often a too costly procedure. Hence the consumer is willing to compromise and use sampling inspection provided it is not likely that the quality of the sample will indicate that the lot is good when, in reality, it contains more defects than he is willing to tolerate. Two such sampling methods for protecting the consumer will now be discussed. A. A Priori Method: The essential element in this method is that, if a lot containing the tolerance number of defective pieces is submitted for inspection, the chance that it will be accepted on the basis of a random sample is a given value P, whereas if the lot contains more than the tolerance number of defective pieces, the probability that it will be accepted on the same basis is less than P. For example, let us assume that a lot of N pieces of product is to be inspected and that the number c of defective pieces found in a sample of n is to be made the basis of acceptance or rejection of the lot. The consumer is perhaps willing to accept a certain amount of defective material provided the number of such pieces thus accepted does not exceed some fixed per- centage of the lot, commonly known as the tolerance pt. In fact we shall assume that, if a tolerance lot, -one containing pt/W defective pieces—is submitted for inspection the consumer wishes to have some assurance that he will accept only a fraction P of such lots in the long run. This fraction P has been called the consumer's risk and it is merely the probability that a tolerance lot will be accepted upon the basis of the sample." * This risk is discussed in an article by H. F. Dodge and H. G. Romig, “A Method of Sampling Inspection,” Bell System Technical journal, October, 1929. 342. ECONOMIC CONTROL OF QUALITY It remains merely to specify the sample size n and acceptance number c in such a way that the probability of finding this number or less of defective pieces in the sample taken from a tolerance lot is a given value P. Mathematically these factors are related by the following equation: I P – ºſcº cº, cº-cº, c."...+ cº, cº", gº 7. where CŞ' means the number of combinations of i things taken j at a time and q = I — pt. Having assigned P a definite value, O.30 r- O, I O H. i O | 2 3 4 5 NUMBER DEFECTIVE IN SAMPLE FIG. I.21.-CoNSUMER’s RISK. say o. Io, it is then possible to find pairs of values of n and c which satisfy (92). To illustrate the meaning of the consumer's risk, let us consider the following simple case. N = IOO, n = 50, pl = 5 per cent, c = I, qi = 95 per cent. The consumer’s risk is then the probability of finding I or o defective pieces in the sample of fifty taken from the lot of one hundred containing five defective pieces. Substituting the necessary values in (92), we find P = o. 1811, which is equal to the sum of the first two ordinates of Fig. 12I. B. A Posteriori Method: This method also offers the con- sumer a certain protection against accepting bad lots, i.e., those containing the tolerance number or more of defective pieces. DETECTION OF LACK OF CONTROL 343 The essential point of difference between this and the method just described is that the present method 1 attempts to find the probability that a lot contains more than X defective pieces if c defective pieces are found in a random sample of n. A little consideration will show that this kind of risk is quite different from the consumer’s risk previously described and that the nature of the assumption that must be made before this risk can be given is quite different from that made in the a priori method. Specifically, it is necessary to assume the a priori existence probability distribution of lots of a given size N in respect to the number of defective pieces contained therein. Having made this assumption, it is then possible to calculate the probability that each of the possible lots would have given the sample. The a posteriori probability that the lot contains just M de- fective pieces is then the ratio of the probability that a lot of size N containing M defective pieces existed and caused the sample to the sum of the probabilities that lots containing o, I, 2, . . . , N defective pieces existed and caused the sample. It follows from this that the a posteriori probability that the lot contains more than M defective pieces is the sum of a series of the above ratios found by allowing the number of defective pieces in the lot to vary from M + I to N inclusive. To illustrate this method, consider again the above example and let us find the a posteriori probability that the lot of one hundred pieces contains more than the tolerance number of defective pieces, assuming that the sample shows only one defective piece. As a very simple a priori assumption we shall assume that all possible constitutions of lots are equally prob- able, i.e., the probabilities of the existence of lots containing o, 1, 2, ..., Ioo defective pieces are all equal to Tºr. Then the existence probability distribution of possible lots is that shown graphically in Fig. I22-a and given in Column 2 of Table 5 I as existence probabilities oo, ol, . . . , oº, . . . , ow. * This method of sampling is discussed in an article by Paul P. Coggins “Some General Results of Elementary Sampling Theory,” Bell System Technical journal, January, I928. 344 ECONOMIC CONTROL OF QUALITY The next step is to calculate the probability that each of the possible lots could have given the observed sample. These are the productive probabilities 30, 31, . . . , 3, . . . , 8w shown in Fig. I22-6 and Column 3 of Table 51. At this stage we should TABLE 51.-CALCULATION of a posteriori PROBABILITY (I) (2) (3) (4) Number A priori A priori A posteriori Defective Existence | Productive Probability in Lot Probability | Probability o:3: M; CY; 6; 20:3; O I/IoI O O I I/IoI o. 5oCOOO o. 252475 2. I/IoI o. 50505 I o. 255026 3 I/IoI o. 378788 o. 191269 4. I/IoI O. 249922 o. 1261.98 5 I/IoI O. I 52947 o.O77231 6 I/IoI o.o88870 o.O44875 7 I/IoI o.O49635 o.O25063 8 I/IoI o.O26838 o.OI3553 9 I/IoI O. O.I.4II2. o.OO7I26 IO I/IoI o.OO7237 o.Oo3654 II I/IoI o.Oo3627 o.OOI831 I 2 I/IoI o.oO1778 o.OOO898 I3 I/IOI o.OOO854 o.OOO43 I I4 I/IoI O. OOO4O2 o.OOO2O3 I 5 I/IoI o.OOOI85 o.OOOO93 I6 I/IoI o.OOOO84 O.OOOO42 17 I/IoI o.OOOO37 o.OOOOI9 I8 I/IoI o.OOOOI6 o.OOOOO8 I9 I/IoI o. ooooo8 O.OOOOO4 2O I/IoI o.OOOOO3 O. OOOOO2 2. I I/IoI O. OOOOOI O. OOOOOI *22 I/IOI O. OOOOOO O. OOOOOO * Probabilities in columns (3) and (4) for M > 22 do not affect the sixth place of decimals. note that certain of the 3's are necessarily zero, +lots of one hundred containing less than one defective piece or more than fifty-one defective pieces could not have produced the sample. For 3's corresponding to number of defects lying between these DETECTION OF LACK OF CONTROL 345 limits, the probability off; that a lot containing just i defective pieces existed and caused the sample is I – i , , i. o:38; = — º IOI 49 1 O. O (a) i | O O 5 O.5Or- O.4 OH- O.30 H. (b) O,2O H. O,! O H. O - le. (C) i i O, |O Fºs I T I l TI T 2O 3O 4O 50 6O 7o so 90 |OO NUM BER DEFECTIVE IN LOT M |. FIG. 122.—RELATION BETweeN PROBABILITIES. The a posteriori probability that the lot contains just i defective pieces is o:6; N 2, oići i = 0 These probabilities are shown in Column 4 of Table 51. Hence, 346 ECONOMIC CONTROL OF QUALITY for the given special case, the a posteriori probability Pi that the lot contains more than the tolerance number (five) of defective pieces is found by summing the probabilities in Column 4 corresponding to M = 6, 7, ..., IOO. Thus 100 2, oft Pl F. †— r= O.O978. 2 offli † = 0 Hence P1 is the consumer's assurance that the lot is bad upon the basis of the given assumption and is represented graphically by the sum of the ordinates of Fig. 122-c from M = 6 to M = Ioo. It is perhaps worthwhile to point out that, if the manu- facturing process is controlled, the probability that a lot of N pieces contains the tolerance or more of defective pieces is known as soon as the equation (58) of control is known. The a priori consumer's risk, however, even under these conditions, has an additional protective feature in that even among the proportion of lots which contain the tolerance number of defective pieces the consumer will accept only a certain fraction P of them. Among those lots containing more than the toler- ance number defective, less than the fraction P of them will be accepted. If the quality is controlled in the sense that the probability of the production process producing a defective piece of appa- ratus is p, it can be shown that the a posteriori method of deter- mining the constitution of a lot of product tells us nothing other than would have been inferred a priori. In fact, if the condition just stated is satisfied, it can be shown that the a posteriori probability that the lot N contains say c + X defective pieces, having found c defective pieces in a sample of m, is precisely X N — m. - N – m — X Cº-"q p This expression, however, is nothing more than the a priori probability that the balance (N – m) of the lot contains just DETECTION OF LACK OF CONTROL 347 A defective pieces and is known as soon as the condition of control is met. It is of importance to note that, in order to be able to state the probability that a lot of N pieces of a product contains not more than X defectives after examining a sample of n in which c defective are found, we must assume something about the constitution of the lot before the sample of n was taken. Now, as we have seen, we approach the condition where we can say something about a lot of size N before the sample of size n is taken as we approach the condition of control. Hence we see that even from the viewpoint of consumer pro- tection, it is an advantage to have attained as nearly as possible the condition of control. PART VII Quality Control in Practice A Summary of the Fundamental Principles Underlying the Theory of Control and an Outline of the Method of Attaining Control of Quality from Raw Material to Finished Product CHAPTER XXII SUMMARY OF FUNDAMENTAL PRINCIPLES I. Introductory Statement The subject of quality control as considered in the previous chapters is comparatively new. The theory is based upon certain statistical concepts—physical properties and physical laws are both assumed to be statistical in nature." With the introduction of statistical theories and statistical laws comes a need for a new concept of causation.” Our understanding of the theory of quality control requires that our fundamental concepts of such things as physical properties, physical laws, and causal explanations undergo certain changes, since indus- trial development rests upon the application of the laws relating the physical properties of materials. The object of industrial research is to establish ways and means of making better and better use of past experience. Insofar as research continues to reveal certain rules or laws. which exist in the production of the finished product whose quality characteristics satisfy some human need, we may expect industry to be interested in research. That industries do have * This development is in accord with modern physics in that statistical theory is basic to a causal explanation of atomic phenomena. For example, Louis De Broglie, recipient of the Nobel Prize for Physics in 1929, says “Consequently there are no longer any rigorous laws but only laws of probability.”—ſ/ave Mechanics, page 9, Metheun & Co., Ltd., 1930. * This is true also in the field of pure physics. See for example Arthur Haas, Wave Mechanics and the New Quantum Theory, published by Constable & Co., London, 1928, He says, “In contrast to the sharply defined causality which is evident in macroscopic physics, the latest theories have emphasized the indeterminate nature of atomic processes; they assume that the only determinate magnitudes are the statistical magnitudes which result from the elementary processes of physics.” 35 I 352 ECONOMIC CONTROL OF QUALITY such an interest in this form of human endeavor seems to be a well-established fact. It is estimated that during the year 1927 upwards of $200,000,000 was spent in industrial research in approximately 1,000 laboratories in the United States. This gives the order of magnitude of the sum of money that is being spent annually in the effort to find out how to do something tomorrow that we do not know how to do today. All effort, however, in this direction is obviously not included in formal research programs. Who, for example, in some way or other has not made use of past experience? It is rather startling to see how much progress was made by that part of the human race which never had any knowledge of applied science as such. Long before any one worried over the physical principles which govern the use of the lever and of the wedge, use had been made of both of these mechanical devices. Long before any one had arrived at the generalization known as the Law of the Conservation of Energy, our fore- fathers had transformed mechanical energy into heat energy to start their fires. These two illustrations are sufficient to indicate that progress in the use of past experience does not depend upon the knowledge of scientific laws as we know them today. The rate of progress on the other hand does depend upon this knowledge. In a similar way, we do not have to know the theory of control to make progress in the improvement of quality of product. But, as the physical sciences have led to useful generalizations which increase the rate of progress, so also does the knowledge of the principles of control. To indicate the relationship which the theory of control bears to exact science, it is interesting to consider six stages in the development of better ways and means of making use of past experience. They are: I. Belief that the future cannot be predicted in terms of the past. 2. Belief that the future is pre-ordained. * Grondahl, L. O., “The Rôle of Physics in Modern Industry,” Science, August 23, I929, pp. I75-183. SUMMARY OF FUNDAMENTAL PRINCIPLES 353 3. Inefficient use of past experience in the sense that expe- riences are not systematized into laws. 4. Control within limits. 5. Maximum control. 6. Knowledge of all laws of nature—exact science. It is conceivable that some time man will have a knowledge of all the laws of nature so that he can predict the future quality of product with absolute certainty. This might be considered a goal for applied science, but indications today are that it is not a practical one. At least we are a long way from such a goal; for years to come the engineer must be content with the knowl- edge of only comparatively few of the many conceivable laws of nature where we think of the term law in the sense of New- ton's Laws of Motion. Furthermore, the engineer is fully aware of the fact that, whereas it is conceivably possible with the knowledge of these laws to predict the future quality of product with absolute certainty, it is not in general feasible to do so any more than it is feasible to write down the equations of motion (were it possible to do so) for a thimble full of mol- ecules of air under normal conditions. The engineer is fully aware that, whereas in the laboratory one may often be able to hold conditions sufficiently constant that the action of a single law may be observed with high precision, this same degree of constancy cannot in general be maintained under what appear today to be necessary conditions of commercial production. In fact, if we are to believe, as do many of the leaders of scientific thought today, that possibly the only kind of objective constancy in this world is of a statistical nature, then it follows that the complete realization of the sixth stage is not merely a long way off but impossible." We have seen that the principle of control plays an im- portant rôle in laboratory research in what is ordinarily termed pure science. We have seen that it is necessary, in general, in all such work to attain as nearly as possible to certainty in the Bridgman, P. W., loc. cit. 354. ECONOMIC CONTROL OF QUALITY assurance that the observations supposed to have been taken under the same essential conditions have actually been taken in this way. As an efficient tool in testing whether or not this condition has been satisfied, we have the criteria of Part VI. We have seen that the criteria for maximum control (Part III) give a test which indicates the limit to which it is reasonable that research may go in revealing causes of variability in a set of observations presumably taken under a constant system of chance causes. We have also seen that many of the quantities with which we actually deal in the so-called exact sciences are but averages of statistical distributions assumed to be given by what we have chosen to term a constant system of chance Call ISCS. - Let us now consider the need for control as an integral part of any industrial program. In most cases we can dis- tinguish five more or less distinct steps in such a program. They are: - I. A study of the results of research to provide principles and numerical data upon which to base a design. 2. The application of such information in the construction of an ideal piece of apparatus designed to satisfy some human want, where no attention is given to the cost. 3. Production of tool-made samples under supposedly com- mercial conditions. 4. Test of tool-made samples and specification of quality requirements that can presumably be met under commercial conditions. 5. Development of production methods. From this viewpoint the results of design, development, and production are grounded on the initial results of research. What is more important in our present study is the fact that often causes of variability enter in the last four steps which by the very nature of the problem are not experienced in the research laboratory. For example, we have the possibility of assignable causes entering through different sources of material, SUMMARY OF FUNDAMENTAL PRINCIPLES 355 the human element, and variable conditions which affect the production process. One possible method of obtaining satisfactory quality under such conditions is to make wherever possible Ioo per cent in- spection of the product at the time it is ready for delivery. In many cases, however, this cannot be done because of the destructive nature of the tests; in any case the cost of inspection must be considered. Furthermore, if indications of the presence of assignable causes of variability are discovered in the quality of final product, it is not easy to locate the causes because the data of final tests may have been taken long after the causes have ceased to function. Even more important, as we have seen in previous chapters, is the fact that the quality may appear controlled in the end and yet there may be assignable causes of variability at one or more steps in production. For these reasons, it seems highly desirable that the measurements made in each of the last four of the steps mentioned above be tested to determine whether or not there is any indication of lack of control. If there is, it may be necessary that a further study be made in the laboratory to assist in finding the assign- able causes of variability. We must emphasize the importance of control in setting standards for the raw materials that enter into the production process. Most physical properties are subject to the influence of presumably large numbers of chance causes. Therefore, if we are to make efficient use of data representing these prop- erties, the data must have been taken under controlled con- ditions. Before we can use experimental results with any assurance of their giving a controlled product, it is highly desirable that we make use of tests to determine whether or not the data have been secured under controlled conditions. Furthermore, in the development of processes of production, it should be of advantage to apply tests to detect lack of control and then to weed out the assignable causes of variability as they occur, with the assurance of the kind already indicated in previous chapters, that after this process of weeding out has once led to a product which appears to be controlled, future 356 ECONOMIC CONTROL OF QUALITY product will remain in the same state unless obvious assignable causes of variability enter. Thus the theory of control plays an important part in the various stages of applied science. It is desirable that the depart- ments of design, development, and production keep the labor- atory research department informed as to evidence of the existence of assignable causes wherever they arise up to the time that product goes to the consumer. The theory is also of value in the study of the life history of product. Obviously, when equipment goes into the field it meets many and varied conditions, the influence of which on the quality of product is not in general known. Such an example would be the varied conditions under which telephone poles are placed throughout the United States. A priori, it is reasonable to believe that the life of the pole depends in a large way upon the service conditions. Among the exceedingly large number of variables which may influence the life of the pole, little information is available to indicate the importance of any one. The value of laboratory research in improving the quality of a pole through life must take into account ways and means of preservation suited to each of the various conditions. Naturally, therefore, it is of interest to know when the vari- ability in the quality of the material at any stage in life is such as to indicate the existence of an assignable cause so that further research may be instituted to find ways and means of effectively removing this cause. Field engineers, therefore, find need for analytical methods of detecting evidence of lack of control in the quality of product at any time as revealed by life data so that they can call this fact to the attention of the laboratory staff. In this chapter we shall discuss briefly such fundamental concepts as physical property, physical law, and cause, basic to every step in control. 2. Object of Control As already stated, the object of control is to enable us to do what we want to do within economic limits. SUMMARY OF FUNDAMENTAL PRINCIPLES 357 As we have seen in Part I, it is necessary to postulate that when we have done everything that we can do to eliminate variability in a quality X, we arrive at a state of statistical control in which we can say that the probability dy of an observed value X falling within the range X to X + d X is given by the equation of control dy = f(X, \, \2, ..., \, ..., \m).dx. (58) 3. Physical Properties In the previous chapters we have seen that perhaps the closest that a physical quality attains to constancy is in the sense that objectively it may be represented by a distribution function (58) characterizing a state of control. It follows that the complete specification of any quality requires the estab- lishment of an equation of control of the form (58) both in respect to functional form f and the values of the m' parameters contained therein. It has been shown in Part V that for most practical purposes it is sufficient to attempt to specify simply two characteristics of this distribution, namely, the average or expected value X and the standard deviation g. Examples: To emphasize the statistical nature of materials still often treated as constants, let us look through a microscope at a cross section of a piece of ordinary steel," Fig. I23. What we see is anything but a homogeneous isotropic body. Why this heterogeneous structure? The answer is—It is produced by chance or unknown causes. & What is the effect of such irregularities upon the physical properties of steel when produced in some useful form as, for example, supporting strand, a piece of which is shown in Fig. I24? The answer is that a physical property, say the breaking load of such strand will, if we are able to eliminate assignable causes of variation, be some distribution function as indicated in Fig. I25. The smooth curve in this figure * Lucas, F. F., “Structure and Nature of Troostite,” Bell System Technical journal, January, 1930. 358 ECONOMIC CONTROL OF QUALITY represents the objective distribution of control (58) for this particular case, as inferred from the study of observed data. Fig. 124.—Piece of Support.ING STRAND. As we have said above, it is usually sufficient to specify merely the average X and standard deviation g of such a distribution. SUMMARY OF FUNDAMENTAL PRINCIPLES 359 Now let us look at a cross section of another important structural material,—wood, Fig. 126. This time we do not need a microscope to see the effects of chance causes upon the structure of the material. Fig. 88 in Part V shows roughly what such irregularities do the modulus of rupture of four kinds of telephone poles. 30 t O 2O O o - l l l 1–1 l l . - O 4,000 8,000 12,000 16,000 20,000 24,000 28,000 TENSILE STRENGTH OF STEEL STRAND Fig. 125-TENSILE STRENGTH Distribution for STRAND SHowN IN FIG. 124, Note the wide spreads of these distributions as compared with their means. These two illustrations are sufficient to show that the variation introduced by constant systems of chance causes into Fig. 126.-CRoss Section of Pole. the physical properties of materials are so large that they need to be taken into account in the use of these materials. - Shortly we shall see what methods are available in the liter- ature for establishing objective distributions for standards of physical properties. 360 ECONOMIC CONTROL OF QUALITY 4. Physical Laws & In Part III we discussed briefly three different kinds of laws, viz., exact, statistical, and empirical. In this section we shall contrast the first two kinds in the hope that by so doing we may take over the part of the concept of exact law that is common with that of statistical law, and that we may see clearly wherein the concepts of the two laws differ, insofar as this bears upon the theory of quality control. Let us consider first the harmonic oscillation of a vibrating system characterized by the equation d? X d X 772 d;2 +k++ x = o where X is a linear displacement, t is the time, m is the mass, K is the frictional force proportional to velocity and 3.x is the restoring force. The solution of this differential equation AAA- V V v FIG. I27-a.—BASIs For ExAcT PREDICTION. gives us the displacement X as a function of the time t. In other words, starting with a knowledge of m, k, s, and X at t = O, we can predict with great precision the displacement at any future time t. Fig. 127-a typifies such a prediction. Let us now consider what is involved in prediction in a statistical sense. Let us contrast with this simple problem that of predicting the number of times that a head will be turned up in m throws of a penny. As was pointed out in SUMMARY OF FUNDAMENTAL PRINCIPLES 361 Part III, the practical method of making prediction in this case is to assume that there is some point binomial (q -- p)" where q + p = I such that the successive terms of this ex- pansion represent the probabilities of occurrence of O, I, 2, 3, . . m heads in n throws. It follows that the standard deviation r| | _l \ O- | \ | - \ \ \ 5 H \ 2 º S.J U T - - - 3 . - T TTT --------------- tº Lº “”TS.*.*.*.......... sa-ºe-aa-ee-seaseºesºeºsºte------~~~~~~ ~ –-mmº Cº. I — — — — — — — — — — — — — — — — — — — u- | . . . . – – — — — — — — — — — — — — -- - - TTTTT UJ _----- > 2 * H. / <ſ / —J / # Fl I º ſ ſ 1—. ſ ſ | f l | ſ ſ ſ | 1 | f f M f fl Jº O 2O 4O 6O 80 |OO |2O 14C) 16O 18O 2OO NUM BER OF TR 1 ALS – ſh FIG. 127-b.—BASIC INForMATION For STATISTICAL PREDICTION. op of the relative frequency p of heads in n trials is given by the relationship |pg O p = \|==. If p = % it follows from what has already been said that approximately all of the observed values of p in future trials should lie within the dotted limits, p0. == —, p=3 W. shown in Fig. 127-b. The dots in this figure indicate the experi- mental results of throwing a penny two hundred times. Now let us compare the results in these two cases. Pre- diction in the first case involves the assumption that the dynamical system behaves in a way such that when we sub- stitute measurable values of m, k, and s in the differential 362 ECONOMIC CONTROL OF QUALITY equation the solution of this equation gives a satisfactory prediction of the future displacement of the mass. In an analogous way, as indicated in previous chapters, it appears that we may expect to find (in the objective sense) a value of p for a given penny such that when used as indicated above, we may establish limits such as those given in Fig. I27–0. The two methods of prediction are alike in that they require the experimental establishment of certain parameters. They differ in that one makes use of these parameters in a differential equation, the other in a binomial expansion. They are alike in that we do not know a priori that the mathematics used in either case is the mathematics that should be used. Now suppose that we were to try to make N dynamical systems to have as nearly as possible the same values for m, K, and s. In the same way let us suppose that we take N pennies that appear to be alike so far as we can determine. If we were to start oscillation in each of the W dynamical systems with the same displacement and observe the resultant displacement, we would expect that each of the systems would follow the curve in Fig. 127-a quite accurately. Similarly, if we were to throw each of the N coins a large number of times, we would expect to get something like the three records shown below, Fig. 128, representing the results of two hundred throws of each of three different pennies. The systems are alike in that the smooth curve in Fig. I27-a represents what we may expect to get on the average when we try to duplicate the dynamical systems as nearly as possible and the straight line p = # in Fig. 127-b represents the expected value for a symmetrical coin. In the statistical case, however, there is a certain indeterminateness as compared with the so- called exact case. Although we can say in the statistical case with considerable assurance that the observed values of p will lie within certain limits and that these limits will decrease proportionately to the square root of n, we cannot say anything determinate about the way the observed values will approach the value p. * In the dynamical case, if t is made indefinitely large, we SUMMARY OF FUNDAMENTAL PRINCIPLES 363 . º e .” |- º •'.” º e © & 9 e a * . o º M- © * * "...". ..”. .*.*.*.*.*.*.*.*..... - g º" ſº ******.*.****“…~...~~~~…”…~~~~º 1—1 I l l h l l 1 l I I l t I º t t † H. •. **... • tº º º *Nº. º ſº s\", º * - e. •. *e "... tº º **•." • * * * * *~~~~A* ..*.*** ~~~~~~~. * *****.*.*.*.* ** }- º * , *.* • © º k- e º º º º A f º I ſº 1 1–3–1 | f— J. n t t | º f n r O 2O 4O 60 8O |OO 12O, 140 |6O 18O 2OO NUMBER OF TRIALs — n FIG. I28.—BASIC INForMATION FoR STATISTICAL PREDICTION. 364 ECONOMIC CONTROL OF QUALITY can say that the corresponding value of X will approach the value zero as a mathematical limit. On the other hand, we can say in the statistical case that for each of the N pennies, the observed fraction p in n throws will approach as a statistical limit the value p. In the first case we can say very definitely how the displacement will approach the value zero. In the second case we can say scarcely anything about the way the value p will approach p. This fundamental limitation of indefiniteness, however, is not solely limited to the statistical case when we come to think of the determination of the parameters which must be found in either case. In Part III we pointed out that our success in being able to predict a phenomenon by means of statistical theory rests ultimately upon the assumption that we can find the parameters in certain functions through use of a statistical limit. In a similar way, the values of m, k, and s can only be obtained in practice through averaging observed values of these factors taken under presumably the same essential con- ditions. In other words the objective values of m, k, and s are in themselves statistical limits. Strictly speaking, all that we can say in the exact case is that the probability of the displacement at a given time t lying within a given range is a certain constant value. Similarly, we can say in the case of throwing a coin under the same essential conditions that the probability of observing a given number of heads in a given number of throws is a constant. In other words in both instances what we really assume to be constant is a certain statistical distribution. In both cases there is the same kind of indeterminateness although it appears in a slightly different way. 5. Causal Explanation We have made much use of the concept of a constant system of chance causes. It is essential that we consider a little more carefully the significant difference between causal explanation as it is usually accepted and causal explanation in the statistical Serl Se. SUMMARY OF FUNDAMENTAL PRINCIPLES 365 It is customary to think of a cause as being an antecedent event which is always followed by one or more definite events or consequents. The antecedent event in such a case is the cause and the consequents are the effects of the cause. For example, the presence of a tubercle bacillus in the lungs of a human individual may produce many different effects, such as a high temperature, change in composition of blood, loss of appetite, and so on. Some of these effects, however, may be produced by other causes. The situation in such a case is indicated schematically in Fig. I29, in which 4 and B are ante- ANTE CEDENT A ANTECEDENT B a b C C € f 9 CONSEQUENT FIG. 129.-SCHEMATIC OF CAUSAL RELATIONSHIP. cedents with corresponding consequents indicated by small letters. If we can state in a given case all of the consequents be- longing to a given antecedent event, it is generally agreed that we may go with certainty from effect to cause. Of course in the practical case we meet with the serious difficulty of not being able to state all of the consequents corresponding to a given antecedent. This point, however, we do not care to con- sider at present. The point that we do wish to make is this. Causal explan- ation in this accepted sense assumes that whenever we have an antecedent A such as indicated above, it is always followed by effects (consequents) a, b, c, e, and f. With this picture of cause let us now contrast the concept of chance cause already 366 ECONOMIC CONTROL OF QUALITY illustrated in some detail in Part III and Appendix I. Imme- diately we note a characteristic difference between the concept of a chance cause and the older concept of cause. For example, in our discussion of Appendix I we treat of very simple systems composed of m different causes. It is assumed that each cause may be followed by one or the other of two events, that is one or the other of two different values of X. In other words, it appears that we can never hope to tie up a chance cause with a given event because for each chance cause more than one event is always possible. Let us go a little further in the amplification of this point. Let us suppose that we have n observed values X1, X2, . . . , Xi, . . . , Xn of some variable quality X taken under controlled conditions represented by the equation of control (58). We may think of the cause of this sample as being the particular equation of control representing the conditions under which the samples were drawn. It is apparent, however, that in general any series of n observed values such as indicated above may have come from any one of let us say N different universes which we may characterize as follows: dy1 = fi (X, \11, \12, • * - 5 N1i, • * - 5 Nim',) dX, dy2 = fº (X, \21, \22, . • • 3 M2i, • • 3 \om,) d X, • 3 > (93) dyj = f; (X, Aji, \j2, . . . . Aft, . . . . Ajm') dX, • 3 dy N - fy(X, MN1, MN2, • * - 5 MNi, • * - 5 ANn',) dx. Under the above condition we may never state with cer- tainty what one of the N universes the observed sample came from. Each of the N universes is a possible cause but our interpretation of a sample must always be indefinite in that under the above conditions we can never be certain as to the origin of that sample. Cause in the older scientific sense, there- SUMMARY OF FUNDAMENTAL PRINCIPLES 367 fore, has a certain determinateness about it which must of necessity be absent in the statistical case. - The two kinds of cause, however, do have this much in common that is very important from the viewpoint of the theory of control—the choice of a cause in a given case largely depends upon the intuitive faculty of the human mind. In other words, we cannot in general write down rules for the correct selection of a cause. It is, however, one of the objects of logic to lay down ways and means of testing postulated causal explanations. An interesting illustration may be drawn from the field of investigation as to the origin of the planets. Two fundamental rival hypotheses are described in a popular way in a compar- atively recent article by F. R. Moulton. The first of these he describes as follows: Laplace started with a heated gaseous mass rotating as a solid. With loss of heat by radiation, it contracted and rotated more rapidly. At various stages of the contraction the centrifugal acceleration at the equator of the rotating mass equaled the gravitational accelera- tion toward its center. At these places the contracting mass left behind gaseous rings which were concentrated into planets by the mutual gravitation of their parts. In six cases, after the contracting rings had assumed approximately spherical forms they similarly contracted and left behind smaller rings, which became satellites. This theory is delightfully simple and can be stated in a few sentences. It makes few demands upon the imagination to conceive of its various steps and it requires no sustained mental effort to organize them into a unified whole. It raises no unanswered questions and arouses no doubts. The account of the creation and the origin of the earth in Genesis is not simpler. He then summarizes the second in the following words: In striking contrast with the foregoing, consider the planetesimal hypothesis. The fundamental point of view adopted in it is that the stars of our galaxy constitute a group of mutually related objects, the evolution of each depending in part upon its relationships to the others. They mix and mingle with one another, in the course of time, somewhat like molecules in a gas. At the time of the dynamic * The Planetesimal Hypothesis—Science, December 7, 1928, Volume LXVIII, Number 1771, pp. 549–559. 368 ECONOMIC CONTROL OF QUALITY adventure of a suitable near approach of one star to another, planets are born from the parent suns. These planets grow up around nuclei by the accretion of countless little planets (planetesimals) born at the same time. Not only in the broad sweep of events leading to the birth of the planets as independent objects does this theory differ completely from the Laplacian, but also all the dynamical considerations involved in the growth and evolution of the planets are wholly different. More than one commentator on the planetesimal hypothesis has regarded with favor the origin of the planets by dynamic approach as being likely, and has then utterly failed to realize that the growth and evolution of the planets could not have been along the lines that are consonant with the Laplacian theory. The new hypothesis gives an entirely new earth and lays down a new basis for the development of dynamic geology. In other words, these two hypotheses may be thought of as A and B in Fig. 129. The effects in this case to be explained are the characteristics of the solar family. Now let us see how the process of checking an hypothesis or cause in the older sense corresponds with that of checking an hypothesis or cause in the statistical sense. The essential dif- ference is this. In the first case we may be able to find that some of the observable phenomena cannot be effects of the postulated cause. In such a case it is customary to reject or modify the hypothesis. For example, this is true in respect to the Laplacian hypothesis as to the origin of the earth referred to above. In the statistical case, however, it is not so easy to reject an hypothesis, as we shall now see. Suppose, for example, that we attempt to test the hypoth- esis that a sample of n observed values of a quantity X came from let us say the first universe of (93). We have already touched upon this problem in Part VI in the discussion of the choice of statistic to be used in a given case and of the choice of method of using this statistic. Let us look at this problem in a more general way. We may represent a sample of size n as a point in n dimensional space. In a similar way we may represent all of the possible different samples of size n that may be drawn from an assumed universe as points in this same space. To get a test of whether or not the observed sample SUMMARY OF FUNDAMENTAL PRINCIPLES 369 came from the assumed universe, it appears to be necessary to establish certain contours in this n dimensional space within which the points corresponding to an observed sample must fall if we are to accept the hypothesis that it came from the assumed universe. Naturally there are an indefinite number of ways of setting up such contours and the choice of any one is quite arbitrary on the part of the individual scientist as was the cor- responding choice of statistic in Part VI. In any given case there are in general an indefinitely large number of possible hypotheses. Hence, in addition to the problem of establishing arbitrary contours upon which to test a given hypothesis, we must consider the problem of judging between alternative hypotheses. Here again we come upon the indeterminateness of the statistical method. It appears that there is no ultimate ground upon which to base our final choice." 6. Measurement of Average X and Standard Deviation o The concept of physical properties and phenomena as frequency distributions introduces the concept of measurement of such distributions. Since for most engineering purposes it is sufficient to know the average X and standard deviation o of such a distribution, we shall consider, the problem of measuring these two characteristics. Assuming that the set of n observed values, X1, X2, . . . , Xi, . . . , Xn, of a quality characteristic X satisfy the equation (58) of control, it follows from the law of large numbers that the observed average X and standard deviation a can be made to approach, in the statistical sense, as close as we please to X and a re- spectively by making the sample size n sufficiently large. In * See in this connection the especially interesting and valuable article by J. Neyman and E. S. Pearson entitled “On the Use and Interpretation of Certain Test Criteria for Purposes of Statistical Inference,” Part I, Biometrika, Volume XX-A, July, 1928, pp. 175–240, Part VII, Biometrika, Volume XX-A, pp. 263-294, December, 1928. Also see pp. 303-314 of A. N. Whitehead's Process and Reality, Macmillan Com- pany, I 929. 370 ECONOMIC CONTROL OF QUALITY other words, it follows from the law of large numbers and Tchebycheff's theorem that, by making n sufficiently large, we can bring as close to unity as we please the objective probability P that the inequality |X – X| < e will be satisfied, e being any previously assigned positive quantity. In practice, however, it is not feasible to take an indefinitely large number of observations. In fact, we must often be sat- isfied with estimates of X and a derived from comparatively small samples. For example, we may wish to determine an approximate standard for a quality X of a given kind of appa- ratus from measurements of this quality on from five to twenty- five tool-made samples. Or again, we may wish to adopt a standard for the physical property of some new material or alloy from measurements made on comparatively few pieces. We shall now consider various ways of doing this. A. A Posteriori Probability Method.—This method has been discussed in a very interesting and novel manner by Molina and Wilkinson.' Assuming that the set of n observed values of the variable X have come from a normal universe - I _(X-X)* € 20-2 ſo- gºv/2t 5 in which X and g are unknown, the a posteriori probability P(X)ax that the true mean lies within the interval X to X + d x is given by § (X; –X)2 * t = } * * * °//(X * 20-2 P(X)dX = AdXſ rºº * do, (94) O where A is a constant and W(X, g).dxdo is the a priori prob- ability, before the observations were made, that the true mean and standard deviation were within the intervals X to X + dx and q to q + do respectively. * “The Frequency Distribution of the Unknown Mean of a Sampled Universe,” Bell System Technical journal, Vol. VIII, October, 1929, pp. 632–645. SUMMARY OF FUNDAMENTAL PRINCIPLES 371 To get a definite answer in a given case, certain assumptions must be made in order to give the parameters in (94) specific values in terms of the statistics of the set of n observed values of the quality X, and in every case one must assume some particular form for the function W(X, 0). In other words, before any measurements are made, one must choose some one function W(X,0) out of the indefinitely large number of possible functions. * Assuming that X and g are independent, we may write W(X, q) = W1(X)//2(0). Making these various general assumptions and certain others of a more detailed nature, the authors then assign to the param- eters in the functions W1 and W2 twenty-one sets of values out of a possible infinite number of such sets, and find as many probable and 99.73 per cent errors for a single example. Their results are shown graphically in Fig. I3O.' The startling and very important thing to note is the great significance that must be attached to the choice of the a priori existence probability functions W1(X) and W2(g) before any measurements are taken. Of course, any one of the twenty-one or, in fact, of the indefinitely large number of probability distributions P(X).dx of (94) gives us only the a posteriori probability that the true mean lies within a specified range, whereas we wish to get usable estimates of X and G. Hence, even though one goes through the a posteriori solution under the conditions stated above, it is likely that he will take the observed average X as his best estimate of X. As for an estimate of G, it will be expressible as a multiple of the observed standard deviation, let us say ca, the value of c depending upon the particular assumptions made in applying (94). B. Maximum Likelihood Method.—In the particular case just considered the probability P of the simultaneous occurrence of the set of observed values X1, X2, . . . , Xi, . . . , Xn within * The authors used the precision constant h instead of or in this paper. However, they have also shown the distribution of or (the dotted lines) for the first seven sets of assumptions. 372 ECONOMIC CONTROL OF QUALITY APF (OR 1 PROBABILITY DISTRIBUTIONS ERRORS IN OBSERVED MEAN PRECISION ME AN PROBABLE 99.73 °/o N consºr ERROR E R R O R w(m)=A(i+B(M-mjºl 2 we(h)=Khºe-ah O O.5 !,0 2 3 O 4. 5 6 7 9 |O l | 12 ! 3 |4 ! 5 ! 6 º s 17 9 20 O.3i 6 w(m); W2(h) 3 2. •l 35 4O 45 50 55 O O.2 OA. O.6 O n h FIG. I.30.-SIGNIFICANCE of a priori ExISTENCE PROBABILITY FUNCTIONs. SUMMARY OF FUNDAMENTAL PRINCIPLES 373 the respective intervals XI to X1 + d X1, X2 to X2 + d x2, . . . , AG to X; + d xi, . . . , Xn to Xn -- d.Kn is 70, I _(Xi-m)* P = II = 6 2a-2 dXi, where m and s are universe parameters. That value of m which will make P a maximum is given by the solution of the equation | 3(log P) = O O777 } since P is a maximum when log P is a maximum. This gives the observed average X as an estimate of X. Similarly the condition 3(log P) = O OS gives the observed standard deviation a as the estimate of G. Since the expected value a of the observed standard devi- ation in samples of size n drawn from a normal universe is less than the standard deviation o of the universe, it is obvious that the estimates of a derived by the likelihood method are too small in the long run, particularly if the sample size n is small. C. Empirical Method.—Assuming, as before, that we are sampling from a normal universe free from assignable causes, there is perhaps no better estimate of X than the average X of the sample. If, however, there is any reason to believe that a few of the observed values were influenced by assignable causes, this fact should be taken into consideration. If we assume that we are sampling from other than a sym- metrical universe, it becomes all the more important that we make use of the average X of the sample of size n as an estimate of the average X of the universe of possible effects. Coming to the estimate of the standard deviation o of the normal universe, we have seen that a posteriori probability theory does not provide a direct method of establishing a specific value as the best estimate and that the likelihood 374 ECONOMIC CONTROL OF QUALITY method leads to an estimate which is too small in the long run. Referring to Fig. 97 indicating the important characteristics of the distribution of an observed statistic 6, say standard devi- ation, we might be led to base our estimate of G on the assump- tion that the observed a is the modal a of the distribution of this statistic. In other words, we might take as an estimate | m O' a = −, 72 — 2. Ó 1 where c1 is given in column 2 of Table 29. To do so, however, means that in the long run estimates made in this way are too large. An estimate that will be consistent in the long run is # where co is also given in Table 29. C2 There is thus some justification under these conditions for adopting # as an estimate of G. In any case the observed standard deviation a becomes the basis of an estimate. Hence it seems reasonable that it should be tabulated together with any correction thereof adopted as an estimate in a given case. The estimate of G of a non-normal universe presents addi- tional difficulties since, in general, we do not know the dis- tribution function of observed standard deviations in samples of n. Here again the observed standard deviations in the long run are too small, in the sense that the expected value in samples of size n from a given universe is less than the standard devi- ation o of that universe. 7. Measurement of Average X and Standard Deviation o- Practical Example Let us consider the significance of previous results in a simple practical case. Four pieces of shoulder leather from a given source were found to have the following tensile strengths expressed in pounds per square inch: 5,29O g 2,950 4,850 5,960 SUMMARY OF FUNDAMENTAL PRINCIPLES 375 Upon the basis of this information, what shall we choose as estimates of the average X and standard deviation o of the tensile strength of leather from this source assuming that this quality is controlled. From what has been said in the previous section it is appar- ent that the answer to this question depends upon many factors. It depends upon more or less arbitrary assumptions as do the answers to many practical questions. In each case, however, it is likely that the average X = 4,762.5 and the observed stand- ard deviation a = I, II 8.6 will be made the basis of the estimate. Furthermore, it is obvious that the interpretation of these depends upon the size n of the sample, in this case four. For these reasons it appears that in the tabulation of results of this character the experimentalist should always record the observed average X, standard deviation a, and sample size n. In general it is perhaps reasonable to believe that the exper- imentalist who is in charge of taking the data is in the best position to make a reasonable assumption upon which to base an estimate. For this reason it is desirable that he record what he considers to be the best values to take as estimates of the average X and the standard deviation g of quality X assumed to be controlled. It is likely in this case that the average X will be taken as the estimate of X. In the same way it is likely that G will be taken as a quantity larger than 0. As we have tº ºg O e o ſº O . º said in the previous section, the estimate – is a consistent es- C2 timate in that in the long run the average of an indefinitely large number of such estimates would give the true value G assuming that the universe of control is normal. Anyone who wishes to make use of these results may use the observed average and standard deviation and the sample size as a basis for his own estimates of X and o, or he may choose to use those selected by the experimentalist himself. In this way he is free to make his own postulates basic to estimating X and o. CHAPTER XXIII SAMPLING—MEASUREMENT I. Place of Measurement in Control In any program of control we must start with observed data; yet data may be either good, bad, or indifferent. Of what value is the theory of control if the observed data going into that theory are bad? This is the question raised again and again by the practical man. Even though it is necessary, as a starting point in the theory of control, to tabulate the results of n measurements of some physical quality X in terms of the average X and the standard deviation a, the engineer often reacts in something like the following way. He will likely admit that this method is an excellent one to follow if, as he says, the data are known to be good, but he will often argue in a given case that the data are not good enough to make it worth while to record more than perhaps the average and the range. He may go so far as to throw out one or more of the observed values before taking even the average and the range. In fact I have heard industrial research men say that they can get more out of a set of data just by looking at it than anyone without their experience can get by the most refined analysis. In discussing this point at a recent round-table conference on presentation of data, one prominent engineer had this to say: Most frequently we are confronted with expressing results that have been obtained by empirical methods in the hands of fallible * Conference held in New York, December 5, 1929, under the auspices of the American Society of Mechanical Engineers and the American Society for Testing Materials. 376 SAMPLING—MEASUREMENT 377 operators on more or less representative samples of generally very heterogeneous materials. When we go to discuss the precision of our methods, we always have three factors which have not been controlled. We have the question of the authenticity of the sample; we have the question of the operator; and we have the question of the method itself. Hence it becomes a very complicated problem to apply the mathematical methods of analysis to these data. A number of years ago I read somewhere an expression which has always struck me. It said something about mathematics being a mill that grinds with exceeding fineness and yet a mill that is no better than the grain that is put in it. So it always seemed that in our work the first thing we had to do was to attempt to develop the limit of precision of our methods after we had at least some- thing to start with; then we could determine the effect of the presence of the operator. From that point we could determine the authenticity of our samples and we would be in a better posi- tion to analyze our crop of results. Not only in the fields of industrial research and engineering do we get such a reaction. We find it also in the field of so-called exact science—for example, physics. Thus in a recent paper by Millikan discussing the value of electronic charge, emphasis is laid upon the importance of the human judgment of the exper- imentalist, as is typified by the following paragraph: This value of the electron is also that at which Birge finally arrives as a result of his survey of the whole field of fundamental constants. It is true that he reanalyzes for himself my individual oil-drop readings and weights them so that he gets from them the value 4.768 + O.OO5 in place of my value 4.77O = O.Ooš, a result that is so much nearer mine than my experimental uncertainty that I am quite content—indeed gratified—but I may perhaps be par- doned for still preferring my own graphical weightings, since I thought at the time, and still think, that I got the best obtainable results in that way from my data. The person who makes the measurements certainly has a slight advantage in weighting over the person who does not, and the graphical method by which I got at my final estimated uncertainty is, I think, in the hands of the experimenter himself more dependable than least squares. In this way we get into the following dilemma: The en- gineer questions the usefulness of refined methods of analysis * Loc. cit., Part II. 378 ECONOMIC CONTROL OF QUALITY because his data are not good; the research man questions their use because he does not need them. The sooner an engineer appreciates this situation, the sooner will he become an influence in getting good data such that he can use in the theory of control to effect certain economies previously discussed. Everyone will admit that in the literature there are numer- ous sets of bad data. As an illustrative case we find the fol- lowing statement in a recent paper on thermionic emission: Most of the observations on emission made up to 1914, and a considerable number of those made since then, are almost worthless because of the poor vacuum conditions under which they were made. 2. All Measurement a Sampling Process An element of chance enters into every measurement; hence every set of measurements is inherently a sample of certain more or less unknown conditions. Even in the few instances where we believe that the objective reality under measurement is a constant, the measurements of this constant are influenced by chance or unknown causes. Hence, the set of measurements of any quantity, even though the quantity itself be a constant, is a sample of a possible infinite set of measurements which we might make of this same quantity under essentially the same conditions. e From this viewpoint, measurement is a sampling process designed to tell us something about the universe in which we live that will enable us to predict the future in terms of the past through the establishment of principles or natural laws. In fact, we may think of the process of examining a subgroup n of a larger group of N things along this same line in the sense that we look at the n things and try to predict what we would find if we were to look at the remaining N – m things. In the measurement of anything four kinds of errors may arise: Theoretical A. Constant { Instrumental Personal 1 Saul Dushman, Reviews of Modern Physics, Volume II, pp. 381–476, 1930. SAMPLING—MEASUREMENT 379 Manipulative B. Mistakes & Observational Numerical C. Effect of Assignable Causes, Type I Methodological D. Effect of Constant Chance Systems { Instrumental Physiological 3. Good Data Three prerequisites of good data are: A. They shall come from a constant system of chance causes—in other words, they must satisfy the criteria of Part VI if they are sufficiently numerous that such tests can be applied. If this condition is not fulfilled, we must rely upon the experi- mentalist’s ability to eliminate all causes of lack of constancy in the chance cause system. B. They shall be free from constant errors of measurement and mistakes. C. They shall provide a basis for estimating the error of In ea.SUlrement. 4. Correction of Data for Constant Errors Let us consider the simplest kind of measurement, viz., that of a so-called physical constant such as one of those in the equation of electron emission as a function of temperature of the form where I = emission per unit area, T = absolute temperature, and a and b are constants characteristic of the emitting SUIT- face. Dushman considers in some detail the constant errors that * [.oc. cit. 38o ECONOMIC CONTROL OF QUALITY must be taken into account in making such measurements. Some of the most important sources of theoretical instrumental error are: A. Error in measurement of surface area at maximum temperature. Temperature gradients along emitting surface. Presence of adsorbed and occluded gases in emitter. Presence of gases in tube. Cooling effect of leads. Effect of anode voltage. Error in measurement of temperature. . Errors (A) and (E) are largely eliminated through design; (B), (F), and (G) are such that the observed data can be cor- rected with the aid of available but complicated theory. Errors from sources (C) and (D) are eliminated by proper bak- ing of bulb, flashing of the filament, and evacuation of the System. - Thus we get a picture of the technique required either to correct for or remove two of the sources of constant error in one very important physical measurement. A more detailed study of this problem of correcting data for constant errors will emphasize the fact that the degree of success will depend among other things upon the intuition, reason, theoretical knowledge, experience and technique of the experimentalist. Is it any wonder that engineering and even research data often fail to satisfy the prerequisite of being free from constant errors of the instrumental and theoretical kinds? It is also true that to correct for personal errors often presents a real problem. Often one finds a set of data revealing the psychological tendency on the part of the observer to favor certain numbers, a case of which was noted in Part VI. One of the most troublesome characteristics of such errors is the fact that many of the psychological errors from their very nature are such that we do not readily detect them in ourselves. Witness for example the tendency for us to feel that the two SAMPLING—MEASUREMENT 381 lines a and & in Fig. 131 are not of equal length although we know better. The method of detecting and eliminating assignable causes has been discussed in sufficient detail in Part VI and hence need not be considered here. It would perhaps be of interest N & / < º FIG. 131.-How Much LoNGER Is a THAN bf to show how mistakes can often be singled out even by analyt- ical methods. To do so, however, is out of place here because the best method of correcting for these is to take care not to make them, or to provide two independent observers. 5. Errors Introduced by Constant Systems of Chance Causes After the state of constancy in the chance cause system has been reached, the problem of correcting data for errors of TRUE EXPECTED OBSERVED VALUE VALUE * | | | | | | F–Ax—- | | | | X X X. FIG. I.32.-PROBLEM OF ELIMINATING ERRORS OF MEASUREMENT. measurement may be schematically indicated as in Fig. I32. In this the true value is represented by X, the expected observed value by X and the average of a sample of size n by X. The distance AX represents the resultant constant error which must be taken care of as indicated in the previous section. 382 ECONOMIC CONTROL OF QUALITY Under the assumption of constancy of the cause system, it follows that - Ls X = X 71 — CO where the limit Ls is statistical. In practice we usually take the observed average X as our best estimate of X and hence make our constant error correction with X as a base. Our problem is not solved, however, until we form some reasonable estimate of the probability that the inequality |X – X| < e is satisfied where e is some preassigned positive quantity. To do this it is necessary to obtain some estimate of the true standard deviation o of the objective dis- tribution of observed values. Except in the case of small samples we usually take the observed value of standard devi- ation as the best estimate of G. If we let 8. 2. F o/v/n then we may, subject to the usual assumption of normality of the distribution of error, use the normal law probability table to estimate the probability that the absolute difference exceeds zo/v/n. Thus we see that the complete discussion of the measure- ment of the simplest kind does involve the use of statistical as well as physical theory. An interesting illustration of such a system of errors attrib- utable to a physiological source is that shown in Fig. 133 repre- senting the distribution of minimum audible sound intensity." It is particularly interesting to note how closely the observed distribution is approximated by the normal law. 6. Correction for Constant Chance Errors of Measurement Let us next consider the case where the thing measured is itself a constant chance variable with average XT and standard deviation oz. Furthermore, let us assume that the error of * For a discussion of these results see “Some Applications of Statistical Methods,” by W. A. Shewhart, Bell System Technical journal, Vol. III, No. 1, January, 1924. SAMPLING—MEASUREMENT 383 measurement is such that the expected value of the measure- ment coincides with XT and that the standard deviation of the error of measurement is Ge. Assuming that the error of measurement compounds linearly with the true value and that there is no correlation between them, it follows 1 that s 2 2 Go = V Oz.” + Onº, (95) 225 2OO H. I75 H. > 150 F- O Z }º 125 II) Oſ i., IOOF- DC * 75 – 5 O H. 25 H. ol l -5 5 FIG. 133.−DISTRIBUTION OF MINIMUM Sound INTENSITY. where go is the standard deviation of the objective distribution of observed values. Fig. 134 shows schematically the rela- tionship between the objective true distribution fr(X) and the objective observed distribution fo(X). Example: Table 52 gives two observed distributions—one is the distribution of single measurements of efficiencies of I5,050 pieces of a given kind of equipment; the other is the distribution of five hundred measurements on a single instru- ment. It had previously been shown experimentally that there * If there is no correlation between the thing measured and the error of measure- ment, we may think of an observed value X as being the sum of a true value XT and an error E. Hence from section 3 of Chapter XVI, Part IV, we get (95). Another way of arriving at this result is given by W. A. Shewhart in an article “Correction of Data for Errors of Measurement,” Bell System Technical journal, Vol. V, pp. 11–26, 1926. 384 ECONOMIC CONTROL OF QUALITY TABLE 52.-TYPICAL CALCULATION INvolved IN ELIMINATING ERRoRs of MeAs UREMENT Measurements on Single Measurement on a Single Instrument Each of a Number of Instruments Cell Midpoint Frequency Cell Midpoint Frequency 2.8 2. O. O I3 3. I I6 O. 5 IO 3.4 46 I . O 8 3.7 88 I. 5 43 4. O I38 2 . O IOO 4.3 II3 2.5 815 4.6 71 3. O 1,761 4.9 22. 3.5 2,397 5.2 4. 4... O 3,431 4.5 3,703 5. O 2,165 5.5 5 Io 6.o 77 6, 5 I 5 7.o 2. n = 500 n = I 5,050 XE = 4.0606 Xo = 4.0251 O E = O. 4423 o'o = o. 81 16 was no correlation between efficiency and error of measurement. Since the numbers of measurements are large, we may assume that oo = go and GE = ap where a0 and ap are the observed standard deviations given in Table 52. With this assumption we get or = Vadº – gº = V(o.81.16)2 — (O.4423)* = O.68o3. 7. Analysis of Bad Data We are now in a better position to consider the practical problem of the engineer in trying to determine how far he shall go in analyzing his results. Again take as a simple illustration SAMPLING—MEASUREMENT 385 measurements of some so-called physical constant such as those considered earlier in this chapter. There is no known method for estimating the true value X of the constant and the true standard deviation o of the error of measurement from a set of m bad data—data that do not satisfy any one of the three prerequisites of good data. We cannot say, however, that the man who took these data cannot intuitively arrive at good (or at least practical) estimates of both X and G. Men of genius such as Poincaré claim often to advance intuitively first and logically afterwards." We have seen how intuition, hypothesis, imagination, and the like are basic to the process of finding and correcting for constant and assignable errors of measurement. If we turn to the history 4 of science and scientific method, we do not / \ OBJECTIVE DISTRIBUTION OF OBSERVED VALUES, fo (X) OBJECTIVE DISTRIBUTION of TRUE VALUES, fºr (X) FIG. I.34.—EFFECT of ERRoR of MEASUREMENT. find, however, many (if any) of the accepted estimates of so-called physical constants that have been obtained by in- tuitive use of bad data. Let us go a little further and see what would happen if we were to accept results obtained from bad data through the * See Dubs, Rational Induction, Chicago University Press, 1930, on this point. Such questions lead us into the fields of logic, psychology, and philosophy in an attempt to reduce to a rational basis the rôle played by each of these in measurement. Other references along this line are given in Appendix III. * References in Appendix III. 386 ECONOMIC CONTROL OF QUALITY intuition of the experimentalist. Immediately the analysis of data would be removed from the field of logic and we would have to accept a result simply on the basis of the authority of the experimentalist. Then we would face the difficult task of de- termining the ultimate authority. Such a method is certainly not scientific, nor does history reveal much ground for belief that it is a method which can be relied upon to give satisfactory results. In the light of this situation it seems reasonable to believe that we are not justified in basing industrial development on intuitive analyses of data. This does not mean that experi- mental science has not profited by hunches that have come to those in the process of collecting data later found to be bad. The very fact that an experimentalist feels that his data are bad is usually an incentive to get good data. A research man is usually concerned with the fact that he may unknowingly get bad data. Here it is that the mathematical theory of detecting the presence of assignable causes (Part VI) comes to his aid." To get the best results through the use of these criteria requires that the data be divided into rational subgroups and that at least the averages and standard deviations of these subgroups be known. 8. Analysis of Good Data Good data in general are expensive. In the process of getting them many measurements are usually taken, from which a few are finally chosen as being good. Furthermore, even though the cost of getting good data is large, experience shows that the cost of making the most effi- cient analytical study of such data is relatively small. In Part VI the problem of choosing statistics to be used and of choosing the best way of using them was considered. * In this connection the following quotation from Mathematics of Life and Thought by A. R. Forsyth is of interest. “Briefly, the science of mathematics cannot be a substitute for essential experiment; but it can show how experiments and observa- tions, duly systematized, can be elucidated so as to discriminate between what is principle and what is detailed consequence of principle.” The criteria described in Part VI help to discriminate between what should be and what should not be left to chance. SAMPLING—MEASUREMENT 387 The fact that one statistic is often much more efficient than another is of considerable economic importance. For example, in general, the standard deviation of n good observations is just as good as the mean deviation of I. I.4 m such data. To take on the average one hundred and fourteen observations where one hundred would do is an unnecessary waste of money which becomes significantly large in extensive industrial research programs. To use the range in such a case instead of the standard deviation effectively results in throwing away a very large fraction of the information in respect to dispersion contained in the observed data. For example, if the sample size n is approximately sixty, the efficiency of the range is only about 50 per cent when compared with the standard deviation. As m increases beyond this value, the efficiency of the range rapidly decreases. In the face of this fact we sometimes find the range instead of the standard deviation tabulated in the literature." Table 53 is taken from an engineering report, and gives the modulus of rupture for three species of telephone poles. To TABLE 53.-Illustratino INEFFICIENT METHOD of TABULATING DATA Number n Modulus of Rupture in psi Efficiency of Species of Poles in Max.-Min. Sample Average | Maximum | Minimum || A . 4 3,985 5,690 2,98o IOO B I6 5,978 7,090 4,460 75 C IOO 5,787 7,790 3,49C 35 have tabulated only the ranges in Cases B and C amounted to throwing away approximately 25 per cent and 65 per cent of the information available in the original data. This state- ment is based upon the assumption that the original data were good and that they came from an approximately normal uni- verse. Of course, the range in bad data may give the experi- * Examples of this kind are Tables 4 and I2 in the first edition of the very inter- esting book, Timber, Its Strength, Seasoning, and Grading by Harold S. Betts, McGraw- Hill Book Company, pp. 34 and 91, 1919. 388 ECONOMIC CONTROL OF QUALITY mentalist some indication of the effects introduced by assignable causes of Type I. As indicated in the previous section, however, the interpretation of the range or any other statistic derived from bad data should be made by the experimentalist and can be accepted by another only upon the authority of the exper- imentalist. For a sample of four, practically all of the information con- tained in the data is retained by using the range. It is for such reasons that efficiency in analysis and pre- sentation of data has been considered so often in the previous chapters. Graphical methods of analysis have not been given any attention simply because experience has shown them to be inferior to and less efficient than analytical ones." 9. Minimizing Cost of Measurement—Simple Example Let us consider the following simple problem: What is the most economical way of measuring a quality X controlled by a constant system of causes to insure with a given probability P that the average of the measurements will not deviate in absolute magnitude from the average XT by more than a pre- assigned quantity e. Let us assume that: a1 = cost of selecting each unit and making it available for measurement, a2 = cost of making each measurement, m1 = number of units selected, n2 = number of measurements made on each unit, os = objective standard deviation of errors of measure- ment and 5 or = objective standard deviation of true magnitudes of the measured characteristic. 1 Whittaker and Robinson make the following statement in the preface of their classic, The Calculus of Observations: “When the Edinburgh Laboratory was estab- lished in 1913, a trial was made, as far as possible, of every method which had been proposed for the solution of problems under consideration, and many of these were graphical. During the ten years which have elapsed since then, graphical methods have almost all been abandoned, as their inferiority has become evident, and at the present time the work of the Laboratory is almost exclusively arithmetical.” SAMPLING—MEASUREMENT 389 Let us take P = o.9973, then the range X, + 30x, includes 99.73 per cent of the observed averages Xo, and hence e = 30x. The average of n2 measurements made on one unit is to be taken as the observed value Xo of the true magnitude XT for that unit. This average has the standard deviation ge/v/m2. Hence, from (95), the objective standard deviation of the observed values is given by on,” go” = q^+ o-pº = g^+--, 712 where oºz, is the objective standard deviation of errors of aver- ages of m2. Thus the objective standard deviation ox, of the average of mi observed values is Oſ g- OT” + E O’Rz = 90 772 (96) X E - Tº F --- 0 v/n: 7? I 5 which gives the relationship on,” OT” + 722 711 = ox.” between n1 and m2. Taking the cost of inspection as C = a1n 1 + 42?! 1712, and using customary methods this can be shown to be a min- imum when OTE 772 = — OT, Nº. The following values correspond to one practical case: e = 0.3 unit a1 = $0.50 GE = O.3 unit a2 = $O.O2 OT = O.9 unit P = O.9973 With the aid of this theory we find that the most economical method of measurement in this case requires two observations 390 ECONOMIC CONTROL OF QUALITY on each of eighty-six units. Here, as in general, we take observed values of gº and at in large samples as estimates of o's and or respectively. Io. How Many Measurements? Perhaps the question most frequently raised by those in- terested in the control of quality is: How many measurements shall be taken? Of course, for such a question to be answerable, it must be understood to mean something like this: How many measurements shall be taken in order that one may have a given assurance that such and such is true subject to certain specific assumptions? When so stated the question usually has an objective answer. Sometimes the question is put briefly as follows: How large a sample shall be taken? When so stated, however, care must be exercised to differentiate between the size of sample, meaning thereby the number of things measured, and the size of sample, meaning thereby the number of measurements, where one thing may be measured more than once. The significance of these remarks will be apparent as we proceed. To introduce the subject, let us ask a very simple question: Assuming that we know that a quality X is normally controlled with standard deviation o, how many measurements of this quality must we make in order that the probability will be, let us say, O.9973 that the deviation of the average of n observed values from the true but unknown arithmetic mean X be not greater in absolute magnitude than some given value AX. From what has previously been said we see that the size n of the sample required in this case is rigorously given by the relation *= Oſ AX = 3–7–. v/n In practice, however, we do not know o'. In fact, this factor is only obtainable as a statistical limit when the sample size n is made indefinitely large. What we can do under such con- SAMPLING—MEASUREMENT 39 I ditions is to estimate o from available data. Calling this estimate or, we may solve for m in the equation * Or Ax = 3; We can then say that the size m of the sample thus obtained is the one required, assuming that a = 0. Perhaps the most important thing to note in this connection is that the standard deviation of the average decreases inversely as the square root of the number of observations, because this indicates the order of increase in the precision of the average with increase in the number of observations under the assumed conditions. In general, if we know that we are sampling from a constant system of chance causes, we can say that the standard deviation of an estimate of any one of the objective statistics, fraction defective p, average X, standard deviation o, and correlation coefficient r, decreases inversely as the square root of the size of the sample, even though we do not know the magnitudes of the respective standard deviations in a given case. Further- more, given the standard deviation as a function of sample size, for any statistic derived from a sample from a specified universe, we have, as indicated, a means of determining the significance of increasing the sample size. It is very important to note that the answer given to the question of how many measurements is in each case limited by the assumption that the variable X is controlled. If we ask a similar question in a case where we are not willing to assume to begin with that the data are controlled, it is first necessary to try to determine by criteria already described whether or not the variable under consideration satisfies this condition. Example: Recent investigations have been made by the American Rolling Mill Company to determine the life of ferrous materials under different corrosion conditions. Data obtained * R. F. Passano and Anson Hayes, “A Method of Treating Data on the Lives of Ferrous Materials,” Proceedings of the American Society for Testing Materials, Vol. 29, Part II, 1929. 392 ECONOMIC CONTROL OF QUALITY from a certain kind of sheet material immersed in Washington tap water showed that the average time of failure of such samples was X = 874.89 days and the standard deviation of the time of failure was a = 85.31 days. One kind of practical question of interest to the research engineer of this company is: What sample size m must be used in order that for similar test conditions, the probability shall be O.90 that the average time for failure determined from the m tests will be in error by not more than 5 per cent of the average of the universe? Assuming that the observed values of average and standard deviation are the true values for the universe, and that averages of samples of n are distributed normally, we may answer this question as follows: The allowable error is 5 per cent of 874.89 days or 43.74 days, and this must correspond to a probability of O.90 or to an error of 1.645 aſ Vn as found from Table A of Part II. Hence n is found by solving the equation 1.645G/Vn = 43.74 having assumed that G = 85.31. In this way, we get m = Io. II. Law of Propagation of Error–Practical Significance Most measurements are indirect in that the quality Y to be measured is derived from measures of let us say m other qualities X1, X2, . . . , Xi, . . . , Xm to which it is either functionally or statistically related. In this section we shall consider the functional case, examples of which are met in everyday work. A simple illustration is the measurement of the density D of a solid by the formula D = 701 701 – 702 where widnd we are the weights of the solid in air and water respectively. § If the solid is such that we can measure its volume / in a SAMPLING—MEASUREMENT 393 more direct way than by determining the difference wi – we we may use the formula - D = # to obtain the density. - - The choice of method of measurement involves at least two things: A. Determination of effect of errors of measurement in each of the m qualities upon the standard deviation of the calculated values of Y. B. Choice of most efficient method of measuring Y. Let Y = F(X1, X2, . . . , Xi, . . . , Xm) be the functional relationship between the quality Y to be measured and the m other qualities upon whose measurements the calculated (measured) value of Y depends, as the calculated value of D depends upon the observed values of wi and wo above. Assuming that F can be expanded in a Taylor's series and that terms containing higher powers in the x's than the first may be neglected, we have - - - * OF (...) ( º) F 23 . . . , X *mºm-s-s-sºm-º. 2 T- * @ s 5 Y = F(X1, X *(*). OX2 x' + X'm. OXm/x, where xi = X. — Xi, and the derivatives are formed for the mean values of the X's. Under these conditions we have as in Part V - Y = F(X1, X2,..., Xm), and O y = Valºgº + ag”o 2* + . . . -- ałºgº + . . . -- amºo m”, (97) where (#) ai = \ – ) , OXi/x, o, is the standard deviation of the measurement of Xi, and Gy is the standard deviation of the indirect measurements of Y. 394 ECONOMIC CONTROL OF QUALITY Equation (97) is the law of propagation of error, and gives us the information called for under (A). If for the simple problem of measuring density we let wi = expected weight in air, W2 = expected weight in water, oi = standard deviation of measurement of wi, oz = standard deviation of measurement of w2, and ord = standard deviation of error of measurement of D, we have on applying (97) I go = —-VW.”gº + wºoº. W1 — W2 By a process exactly similar to that used in Paragraph 7, Chapter XVII of Part V, we can determine the mean values X1, X2,..., Xi, ..., Xn (if they exist) which will minimize gy. By comparing the minimum values of oy obtainable by different methods we can arrive at the most efficient method of measuring Y. 12. Measurement through Statistical Relationship Let us consider the problem of measuring some physical quality such as tensile strength which cannot be measured except through the use of some statistical relationship unless we resort to a destructive test. Let us start with a simple question. How can we be sure as to whether or not the tensile strength of the bar in Fig. 135 lies within specified limits Yi and Y2: The answer is: Break it and find out. However, since we cannot break it and use it too, we must be satisfied with the answer to a slightly different question: How shall we test the bar indirectly through sta- tistically correlated variables? Let us start with the illus- tration introduced in Part I, Fig. I4. Let us consider first the correlation between tensile strength Y and hardness X. We can never expect to be sure that the tensile strength of tested SAMPLING—MEASUREMENT 395 material will lie within two specified limits Yi and Y2 by making sure that the hardness lies within some two limits X1 and X2. The situation is shown schematically in Fig. 136, for the - Fig. 135–Test BAR. data of Fig. 14-a. In such a case values of tensile strength may be expected to be found in the shaded area of the figure between the limits X, and X2 and outside the limits Yi and Y2. Y2 Y HARDNESS Fig. 136.-WHY ONE CANNot BE SURE THAT STRENGTH LIEs witHIN SpecIFIED Limits. If, and only if, the product is controlled in respect to the two correlated variables Y and X, can we predict how many pieces * Mathematical details considered in Part II. 396 ECONOMIC CONTROL OF QUALITY of material having quality X within the range X, to X2 will have quality Y within the range Yi to Y2. In other words, the use of indirect statistical measures must be based upon the assump- tion that the probability P that the point corresponding to an observed pair of values X and Y will fall within a given rectangle is constant. - A. Calibration.—Suppose one has a lot of N. pieces like the One shown in Fig. IS5, and wants to mark each of them with a F - A C . A - LIN E OF REGRESSION X ON Y B - LINE OF REGRESSION Y ON X C - LINE OF BEST FIT | | I l I | H A R D N E S S - X : FIG. 137.-SHALL ONE of THESE LINEs BE USED For CALIBRATION? value of tensile strength derived from the corresponding hard- ness measure. What functional relationship between Y and A shall be taken as a basis? In other words, how shall we calibrate Y in terms of X assuming that these two variables are normally correlated? Shall we take one of the three lines illustrated in Fig. 137? Let gy = standard deviation of objective distribution of ten- sile strength Y, SAMPLING—MEASUREMENT 397 or = standard deviation of objective distribution of hardness X, r = correlation coefficient between X and Y in objec- tive distribution of X and Y. It follows from the discussion of lines of regression in Part II that the line of regression of tensile strength on hardness Oy = r— X: y or * where y = Y — Y and x = X – X, gives the expected or aver- age value of y to be associated with a given value of x. In other g . . IOT words, if we were to mark with *x each of a very large number O'a, n test pieces that gave a hardness value X + x, and then we were to break these to determine their tensile strength, we should expect to find that the average tensile strength of the g roſ e m pieces would be *.x, although the observed tensile strengths O'a, would be distributed about this value. Furthermore we should expect 99.73 per cent of the m pieces to have tensile strengths measured in terms of deviations, within the limits roſ, - .* + 30 y\/ I – rº, 3C since as we have seen in Part II, the standard deviation of any y array about its mean in this simple case is Sy = gºv/ – rº. (98) In fact, if the regression of y on x is linear and the scatter of points is homoscedastic, then the standard deviation of each On! . . array of y's about the mean rºx is given by (98) and we can Oa: say by virtue of Tchebycheff's inequality that more than 398 ECONOMIC CONTROL OF QUALITY I e iceſ - }) per cent of the y values may be expected to lie within the band Oy r-ºx == tsy. (99) Oa: Where the correlation surface is normal, the number of points lying within such a band is given by the normal law integral. Under the same conditions, similar statements hold with respect to the regression of x on y. It is sometimes argued that some line other than the line of regression should be used as a measure of y in terms of x. One such suggestion is that line for which the sum of the squares of the perpendicu- lar distances of the points in the xy plane to this line is a minimum. The reason for choosing the line of regression instead of this or any other line is that this is the only line about which we can make the general statements previously made in connection with the range (99). In the discussion of Fig. I4 it was pointed out that the use of the plane of regression of tensile strength Z on hardness Y and density X is a better measure of tensile strength than either the line of regression of Z on X or Z on Y. This follows because the standard deviation, © 9 o I I – rary” – rºz” – Tzz” + 2r ** Oz. zy = •l ... 2 I — Tay” of the values of tensile strength from the plane of regression is less than either - S2a: = OT2 V I – raz”, Or Szy = gav'ſ - ryzº, where Sac and Say are the standard deviations of tensile strength from the lines of regression of z on x and z on y respectively. B. Effect of Error of Measurement.—Thus far we have con- sidered the problem of measuring some quantity such as tensile strength Y through its statistical relationship with some other SAMPLING—MEASUREMENT 399 quantity, let us say hardness X. In general, the observed values of both tensile strength and hardness are in themselves subject to error. Let us assume for example that or = The standard deviation of the objective distribution of the observed values of hardness, gy, = The standard deviation of the objective distribution of the observed values of tensile strength, or, − The standard deviation of the objective distribution of true values of hardness, gy, = The standard deviation of the objective distribution of true values of tensile strength, ro = The true correlation between the observed values of hardness and tensile strength, and r = The true correlation between the true values of hardness and tensile strength. It can easily be shown that under these conditions Oſ **urr (IOO) Oa,0-yo * From this relationship we see that the correlation between the observed values of two correlated variables is always less than the correlation between the true values, unless the error of measurement of each of the variables is zero. In other words, the smaller the error of measurement for each of the variables, the more precise will be the regression method of measuring one in terms of the other. C. Conclusion.—To be able to measure through the use of statistical relationship, it is necessary that the variables be con- trolled. In the simple case of normally correlated variables the line or plane of regression has certain advantages as a calibration line or plane over any other. It should be noted, however, that the use of a statistical calibration curve involves the introduction of a concept quite different from that underlying the use of a calibration curve 4OO ECONOMIC CONTROL OF QUALITY based upon a functional relationship. The use of statistical relationship introduces a certain indeterminateness not present in the use of the functional relationship. To make this point clear let us suppose that we had, say one hundred bars, such as shown in Fig. I 35, and let us suppose that we wished to test these for tensile strength indirectly through the use of the Rockwell hardness measure. If we assume that tensile strength Y is functionally related to X, as - Y = f(X), (IoI) where f is a single valued function, then for every X there is one and only one value of Y. If we use such a calibration curve, we can mark each of the one hundred bars with a value Y which will be the tensile strength of that bar except for errors of measurement. If, however, the two quantities Y and X are related sta- tistically and we use a line of regression y = r-x, (IO2) QC where y = Y – Y and x = X – X, then we cannot say that for a given value of X there is only one value as given by the line of regression of y on x. Instead for every X there is an array of Y’s, the mean of which under controlled conditions will be the value of Y given by (IO2). Here we run into the kind of indeterminateness discussed in the last chapter. Equation (80) expressing r as a measure of the commonness of causation under simplified and controlled conditions may help one to form a better picture of the significance of the line of regression (IO2) as a calibration curve. Unless r is unity there are always causes of variation in Y that are not present in X. Even under these simple conditions if we could be sure that the correlation coefficient r and the variable X were con- trolled, we could not be sure that Y was controlled and we could not be sure of the interpretation of y as given by (102) except in the sense that the mean value of y for a given X would SAMPLING—MEASUREMENT 4OI be given by (IO2). A priori, however, it seems unlikely that Y will be uncontrolled if both r and X are controlled. At least it appears that the best we can hope to do in trying to control Y through the measure X is to try to control r and X. D. Example: Since the use of statistical relationship plays such an important rôle in measurement, it may be of interest to consider another simple problem. Many machine measures of quality depend upon the use of statistical relationship. A very important type of machine in the telephone plant is that introduced to supplant measures depending upon the human H. Z lil OC OC J O I | f 1 | | | | | | | ſ O | O 2O 3 O 4 O 5 O 6O TIME IN SECOND S X | O ~3 FIG. I.38.-OscILLogRAM of “Noise CURRENT.” ear, such as in testing the quality characteristics of telephone instruments. Fig. Ig8 shows the oscillogram of a greatly magnified “noise current” attributable to chance fluctuations in the resistance of a certain kind of telephone instrument. It is obviously desirable to go as far as one can in reducing such noise to a minimum and in controlling the effect of this kind of distortion as measured by the human ear. Consequently, all instruments of this type are tested to make sure that they meet specification requirements in respect to this kind of distortion. Of course, the cost of doing this by ear would be prohibitive; therefore it is desirable to secure the economic advantages of a machine 11162a SUlſe. 4O2 ECONOMIC CONTROL OF QUALITY A little consideration will show, however, that it is almost hopeless to expect to be able to find a machine measure of such fluctuations in current that will be functionally related to the measures of the human ear. The best we can hope to do is to find some machine measure X which is statistically related to the ear measure Y. 28.95 24.35 19.75 | 5.15 HO.55 5.95 1.35 O. 1493 O.1637 O. 178 | O.I.925 O.2O69 O.2213 O.2357 O.25 Ol M A C H | N E M E A SU R E FIG. I.39.-How SHALL WE CALIBRATE MACHINE MEASURE IN TERMs of EAR MEASURE; Fig. I39 shows the calibration scatter diagram of a machine measure X and ear measure Y on 942 instruments. These data were obtained under conditions of control as determined by the criteria described in Part VI. The solid line in this figure represents the line of regression of the ear measure Y on the machine measure X. The fact that the difference my.” – r" is approximately zero indicates that we are justified in assuming linear regression. This incidentally is what we should expect to get for reasons outlined in Part III. SAMPLING—MEASUREMENT 4C3 For reasons given previously in this section, it appears that there is good ground for the belief that we may control the quality Y determined by the ear by controlling the quality X determined by the machine in respect to both the average X and the standard deviation or in samples of size n. To check the calibration of such a machine, it is necessary that the cor- relation r between the ear measure Y and the machine measure X for a sample of n instruments be controlled in the sense of the criteria of Part VI. CHAPTER XXIV SAMPLING I. Fundamental Considerations Table 54 gives the results of measurements of modulus of rupture on twenty-four telephone poles of species D. Based upon these data, what can we say as to the strength of this species? Assuming that no assignable causes of variation of Type I are present, or in other words, assuming that these poles came from a constant system of chance causes, it follows from the discussion of the previous chapter that reasonable estimates of the average X and standard deviation o of the distribution of TABLE 54.—MoDULUs of RUPTURE of Twenty-Four TYPE D TELEPHONE Poles - Pole Modulus of Pole Modulus of Number Rupture Number Rupture I 3,643 I 3 5,385 2. 5, IQ5 I4. 5,843 3 3,925 I 5 6,905 4. 4,595 I6 5,696 5 4,482 17 7,392 6 6,248 I8 6,184 7 - 6,or 2 I9 4,885 8 6,697 2O 6,182 9 7, II 7 2. I 6,2OI IO 5334O 22. 7,334 II 8,712 23 5,497 I Q. 5,819 24 4,621 Average. = 5,829 psi a = 1,159 psi 4O4. SAMPLING 4O5 modulus of rupture given by the assumed constant system of Call SCS alſº e - A = X = 5,829 psi * = g = I,197 psi. C2 So far as the distribution of the twenty-four observed values is concerned, there is no definite evidence of lack of constancy in the chance cause system. Under these conditions one would be led to the conclusion that the average strength and standard deviation of this species of telephone pole are 5,829 psi and 1,197 psi respectively. Any one who knows anything about the strength charac- teristics of timber would likely and justly challenge such a conclusion. For example, such a one would likely ask what effect moisture content has on the strength of poles of this species, knowing as they would that moisture is at least for most species an assignable cause of variation in strength. Dividing the poles in respect to moisture content in this case leads to the results shown in Fig. I40. There can be little doubt that moisture content is an assignable cause in this case. How then, does this effect the validity of our conclusion arrived at upon the assumption of constancy? . From Fig. 140 it appears that there is a difference of the order of magnitude of 1,000 psi between the strength when the pole is dry and that when it is wet. What strength one may expect to find in the future then may be something nearer 5,000 psi than the predicted 5,829 psi if the poles to be tested are wet. It appears that prediction based upon a sample coming from a non-constant system or non-controlled system of chance causes may differ widely from what the future will reveal. What reliance then, asks the engineer, can be placed on sampling results? The answer is that pre- diction based upon a sample from a non-controlled universe in which the causes of lack of control are unknown is likely always to be in error just as a measurement uncorrected for constant errors always in the long run is in error. Sampling theory 4O6 ECONOMIC CONTROL OF QUALITY applies to samples arising under controlled conditions. Too much emphasis cannot be laid upon this fact. To be able to make accurate predictions from samples, we must secure control first just as to make accurate physical measurements, we must eliminate CO725/47?? 6.7°7'07"J. In this section we have approached the problem of inter- preting a sample from a practical angle, and in so doing, have been led to see the importance of control. Having read Parts III, IV, and VI, one sees that the only theoretical basis of 9 ſ © , 8 H 2. til or CŞ O O D J) O H. “ 7 H. © 5 of O º lu Ol. 8 e O U- . |-sº O O 6 CD O O wo- O P2 e *e O O ID º Po 5 O O O © > z © 4 H. O O O L. | | | 1– | | º - | f O | 2 | 6 2O 24 28 32 36 4 O 44 4.8 M O | STURE CONTENT IN PER CENT FIG. I.4o.—WHY ConstANCY of CAUSE SYSTEM Is EssentIAL FOR PREDICTION. interpreting a sample is the assumption that it arose under controlled conditions characterized by (45) in the most general case, or in other words, by the fact that the sample was taken under the same essential conditions that will maintain through- out the future so that the universal physical law of large num- bers applies. 2. Random Sample A sample taken under conditions where the law of large numbers (45) applies will be termed a random sample. This concept of SAMPLING 4O7 random is of fundamental importance in the theory of control. By simple illustrations we shall now try to make clear how this concept differs from that of some of the prevalent definitions of random in order that no confusion may arise in the use of the term in this book. Yule in that treasure house for statisticians, An Introduction to the Theory of Statistics, indicates that the usual concept of random sample is one drawn with replacement, though he crit- icizes the use of the term random because it is so often taken to be synonomous with haphazard. Caradog Jones 1 apparently would also have us believe that a random sample is one drawn with replacement. For example, he says in effect: To select 99 sheep from 999, number each sheep and place in a box 999 tickets numbered I to 999, one to correspond to each sheep, then pick out 99 tickets in succession being careful to replace each and shake up the box before picking out the next; if there were absolutely no difference between the tickets such as would cause one to be picked more easily than another, the selection made in this way would be random. Now, if a random sample were only that kind of a sample and if the theory of sampling had to start with that kind of a sample, one can well imagine how enthusiastic a purchaser of 999 sheep would be about the theory. To such a man that method of sampling would be foolish. - Not only is it foolish from a practical viewpoint in certain cases to try to take this kind of a sample—very often indeed, it is impossible to take a sample with replacement. As an illus- tration: How would you take a sample tensile strength test with replacement from the coil of wire in Fig. I41 The kind of sample described by Yule and Jones is random, of course, but so are other kinds of samples as will be apparent from a study of the generalized law of large numbers (45). Thus either a sample without replacement or a Poisson sample may be random in this general sense. * A First Course in Statistics, G. Bell & Sons, Ltd., London, 1924. 408 ECONOMIC CONTROL OF QUALITY 3. Sampling for Protection Various methods of setting up sampling schemes to give definite consumer's risks were outlined in the last chapter of Part VI. A study of the subject matter of the references there given shows that the conclusions drawn rest upon the assump- tion that samples are selected at random. In other words, assuming that there are N items in the lot to be inspected, it is necessary that the sample of n required by one of these sampling Fig. 141.-How SHould WE Choose A RANDoM SAMPLE of THE TENSILE STRENGTH of THIs Coil of WIRE” schemes, for a certain consumer's risk, be drawn at random. The risks calculated in this way apply so long as the samples are random. If, however, the samples are not random, the risks do not necessarily hold." The kind of random sample required by the risk theory can be obtained by sampling without replacement from a bowl containing N identical chips marked I to N where it is assumed that the chips have been thoroughly stirred before the sample of n is drawn. We can see, therefore, the nature of the dif- ficulties involved in getting a random sample of the poles from the pole yard of Fig. 142. As another illustration let us consider the problem involved in drawing a random sample of soldered terminals from fifty panels such as the one shown in Fig. 143 where there are 4,500 terminals on each panel. Cf. Sec. I of this chapter. SAMPLING 4O9 We need not go further to see that it is very seldom feasible to draw a sample in which the experimental conditions requisite for randomness have been secured. Therefore we must rely upon the engineering ability of the inspector to divide as in Part VI the total lot N to be sampled into, let us say, m sub- groups which a priori may be expected to differ assignably. A sample may then be drawn from each subgroup of the right size to insure that the chosen risk is met by the sampling test for each particular group. These remarks are sufficient to emphasize the importance of a priori information about the lot prior to the taking of a sample. Now let us consider the problem of selecting a sample from a shipment of ten carloads of boxed material, there being twelve items in a box and roughly 1,000 boxes in a car. Obvi- ously it is not feasible to arrange experimentally for a random sample to be drawn. The next best thing is to try to divide the total of N = 120,000 items into m rational subgroups. If, however, we know nothing about the manufacturing process or the conditions under which the lot was produced, we are faced with the necessity of doing something that we cannot do; yet we know that unless the sampling is done as it should be, sampling theory does not apply. Fig. 142.-How Should We Choose A SAMPLE of THE Poles IN THIs YARD? 4 IO ECONOMIC CONTROL OF QUALITY Thus we see how important it is that the consumer know assignable causes of variation if he is to devise a sampling plan to insure that the product accepted is of satisfactory quality. If the product is controlled, one can easily set up a satisfactory sampling plan, but if it is controlled, the plan is often not needed. If the product is not controlled, the consumer needs to know the assignable causes of variation so as to establish an adequate sampling scheme. - - - - - - … - . - -- - -º-º-º-º-º: º º Fº º º: º . . . . . . º º º º - - - - - - - - s - - - - - - ºl ---…-º-º-º-º-º-º-º-º-º-º-º: º: º | º º | | || º º º - - - - º - º º º --- - - - - - - - - - - - - - - - - - --- - ſº . . . . . ſº - º * º --- º . . . . . . . . . | - º - º - . . . . º . . . . . º - - º º … - ºf º - gº - . - - º:---------------º-º-º-º-º: == Fºº . - - - - º ºººººººººººººººººººººººººººº. | "…º. º . . . . . . . . . . º . . . . . . -- - - - - --- º - º - ººººººº. º ---------~~ --> --> - -º- Fig. 143.-How SHould WE Choose A SAMPLE of THE SoldeRED TERMINALs in THIS PANEL2 In this way we come to see the advantage of control to both consumer and producer. Just as each of these now secure advantages through cooperating in laying down specifications' for quality, it is reasonable to believe that each will soon try to obtain the mutual benefits of control. 4. Representative Sample A sample that is representative of what we may expect to get if we take additional samples, is one satisfying the general * On this point see H. F. Moore's Text-Book of the Materials of Engineering, 4th Edition, Chapter XVII, McGraw-Hill Book Company, 1930. SAMPLING 4 II condition (45) of the law of large numbers. In other words, if we let N be the total universe, finite or infinite, to be sampled, we should try to divide the universe on an a priori basis into m objective rational subgroups as represented schematically in Fig. 144. The total sample of n should then be divided between these m subgroups in such a way as to give some indication of what we may expect to get from each group. FIG. I.44.—SchemATIC of DIVISION INTo RATIONAL SUBGROUPs. 5. Size of Sample We have seen in the previous chapter and in the last chapter of Part VI that the size of sample always depends upon what we assume a priori to be the conditions under which we are sampling. In any case the interpretation of the sample rests upon the assumption of control, or upon the assumption that the law of large numbers holds in the particular case. Thus we need to know if the quality of the product gives evidence of control, and in this way we are forced to come back to the problem discussed in Part VI. A very simple case will illustrate this point. Several years ago an engineer reported trouble on the job because the width of saw-slots in the screw heads was under minimum require- ments so that the available screw-drivers could not be used. The question was raised as to how large a sample n should be inspected in each lot of size N to protect against the recurrence of this trouble. Investigation revealed that a sampling plan 4. I2 ECONOMIC CONTROL OF QUALITY was already in use in which a certain fraction was taken from each lot of N. Just a little engineering investigation showed that the only assignable cause of the kind of trouble reported was wearing of the saw blade that made the slot. The obvious thing to do was to inspect the blade and not the screws. The important question was not “how many,” but rather “how.” A few measurements of the saw blade to control the product were worth far more than many measurements made blindly, as it were, on the screws to find trouble that should have been, and could easily have been, eliminated. 6. Size of Sample—Continued To summarize, we may say that the answer to the question as to size of sample depends first of all upon whether or not we can assume that the product is controlled. However, to deter- mine whether or not the product is controlled, it is necessary to use the sampling process after the manner discussed in detail in Part VI. The answer to the question—How large a sample? depends upon the following five important things considered in that chapter: A. Ability of engineer to divide data into objective rational subgroups. Choice of statistics. Choice of limits for statistics. Choice of method of using statistics. : The way control is specified. Illustrative examples showing the importance of each of these five factors have already been considered in Part VI. It may be of interest, however, to give one more illustration here to show the importance of choosing the right statistic in de- tecting lack of control. Fig. I 45-a shows the observed fraction defective in a certain kind of apparatus over a period of ten months. Beginning about April, the rejections for this kind of apparatus became SAMPLING 4 IS excessive. It is of interest, therefore, to see how this trouble could have been detected through the use of a control chart on fraction defective. Such a chart (Criterion I, Part VI) is shown 50 ri 4O H. O O |-e z 3O H. til © O O O O uſ 20 H. O Cl- O Q O I O He O O O O O e e O O O O O O. © Ye ee e O O O O e e o e •e *e e - e. • ? . | : | | | ſº O' N D J F M A M J M O N T H FIG. I.45-a.—WHEN DID TRouBLE ENTERP in Fig. I45–6. An indication of the presence of assignable causes of variation is given by this chart eight weeks in advance. Investigation revealed that it was very likely that the assignable cause at this particular time was the same as that found to have 5 O r- 4 O H. O O H. 2 30 H. Lil O O O • e * OC O O up 20 H 0. rºoveLes O O O I O H. O O –––––-s-— — — — gº — — — se——e e-—e-2– — — — — — — —*—e. e ee ee, e - * * * * e o e " O O — — ?---- *——e— — — — — — — — — — — — — — — — — — — — — O L- | I \ | ſ | ) | | —-f O N D J F M A M J J M O N T H FIG. I.45-b.—Even AN INEFFICIENT ContRol. CHART CAUGHT TROUBLE EIGHT WEEKS IN ADVANCE. 414 ECONOMIC CONTROL OF QUALITY caused the trouble beginning about the second week in April. - As shown in Parts V and VI, the average is usually a much more sensitive detector of assignable causes than is the frac- tion defective. It so happened that the quality of a few instru- ments of this particular kind had been measured as a variable each week over this same period. Applying Criterion I to these data, we get the results shown in Fig. I 45-c. Evidence 2 .C. ~5 2 TROUBLE Li J > O g Q * - e o e O O O LU O © C e O O e O O O § ºf O • * ©e e O CC O g “-1 L I f ſ f º I | f O N D J F M A M J J M O N TH FIG. I.45-c.—EFFICIENT ContRol CHART CAUGHT THE TROUBLE SIXTEEN WEEks IN ADVANCE. of lack of control is given by this chart sixteen weeks prior to that given by Fig. I 45-b. Such results are typical of those experienced every day in the analysis of inspection data to detect lack of control. Having assured ourselves that the product, is controlled about a certain level of quality, it may be desirable in some instances to set up sampling limits to give a certain assurance that the quality in a given lot meets certain limits. From what has been said in previous sections, it appears, however, that the size of sample required to give the desired assurance depends upon the following factors: (2 . Kind of risk. Magnitude of risk. Kind of sampling scheme. . Kind of specification. º Previous information as to the quality of product. º 6. SAMPLING 4I 5 Obviously, therefore, the answer to the question—How large a sample?—even when product is controlled—depends upon several factors. Of course, the need for protective sampling schemes is very much reduced when we have the assurance that quality is being controlled. 7. Size of Sample—Continued To emphasize the importance of the conclusions stated in the previous section, let us consider very briefly four typical problems. A. Quite recently, the head of a large organization inter- ested in the production of linseed oil raised the following question. Three shiploads of flaxseed constituting a lot of approximately 65,000 bushels had been received. A test sample for chemical analysis had been taken from each shipload, the manner of taking being unknown. An order had been accepted for several thousand dollars’ worth of oil at a price based upon the results found in the sample. When sufficient oil to fill the order was extracted from a portion of the flaxseed, it was found that the average oil content was so much less than that of the sample that the producer suffered considerable loss. The question asked was: How many samples should be taken under similar circumstances in the future in order to prevent the recurrence of such loss? If we turn to almost any book on the specification of prop- erties of materials for design purposes, we shall find problems of which the following three are typical. B. Given the observed distribution, Table 55, of resistance of a sample of 904 pieces of a given kind of apparatus, what is the tolerance limit X2 that will not be exceeded more than, let us say, O.5 per cent of the time? C. The tensile strength of Code A wire shall not be less than 21,000 pounds per square inch. How many samples shall be taken in order to insure that the specification is being met on a carload lot? D. Fig. I46 shows a typical cross section of a coating mate- rial. One of the specification requirements is that this coating 416 ECONOMIC CONTROL OF QUALITY shall have an average weight between twenty-five and fifty milligrams per square inch. The question is: How shall we sample this product to insure that this quality specification is being met? TABLE 55.-How Should WE CALculate Tole RANce LIMITs? Resistance Number of Resistance Number of in Ohms Pieces in Ohms Pieces 31.25 2 SI - 25 Go 33-75 3 53-75 3o 36.25 37 56.25 IO 38.75 99 58.75 II. 41 - 25 189 61.25 9 43-75 228 63-75 3 46.25 I75 66.25 I 48.75 76 76.25 I It follows from what was said in the previous section that we cannot give definite answers to these questions in their present form. It will be noted that in no case are we justified in assuming that the material is controlled upon the basis of FIG. I.46.-Typical CRoss Section of A PRotective CoATING—Note IRREGULAR LINE of DEMARKATIon between CoATING AND METAL. the information given. On the contrary, questioning revealed in each of these typical cases that a priori there were good grounds for the belief that the quality was not controlled. In not one of the four cases did the engineer proposing the problem SAMPLING 4 I? know what assignable causes were likely to influence the par- ticular set of data giving rise to the question. Without this kind of information, any answer to the question —How large a sample?—is likely to be greatly in error because, as we have seen, the presence of unknown assignable causes may play havoc with the conclusions derived upon the basis of any sampling scheme which tacitly assumes, as it must, that the sample is random or, in other words, that it has come from a controlled system of chance causes. Before any one of the four questions previously proposed can be given a reasonable answer, it is therefore necessary to know whether or not we are justified in assuming control, and if control cannot be assumed, it is necessary that we employ the sampling scheme that will make the best use of a priori knowledge of assignable Call ISCS. 8. Sampling in Relation to Specification of Quality In Part V the advantages of specifying control of quality were considered in some detail. It was pointed out that wherever possible we should specify the average X and stand- ard deviation o of the objective distribution of control. It is of interest to note that we are led to this same conclusion from the viewpoint of sampling theory because, strictly speaking, it is only under the condition of control that we have a basis for interpreting samples. CHAPTER XXV THE CONTROL PROGRAM I. Résumé Five important economic reasons for controlling the quality of manufactured product were considered in Part I. In Chap- ter XXI of Part VI, we saw that, from the viewpoint of con- sumer protection, it is also advantageous to have attained the state of control. If only to assure the satisfactory nature of quality of product which cannot be given IOO per cent inspec- tion, the need for control would doubtless be admitted. In a very general sense, we have seen that the scientific interpretation and use of data depend to a large extent upon whether or not the data satisfy the condition of control (58). The statistical nature of things and of relationships or natural laws puts in the foreground this concept of distribution of effects of a constant system of chance causes. For this reason, it is important to divide all data into rational subgroups in the sense that the data belonging to a group are supposed to have come from a constant system of chance causes. We have considered briefly the application of five important criteria to check our judgment in such cases. We have seen, however, that such tests do not take the place of, but rather supplement, the inherent ability of the individual engineer to divide the data into rational subgroups. Thus we see clearly how statistical theory serves the engineer as a tool. 2. Control in Research Since observed physical quantities are, in the last analysis, statistical in nature, it is desirable that the results of research be presented in a form easily interpreted in terms of frequency distributions. As a specific instance, the design engineer 418 THE CONTROL PROGRAM 4 IQ must depend upon the results of research to give him a basis for establishing the requisite standard of quality character- ized, as we have seen in Part V, by the arithmetic mean X and the standard deviation o of a controlled quality X. Naturally the research engineer is always interested in detecting and eliminating causes of variability which need not be left to chance. Hence the criteria previously discussed often become of great assistance as is shown in Part VI. The data of research are good or bad, depending upon whether or not assignable causes of variability have been eliminated. In most instances the data which have been divided into rational sub- groups can best be summarized by recording the average, stand- ard deviation, and sample size for each subgroup. 3. Control in Design Our discussion of this phase of the subject in Part V in- dicated the advantages to be derived through specification of the condition of control in terms of the arithmetic mean X and standard deviation o of any prescribed quality characteristic X. 4. Control in Development From the results of measurements of quality on tool-made samples supposedly produced under essentially the same con- ditions, we may attain tentative standards of quality express- ible in terms of averages and standard deviations. These tentative standards may then be used as a basis for the con- struction of control charts in accord with Criterion I for the purpose of detecting and eliminating assignable differences of quality between tool-made samples and those produced under shop conditions. 5. Control in Commercial Production It is obviously desirable that a method of detecting lack of control be such that it indicates the presence of assignable causes of variability before these causes have had time to affect a large per cent of the product. For this reason, the method 42O ECONOMIC CONTROL OF QUALITY to be used on the job should involve a minimum number of computations. Here again Criterion I usually proves sat- isfactory. 6. Control in the Purchase of Raw Material As is to be expected, a prevalent source of lack of control is selection of raw material. It is not necessary that the dif. QUALITY REPORT PRODUCT X Y. Z "r Y p E A "r Y P E B CHARACTERISTIC ſe •5 .5 i O i •. 3. * $. S | > X- 5 Nel-A. Aſ ºf::::::::::::::: E = . . E-g---T---------------- =! º CO sº s ſº % o sº > M. J S D J F M A M J J A 5 O N D > CHARACTERISTIC 2 |250 § s § lazs |Oo § 3. tal g 3 OOO > 1200 « < I 175 º "22s & O $. S | X- I75 t 40 H. tº- J =! x 125 ºn 3 O º sº o: º: < 75 $ ro > M. J S D J F M A M J J A S O N D M. J S D J F M A M J J A $ O N 9 & STANDARD ou ALn LEvtu DATA ºr Rom sounct MMA FIG. I.47. ferent sources of material come from what could be considered to be the same constant cause system, but it is desirable that each source of a given material be controlled within itself. As an example, a physical property such as the tensile strength of a given species of timber may be assignably different for different sections of the country although within one section THE CONTROL PROGRAM 42 I TABLE 56.-SchEMATIC ForM of SUMMARY QUALITY ContRol REpoRT Quality Indication Action Taken Quality Nature of Cause or Called for Controlled | Not Controlled X1 V A 2 V New source of raw No other source of material. raw material availa- ble. Nothing can be done unless we change the kind of raw mate- rial called for in the design specification. A3 V Raw material Should secure mate- comes from sources | rial only from sources assignably different. A, B, and C. A 4 V Poor assembly oc- | Source of trouble casioned by new oper- eliminated. atOrS. A 5 V Unknown. Further investiga- tion under way. A 6 V Low insulation Source of trouble caused by improper eliminated. washing of insulation material before as- sembly. 28 m * * * * ~ || 4 - © tº $ tº & © & - - tº $ & - - - - - & G & º e this variability may be such as to be attributable to a constant system of causes. In the same way, we may have sources of supply of piece-parts produced by different units of an organ- ization or different manufacturers wherein there are assignable differences between the product coming from different sources even though each source represents a controlled product in 422 ECONOMIC CONTROL OF QUALITY itself. Such a condition can easily be taken care of in the use of the material, since the object of securing control from a design viewpoint is, as we have seen, the prediction of variability in the finished product. 7. 9tuality Control Report The quality report should, in general, do two things: a. Indicate the presence of assignable causes of variation in each of the quality characteristics, b. Indicate the seriousness of the trouble and the steps that have been taken to eliminate it. Fig. I47 is a page from a typical quality report which ful- fills the first requirement. Information similar to that shown schematically in Table 56 meets the second requirement. Appendices APPENDIX I RESULTANT EFFECTs of ConstANT CAUSE SYSTEMs I. Introductory Remarks Our discussion of the problem of establishing the necessary and sufficient conditions for maximum control was based upon the following three assumptions: A. The resultant effect X of the operation of the m causes is the sum of the effects of the separate causes. B. The number m of causes is large. C. The effect of any one cause is finite and is not greater than the resultant effect of all the others. It was stated that under these conditions the distribution of resultant effects of a cause system approached normality as the number m of causes was increased indefinitely, at least in the sense that the skewness VBºx and the flatness B2xx of this distribution approach o and 3 respectively. We shall now con- sider the basis for this statement in more detail. To start with it will be found helpful in trying to get an appreciation of the significance of the three limitations to carry through the details of finding the distribution of resultant effects of a few simple systems. For this purpose we shall consider eight such systems characterized as follows: m = 5 (a) { x: o I; o I; o I; o I; o I. . 5. l. 5. l. 5. l. 5. l. 5, 1 p: # #; ; ; ; ; ; ; ; ; ; ; #. m = 5 (b) { x: o I; o 2; o 3; o 4; o 5. . l. 5. 2, 4... 3. 3. 4. 2. B. l. p: ; ; ; ; ; ; ; ; ; ; ; ; ; 5. 42.5 426 ECONOMIC CONTROL OF QUALITY m = 5 (c) 3 x: o I; o 2; o 3; o 4; o 5. e B. l. 5. l. 5. l. 5. l. 5. l. p: § 3; # 3; # 33 & #3 & #. m = 7 (d) { x: o I; o 2; o 3; O 4; o 5; o 6; o 7 e 5. l. 5. l. 5. l. 5. l. 5. l. 5. l. 5. l. p: 6 G 5 Öſ 6 x 6' 6 (; ; G G : G G 5 Öſ G 777 = IO (e) { x: o I; o 2; O 3; o 4; O 5; o 6; o 7; o 8; O 9; O IO . 5. l. 5. l. 5. l. 5. l. 5. l. 5. l. 5. l. 5. l. 5. l. 3. l. p : é à: ; Ś j (, ; } {; ; ; ; ; ; ; ; Ś (; ; Ś (; ; ; ; ; ºff (, m = 5 (f) { x: o I; o I; I; O I; O I 5 4. 2. 3. 3... 2 4... l. 5. p: á č; ; ; ; ; ; ; ; ; ; ; ; m = 6 (g) { x: o I; o 2; o 4; o 4; o 2; o I. e 1. l. l. l. l. l. l. l. l. l. l. 1. p: ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; #. m = 5 (h) { x: o I; O 2; o 4; o 8; o I6. e l. 1... l. l. l. l. l. l. l. l. p: ; ; ; ; ; ; ; ; ; ; ; ; ; #. The notation used in describing the cause systems can be made clear by considering only the first one. Here we have a system of m = 5 causes. Each of these five causes may produce an effect of either o or I. For each cause the probability of zero effect is # and that of unit effect is #. Using this cause system we may illustrate the method of finding the distribution of resultant effects. Obviously the magnitude of this effect may take on values o, I, 2, 3, 4, 5. The probability that the resultant effect will be zero is the compound probability of each component cause producing zero effect or (#)". In a similar way the probabilities of getting a resultant effect equal to 1, 2, 3, 4, or 5 are respectively 5(#) (#)*, Io(#)*(#)*, Io(#)* (#)”, 5(#)*(#), and (#)". In this way we get the following distribution: Resultant Effect X O I 2 3 4. 5 Probability o. 401878 o. 401878 o. 160751 o.o.321 5o o. Oo32.1 5 || O.OOOI 29 APPENDIX I 427 This is shown graphically in Fig. I-a. The distributions of the resultant effects of the seven other systems are also shown in Fig. I. What significance do these results have? O.4 r. gºe O. 3 }* }* & b | || bºss | i O 2 ºrms O O| re ſt 34 H.Pºme }*me i O 2 | }* O O| º:me | 1–a–a—a a a illum. O. 4e Fº O. 3 º |- : O. 2 }= }* O. }= |- |- 1–1–a–a–a a a | | -A—º O i 234. smººFº hºme}*F- O. OL |. - L| || || || || || || || || || || || || | | | | | | O .. 5 O 5 |O 15 20 25 3O |O |5 2O 25 3O RESULTANT EFFECT FIG. I.-DISTRIBUTION of RESULTANT EFFECTS OF SIMPLE CAUSE SYSTEMS. In the first case we see that the distribution of resultant effects will always be characterized by the point binomial. Hence it will always monotonically decrease on either side of 428 ECONOMIC CONTROL OF QUALITY the mode—in other words, it is a smooth distribution. Dis- tributions b, c, d, and e indicate the effect of lack of uniformity among the component causes. From this viewpoint smooth- ness is a necessary condition. That it is not, however, a suffi- cient condition is evidenced by Systems f, g, and h. As long as the component causes are the same, we have already seen (Fig. 53) that the distribution of resultant effects approaches normality as the number of causes is increased. The condition that there shall be an indefinitely large number of causes is, however, certainly not sufficient as is shown by systems g and h, for in these cases the shapes of the distributions will always be those shown in Fig. I-g and h. Of course, if we admit that the effect of any cause must be finite, systems such as g and h with an indefinitely large number m of causes are ruled out. 2. Practical Significance of Results In practice one is confronted with an observed distribution and from its nature must often decide whether or not it is worth while looking for assignable causes of either Type I or Type II. We shall concern ourselves here only with the problem of deciding whether or not an observed distribution TABLE I.—THERMAL UNITS PER CU. FT. OF GAs I,391 I,318 I,2O3 I,29 I I,416 I,268 I,38o J,273 I,367 I,294 I 3349 I,242 I,258 I,368 1,360 I,23 I I,289 I,33O I 33 I 3 I,32O I, I 99 I,254 I 335 I I 334O I,275 I,226 I,289 I,42O gives evidence of the presence of a predominating cause, that is, an assignable cause of Type II. Let us consider a typical problem. The operation data for a certain gas plant for one month expressed in terms of arbitrary thermal units per cubic foot of gas produced from oil by cracking are those given below in Table I. The data are tabulated in the order in which they were taken. Ideal operation calls for APPENDIX I 429 as high and as nearly constant value as can economically be attained. The following question was raised by the Director of Research of the large organization interested in these results: If I understand the methods of statistics correctly, it should be possible to determine from these data whether or not there is a pre- dominating cause of variation, and hence to determine whether or not it should be reasonable to expect that a marked improvement in product can be made by controlling one or at least a few causes of variation. Am I right in this interpretation of the possibilities of statistical methods? In answer to such a question we can at least say something like the following. If we divide the data into subgroups of four in the order in which they were taken and apply Criterion I of Part VI, we get no evidence of lack of control, as may easily be verified by the reader. Assuming that the quality is con- trolled, we may now consider the evidence for the presence of a predominating effect. An examination of these data shows that they are more or less uniformly distributed over the range of variation as one might expect with a cause system such as (h). In other words, the observed results are consistent with the hypothesis that a predominating cause was present. Need- less to say such evidence is not conclusive: it is suggestive. 3. Analytical Results Let us now find expressions for the skewness VBs, and flatness 32xx of the distribution of resultant effects under simplifying assumptions. If we let pi, represent the ith moment of the effects of the jth cause about their expected value, it may be shown that o” = 1.2 = 0.1° -- 0.2°,+ . . . -- 0:2 + . . . -- Gm”, Pºs = 13, + H8, -- . . . -- 13, + . . . -- 13, and ???, |14 = 24,- 30.3%) + 3|12°, J = * See for example Elements of Statistics, by A. L. Bowley, published by P. S. King & Son, Ltd., 1920, pp. 29 I-292. 43O ECONOMIC CONTROL OF QUALITY where pi is the ith moment of the resultant effect of the m causes about the expected value of the resultant effect. From these results we get Pixx - º -: (I) and X (114; – 305°) + 312° X (94; – 30 j') £2 - * ~ * = + + 3. (2) X X |12” |12” |12” As a simple case let us assume that the distribution of effects of (m — I) of the component causes are the same, at least in respect to their second, third and fourth moments, all of which are assumed to be finite, which we shall denote by M2, M3, and M4. Let us assume that the remaining cause is pre- dominating in the sense that the corresponding three moments of its effects are bol\ſ2, bºl\ſ3, and bal/IA, where b2, Ö3, and b4 are all positive and greater than unity. Under these conditions, we get *- (m — I + b3)? M32 Pixx *- (m — I + b2)* M23 and §2xx = (m — I + ba)M4 – 3(m — I + bo”) M2” 2XX (m — I + h2)*M2” Evidently these two expressions approach o and 3 respec- tively as the number m of causes becomes indefinitely large, assuming that be, b2, and b4 are finite. In this way we come to see that the skewness and flatness of a distribution of re- Sultant effects will, in general, be approximately oand 3 if the number m of causes is very large. 4. Economic Significance of Control from a Design Viewpoint In Chapter III of Part I we called attention to the fact that as a result of control we attain maximum benefits from quantity APPENDIX I 43I production. Only general statements as to obtaining these benefits were given at that time. In Part III, however, we developed the theoretical basis for control, making it now possible to show specifically how control enables us to attain these benefits. We shall consider here only the simplest kind of examples. A. Example J.-Suppose that an assembly is to be made in which two washers are to be used, one brass and the other mica. Assume that it is desirable to maintain as closely as possible a uniform overall thickness of these two washers. This could be done, of course, by selecting the pairs of brass and mica washers to give the desired thickness. Such a process, however, would tend to counterbalance the benefits of quantity produc- tion, since the economies rising from assembly processes result from interchangeability of piece-parts. Table 2 gives the results of measurements of thickness on one hundred tool-made samples each of mica and brass washers to be used in the manner previously indicated in the assembly of an important piece of telephone equipment. The reader may easily satisfy himself that both of these distributions are sufficiently near normal to indicate that each of the piece-parts was controlled, and we shall therefore assume this to be the case. For this size of sample we are perhaps justified in assum- ing that the observed standard deviations of these two dis- tributions may reasonably be taken as the standard deviations o, and G2 of the objective controlled distributions of mica and brass washers respectively. The theory of the previous section shows that under these conditions the standard deviation of a random assembly of two washers, one of each kind, is O = \/ o 12 + 0.22. Furthermore, it follows that the distribution of the sum of the thickness in such a random assembly will be normally dis- tributed about a mean value which is the sum of the mean values of the two objective distributions. Upon this basis, therefore, the design engineer is justified 432 ECONOMIC CONTROL OF QUALITY TABLE 2.--TYPICAL DISTRIBUTION REQUISITE FoR EFFICIENT DESIGN Thickness Number of Thickness Number of of Mica Wash of Brass Wash e alST1CIS º alSIleſ S in Inches in Inches o, Oo38 I o.o. 82 I o.OO89 I o.OI85 I o.OO92 I o, or 86 2. o.OO93 I o.ol 87 2. O.OO94 I o. of 88 2. O. OO95 I O. O.I.90 2 o.OO98 2. o.OI 91 3 o.OO99 I o.OI 92 3 O. OIOO 2. O. O.I.93 3 O. OIOI 5 O. O.I.95 5 O. OIO2. T 2. o.O.I.96 6 o.OIO3 3 o.OI97 5 O. OIO4 7 o.OI 98 4 o.OIOS 5 O. O.I.99 I o.oloé 8 o.O2Oo 3 o.OIO7 IO O. O2OI 8 o. olo? IO O. O2O2 4. o.OIo9 7 o.O2O3 5 O. O.I IO 5 O. O2O4 7 O. OIII 3 o.O2O5 4. O. O.II 2 5 o.O2O6 3 o.OI 13 6 o.O2O7 3 O. OII.4. 6 o.oOo.8 6 o.OII 5 3 O. O2 IO 3 o. of 16 3 O. O2 II I o.OI 19 I O. O.Q. I 2 I o.O2.I.3 3 O. O2 I4. 2. o.O2I 5 3 o.O2.I6 2. O. O.Q.2O I O. O222 I in predicting that the overall thickness of random assemblies of mica and brass washers will be distributed as shown in Fig. 2. The dots in this figure show how closely the first one hundred assemblies made from manufactured product check the pre- diction. Furthermore, if the observed average thickness of APPENDIX I 433 each distribution is taken as the expected value of the distribu- tion, the design engineer can easily calculate the percentage of assemblies that will be defective in respect to overall thickness subject to the assumptions that have been made. 4. Or e OBSERVED DISTRIBUTION EXPECTED DISTRIBUTION 3 5 }* 3 O }* 2 5 }* 2OH. O L Ł l | l I -ºr ſ O.O27 O.O29 O.O3 | O.O.33 O.O.35 TH | CH&N ESS IN INCHES FIG. 2.-STATISTICAL METHOD MAKES PREDICTION IN DESIGN PossIBLE. B. Example 2.-For a shaft to operate in a bearing it is, of course, necessary to have a certain clearance. Thus, if p1 and ps represent the radii of the bearing and shaft respectively, then the specification will, in general, state that the difference pi – p2 must satisfy the inequality di < p1 – p2 < d2, where di and dº are both positive. This situation is represented schematically in Fig. 3. In most instances the shaft and bearing are fitted. Some- times, however, it is of economic importance to be able to product shafts and bearings separately and to assemble these on the job. The question, of course, that is always raised is: What will be the expected rejection of such assemblies because of failure to satisfy the clearance specification? 434 ECONOMIC CONTROL OF QUALITY From the theory of the previous section we see that this question can be answered readily, at least if we assume that radii of bearings and shafts are normally controlled with stand- ard deviations of and 02 respectively. Under these conditions the difference pi – p2 between any bearing and shaft chosen at random will be distributed normally about a mean value pi – po with standard deviation O = V of 2 + q22. I ! 1 sº sº d O di Pl - P2 2 SHAFT AND BEARING Distribution of DIFFERENCES BETWEEN RADII OF SHAFT AND BEARING FIG. 3–How MANY Rejections SHOULD WE Expect IN Assembly? Hence, the probability of a random assembly being rejected because the clearance fails to come within the required limits is given by 22 I I — =e-*/2dz, ſ v/2t 2 = (p1 – p2) — (pi – P2) where O' di - (PI = P2) z1 = — O' d2 – (pi – p2) 22 = —, O' and the value of the integral can be read directly from Table A. C. Example 3.−We shall now consider a problem involving maximum control. Many instances arise in production where APPENDIX I 435 materials must be covered with protective coatings. Of such are the various kinds of platings, nickel, chromium, zinc, etc. In other instances we have coatings of paper or lead. In practically every instance of this kind it is very desirable to maintain a uniform coating that is never less in thickness than some prescribed value. It is obviously desirable from the viewpoint of saving to reduce the variability to a minimum. Table 3 gives an observed distribution of one such kind of TABLE 3–Do THE WARIATIONS IN THIcKNEss INDICATE A PossIBLE SAVING: Thickness in * || Thickness in * Inches of º Inches of - Observations Observations O. 125 2. O. I 3 I 2O o. 126 I 2 O. I 32 5 o. 127 ºn I O. I 33 3 o. I 28 I 8 O. I 34 O O. I º 33 O. I 35 3 O. I 3O 33 coating supposed always to be more than O.I.24 inch in thick- ness. The histogram in Fig. 4 shows this distribution. What 4.O ECONOMIC GOAL 3 5 - 3 O - 2 5 - T_ 5 - |O - O | 1– 1 I l l t l I —1– —1– *I f O.123 O.H25 O. 127 O. 129 O.13] O, 133 O. 135 TH | CKNESS IN IN CHES FIG. 4.—How MAXIMUM ContRol SAves Money. 436 ECONOMIC CONTROL OF QUALITY does the theory of maximum control tell us about the uni- formity of coating? In the light of the previous section the lack of smoothness in this distribution is indicative of the presence of assignable causes of variation which can be removed. In fact, an investigation revealed assignable causes of variation, and on removing these, the resulting quality approached the distribution shown by the smooth curve of Fig. 4, representing the state of maximum control for this particular kind of coating. By attaining this state of maximum control, it is apparent that the average thickness of coating is materially reduced without increasing the probability of obtaining a defective thickness. Not only does control lead to a saving of material in such cases but it also leads to a more uniform product because as shown in Chapter XXIV of Part VII, it is practically impos- sible to sample for protective purposes unless the quality is controlled. APPENDIX II PRESENTATION OF ORIGINAL ExPERIMENTAL RESULTs USEFUL IN OBTAINING AN UNDER- STANDING OF THE FUNDAMENTAL PRINCIPLE's UNDERLYING THE THEORY OF QUALITY CONTROL The six tables in this appendix give in detail the results of 4,000 drawings from each of the three experimental universes referred to in the text. Tables A, B, and C give the original drawings divided into groups of four in the order in which they occurred. Tables D, E, and F give various statistics for these samples of four. It should not be inferred that these statistics are arranged to correspond to the samples as this is not always the case. We have made extensive use of these data in our discussions of the theory of quality control, and it is advisable to reproduce these data if for no other reason than that the reader may wish to carry out for himself com- putations similar to those referred to throughout the text. There is, however, a far more important reason for present- ing these experimental results. It will have become apparent by this time that statistical theory rests upon a fundamental natural law—the law of large numbers. In the last analysis we must always appeal to experimental evidence to justify our belief in such a law and to give us a feeling for its physical significance. For example, in the discussion of the theory of statistics, we always have to talk about doing something again and again under the same essential conditions; or, as we have said, under a controlled condition where the chance cause system is constant. We have used these data in various places throughout the book to illustrate a controlled phenomenon. In particular we 437 º, O.9 O.8 O 7 O.6 O.5 O.4 O.3 O.2 O. i – O.6 O,5 O.4 O.6 O.5 O.4 O.6 O.5 •,•"e ... ---efa.” sTess O * - e. ." **, •,•'s 5O 450 *. 1 w wº y - w w - - we esw - 650 85O O.6 O.5 H. 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T || Ogt“ - 208 ç35 ° GOO" I g2*, * 2O4. 993 * T- || Ogg. * Ogi; " - 209 £80° 2gT * T | OOT * 2OG 922° 6go*T Oga • TO6 $22°- gTT *T || Oga • - TOB 262" - || 628° gaç” – TOL 200” [- || 6′5 * Og: " - TO9 £4.1° - || Ogo” T | Gº, I*- TOG 2 JO X J.9qumn 2 JO X. ..I equmM z JO X J equmn 2 JO X IequmM 2 f) X Joquinn 9Tâmeg etdures - otdures * otdmbs - otdmeS (p2nutſuo))—3SYIGAINſ) XIVT neNVII L J.HOIYI JWOYIJI SoNIAvXICI YIOJI SOILSILVLS-'I 3Tav L § gºz”- Ogz” OTs "T- 99.8°- ggy*T- çSO"T- OZO"T- 34.3° (- 2gy"T- OTº “T gº, I*- 949° 39 y”T 29**t O2g"I 28w’t Agy"T O9W" I Q ºt" Gº" - O9A," " gT2 *- get" - T9g." T66* 268° OOg” APPENDIX III A BIBLIOGRAPHIC GUIDE witH SUGGESTIONS For STUDY IN THE FURTHER DEVELOPMENT OF A SCIENTIFIC BASIs For THE EconoMIC Con- TROL OF QUALITY OF MANUFACTURED PRODUCT As stated in the preface, the present book is but an initial step toward the formulation of a scientific basis for securing economic control. Much remains to be done. In presenting a list of references for further study, an attempt has been made to include those sugges- tive of what appear to be profitable lines of further development. Throughout the book we have had occasion to give many specific references. The object of the present bibliography is to suggest refer- ences of a more or less general nature to be read in connection with each of the seven parts. It is hoped that in many instances these ref- erences will be suggestive of work which may be profitably done in extending the theory of quality control, particularly in the direction of the development of improved ways of securing good data through the more thorough application of the scientific method. REFERENCEs For PARTs I AND III I. Exact and Statistical Laws In Parts I and III the rôles of exact, empirical, and statistical laws in helping us to do what we want to do are touched upon. The recent book, A History of Science, C. D. Whetham, 2nd edition, Macmillan Company, New York, 1930, gives an inter- esting and up-to-date survey of the results of human effort in estab- lishing laws of nature. To get a more exact picture, however, we must turn to some such book as Introduction to Theoretical Physics, A. Haas, 2nd edition, Constable & Company, London, Vol. I, 1928, Vol. II, 1929; or the book of the same title by L. Page, D. Van Nostrand Company, Inc., New York, 1928. 473 474 ECONOMIC CONTROL OF QUALITY With the development of the atomic structure of matter and electricity, it became necessary to think of laws as being statistical in nature. The importance of the law of large numbers in the inter- pretation of physical phenomena will become apparent to any one who even hastily surveys any one or more of the following books: Statistical Theories of Matter, Radiation, and Electricity, K. K. Darrow, The Physical Review Supplement, Vol. I, No. 1, July 1929, also published in the series of Bell Telephone Laboratories' reprints, No. 435; Introduction to Statistical Mechanics for Students of Physics and Physical Chemistry, J. Rice, Constable & Company, Ltd., London, 1930; Statistical Mechanics with Applications to Physics and Chemistry, R. C. Tolman, Chemical Catalog Company, New York, 1927; Kinetic Theory of Gases, L. B. Loeb, McGraw-Hill Book Com- pany, New York, 1927; The Kinetic Theory of Gases, E. Bloch, Methuen & Company, Ltd., London, 1924; Introduction to Modern Physics, F. K. Richtmeyer, McGraw-Hill Book Company, New York, 1928; Modern Physics, H. A. Wilson, Blackie & Son, Ltd., London, 1928; Introduction to Contemporary Physics, K. K. Darrow, D. Van Nostrand Company, Inc., New York, 1926; and Atoms, Molecules and Quanta, A. E. Ruark and H. C. Urey, McGraw-Hill Book Com- pany, New York, 1930. One cannot return from even a brief excursion into the field of modern physics and chemistry without having caught a glimpse of the importance of the concept of the statistical limit in all of the latest developments. Even in this field of exact science nothing is exact. In the last analysis the influence of chance causes is felt. Almost the only things that appear to be constant are distribution functions or statistics of these functions—and this constancy is only in the statistical sense. For example, one interested in the specifica- tion of quality of materials need read only Chapter III of The Physics of Solids and Liquids, P. P. Ewald, Th. Pöschl and L. Prandtl, Blackie and Son, Ltd., 1930, to see how far we are from being able to explain some of even the simplest mechanical properties in terms of atomic physics. 2. Empirical Laws To contrast the way in which the so-called exact and statistical laws enable one to predict with the way in which an empirical law does, the recent excellent book Business Cycles, W. C. Mitchell, Na- APPENDIX III 475 tional Bureau of Economic Research, New York, 1927, should prove to be of interest. The author of this book discusses in a critical manner the very extensive amount of work that has been done in trying to develop a rational basis for predicting cyclic movements with a net result that is not so very encouraging. Even a casual reading of this book must impress one with the serious hopelessness of trying to predict the future in terms of the past when the chance cause system is not constant. In the present state of the scientific method of induction, it appears that empirical relationships such as time series give little basis for prediction. This conclusion is con- sistent with that so admirably presented in a recent paper by S. L. Andrew in the Bell Telephone Quarterly, Jan., 1931, and also with conclusions set forth in the recent book Business Adrift, by W. B. Donham, Dean of the Harvard Business School. Such reading cannot do other than strengthen our belief in the fact that control of quality will come only through the weeding out of assignable causes of variation—particularly those that introduce lack of constancy in the chance cause system. 3. Frequency Distribution Functions In Part III we considered very briefly the problem of determining the kind of frequency distribution function or functions that we might expect controlled quality to follow. In this connection we touched upon the philosophy of frequency curves as laws of dis- tribution. $ Two systems of curves were mentioned in particular, namely, the Pearson and the Gram-Charlier systems. Although we have not had occasion to make much use of these functions as such, a serious student of control of quality will find it greatly to his advantage to read some of the original memoirs dealing with these two systems of curves. Those of Pearson are naturally available in English and cannot help but prove stimulating. The more formal part of Pearson's work in this field has been summarized by Elderton in the interesting book, Frequency Curves and Correlation, second edition, Layton, Lon– don, 1928. T. L. Kelley, a former student of Pearson, also has much of interest to say about this system of curves in his book, Statistical Method, Macmillan Company, New York, 1923. Very interesting and stimulating accounts of the significance of the Gram-Charlier series have been given by Arne Fisher, Mathe- 476 ECONOMIC CONTROL OF QUALITY matical Theory of Probabilities, 2nd edition, Macmillan Company, New York, 1922; by F. Y. Edgeworth in a series of articles referred to in his article Probability in the 13th edition of the Encyclopedia Britannica; and by T. N. Thiele, Theory of Observations, London, 1903. J. F. Steffensen in Some Recent Researches in the Theory of Statistics and Actuarial Science, Cambridge University Press, 1930, makes some very interesting and pertinent remarks on the theo- retical foundation of certain types of frequency curves. It is of particular interest to note the way in which Edgeworth arrives at the Gram-Charlier series as a method of expressing the results of the joint action of a complicated system of causes. Of course, the Pearson system can be given somewhat similar causal interpretation although great emphasis has not been laid upon this point by many of those writing about the Pearson system. The sythentic building up of a frequency curve in terms of the effects of component groups of causes forms a basis, as we have seen, for our discussion of the necessary and sufficient conditions of max- imum control. We have emphasized the significance of the fact that, as the number of causes of variability is increased, we seem to ap- proach closer and closer to what we have termed the point (o, 3) of maximum control in the 31 32 plane. In this connection The Behavior of Prices, F. C. Mills, National Bureau of Economic Research, Inc., New York, 1928, should prove interesting reading, particularly that part having to do with the march of the 3’s back to normalcy, as he puts it. 4. Probability Probability and its Engineering Uses, T. C. Fry, D. Van Nostrand Company, New York, 1928, and An Introduction to Mathematical Probability, J. L. Coolidge, Oxford University Press, New York, 1925, contain interesting discussions of the meaning of probability and the difficulty involved in defining it. 5. Quality Control The only book touching upon the subject of quality control in anything like the sense of the present text is that by Becker, Plaut, and Runge, referred to in Chapter I of Part I. APPENDIX III 477 REFERENCEs For PART II I. Economics The problem of economic control of quality in its broadest sense is, as we have seen, that of doing what we want to do within limits which are economical. To do this, we must establish economic standards of quality. A brief outline of the economic considerations which must be taken into account in attempting to establish such standards of quality is given in an interesting article, “Standard Quality,” G. D. Edwards, Bell Telephone Quarterly, Vol. VII, pp. 292–303. For example, in establishing such a standard, we must consider the relationship between cost and value. Value, however, is not so easily defined in a way that will cover all of the prevalent concepts of this term. To attempt to do so leads us into difficulties touched upon in our discussion of the definition of quality. Naturally, value in some way or other depends upon the degree to which a given quality satisfies human wants; but, in turn, human wants are not constant even for the same person. Furthermore, the degree to which a thing having several quality characteristics tends to satisfy the human wants of even a single person is to a large extent a complicated and unknown function of the magnitudes of the phys- ical characteristics of the thing. Even assuming that the value determined on the basis of the wants of a single person is a constant, it is apparent that the values for different people differ among them- selves so that, in the last analysis, value, if it can be expressed quanti- tatively, is presumably a frequency distribution function. A brief, terse exposition of the fundamental economic problems involved in attaining a dynamic measure of value will be found in the Mathematical Introduction to Economics, G. C. Evans, McGraw- Hill Book Company, New York, 1930. Having obtained a picture of the complicated nature of this problem, one may feel inclined to despair of its solution. However, for some time to come, it is likely that we shall not get away from the desire on the part of all of us to find some measure of quality which is common to all qualities. In our discussion of economic control, we left out any detailed consideration of this problem of finding an adequate measure of value, even though such a measure apparently would serve a very useful purpose. We started with the tacit assumption that when such a measure of value can be found, it will have two characteristics: it will 478 ECONOMIC CONTROL OF QUALITY be a statistical quantity, and it will be statistically related to the measurable quality characteristic of the product. Beginning at this point, we have shown, particularly in Part I, that certain economic advantages can be attained in the production of a controlled quality. This means, of course, as previously stated, that the quality standard is some frequency distribution function. We emphasized the importance of at least two characteristics, namely, the average X and the standard deviation o of this function. To insure that the specified parameters in a given case are economic standards would require a consideration of the fundamental prob- lems involved in establishing measures of value already referred to. In such cases we must choose standards which to the best of our knowledge at the present stage of the development of the subjec appear to be reasonable estimates of economic standards. - 2. Texts on Statistical Theory The ninth edition of Yule's An Introduction to the Theory of Statistics, C. Griffin & Company, Ltd., 1930, should prove to be a veritable storehouse of knowledge in respect to many of the things discussed in Part II. This is particularly true in respect to measures of central tendency, dispersion, and correlation. As supplementary reading for the more technical part of the discussion, Mathematical Statistics, H. L. Rietz, Open Court Publishing Company, Chicago, I917, should prove of great value, particularly in connection with the consideration of the analytical aspects of correlation. A. L. Bowley’s Elements of Statistics, Chas. Scribner's Sons, New York, 1926—in particular the second volume—contains much of interest in regard to the point binomial and the second approximation (23). The Mathe- matics of Statistics, R. W. Burgess, Houghton Mifflin Company, New York, 1927, will be found helpful as a general elementary text. It also contains references to several elementary books dealing with statistical methods and their application in other fields such as eco- nomics. Two of these should be mentioned here: Statistical Methods Applied to Economics in Business, F. C. Mills, Henry Holt & Com- pany, New York, 1924, and Principles and Methods of Statistics, R. E. Chaddock, Houghton Mifflin Company, Boston, 1925. Attention should also be called to the recent book, The Mathematical Part of Elementary Statistics, B. H. Camp, D. C. Heath & Co., New York, I93I. APPENDIX III 479 3. Curve Fitting In connection with our discussion of the derivation of empirical formulas to represent relationships, the little book, Empirical For- mulas, T. R. Running, Wiley & Sons, New York, 1917, is of interest. The method of moments is discussed in some detail in Elderton’s book, Frequency Curves, previously referred to. The method of least squares is admirably treated in the Calculus of Observations, E. T. Whittaker and G. Robinson, 2nd edition, Blackie & Son, London, 1926. REFERENCEs For PART IV In 1922, R. A. Fisher presented in The Philosophical Transactions of the Royal Society in London an article, “The Mathematical Foun- dations of Theoretical Statistics,” in which he characterized three fundamental problems, namely, specification, distribution, and esti- mation. At least the first nine paragraphs of this paper should be read by any one interested in the application of statistical theory in the control of quality. In Part IV, we are particularly interested in the theory of distribution which has been developed to a marked extent during the last few decades at the hands of R. A. Fisher, “Student,” J. Neyman, L. Isserlis, A. E. R. Church, V. J. Romanovsky, J. Wishart, E. L. Dodd, B. H. Camp, H. Hotelling, Karl Pearson, E. S. Pearson, L. H. C. Tippett, P. R. Rider, A. A. Tchouproff, A. A. Markoff, M. Watanabe and E. Slutsky. Perhaps one of the best ways for a newcomer to orientate himself in this field of investigation is to read the excellent “Report on Sta- tistics” by H. L. Rietz, published in the Bulletin of the American Mathematical Society, October, 1924, pp. 417-453. References to later work of the men mentioned in the previous paragraph and others on the theory of distribution will be found in the bibliographies of the books by Yule, Rietz, and Kelley, already referred to. In connection with the discussion of Tchebycheff's theorem, one of the most inter- esting articles is that of A. A. Tchouproff, “Asymptotic Frequency Distribution of the Arithmetic Means of n Correlated Observations for Very Great Values of n,” journal of the Royal Statistical Society, Vol. LXXXVII, 1925, pp. 91—IO4. This article gives detailed references to the work of Watanabe, Markoff, Slutsky, and others touching upon this same problem. A recent paper, “British Statistics and Statisticians Today,” H. Hotelling, journal of the American Statistical Association, June, 48O ECONOMIC CONTROL OF QUALITY 1930, pp. 186-190, gives an interesting brief account of what is going on in England today in the development of statistical theory. If one is interested in tracing the development of the theory of dis- tribution or, in fact, any part of statistical theory back through the ages, Studies in the History of Statistical Method, Helen M. Walker, Williams & Wilkins Company, Baltimore, 1929, will be found helpful. Perhaps our best general source of information on the important work of the Scandinavian School of statisticians is the book by Arne Fisher previously mentioned. REFERENCES FOR PARTs VI AND VII I. Estimation Two fundamental statistical problems are touched upon in Parts VI and VII. One is that of going from a random sample of size n to its universe. Today there are in the literature the following three general methods of going from a sample to its universe: (a) The a posteriori method. (b) The method of maximum likelihood. (c) The empirical method. To mention these three in the same breath in the presence of a group of statisticians is almost certain to start an argument, for there is a wide divergence of opinion as to the comparative validities of these methods. For this reason, the reader will find it advantageous to consider in some detail the original memoirs dealing with these separate meth- ods. The a posteriori method is tied up with the theory of causes and the name of Bayes. The recent important article, “Frequency Dis- tribution of the Unknown Mean of a Sampled Universe,” E. C. Molina and R. I. Wilkinson, Bell System Technical journal, Vol. VIII, pp. 632–645, October, 1929, should prove an interesting starting point for the consideration of this method, although the reader will doubtless wish to read other original memoirs referred to in connection with the discussion of Bayes' theorem in the general bibliographies mentioned in a previous paragraph. The method of maximum likelihood is tied up largely with the work of R. A. Fisher, starting primarily with his article in the Philo- sophical Transactions previously mentioned. APPENDIX III 481 A recent article, “On the Use and Interpretation of Certain Test Criteria for Purposes of Statistical Inference,” E. S. Pearson and J. Neyman, Biometrika, XXA, pp. 175–240, 1927, and XXA, pp. 263– 294, 1928, is perhaps the best critical discussion of the available methods of solving the problem of estimation. It should certainly be read by any serious student of this subject. The third edition of Statistical Methods for Research Workers, R. A. Fisher, summarizes most of the detailed methods of estimation de- veloped by him. It is a book of particular value to scientists and engineers, although one must keep in mind the serious limitations of all methods of estimation based upon small samples as noted in the text and discussed in such references as that of Pearson and Neyman. It is of interest to note that a divergence of opinion is expressed in the literature as to the usefulness of the theory of the so-called small sample. Perhaps most of the critical remarks are based upon the assumption that this theory is to be used as the basis of estimation, and that it may give the impression that we can replace large samples by small ones. In the first place, a careful reading of the available literature does not reveal any specific suggestion to substitute small samples for large ones. In the second place, it should be noted that the application of small sample theory used in this text is required in handling large numbers of data in a rational way by breaking them up into rational subgroups. In this work the distribution theory for small samples plays a prominent rôle. In general the problem of estimation presents the universal dif- ficulties involved in all induction. If one reads such a book as A Treatise on Probability, J. M. Keynes, Macmillan Company, New York, 1921, he may feel at first very much discouraged, because his attention will have been directed to many of the serious difficulties involved in the application of probability theory. A useful tonic in such a case is to read any one or more of the following books: The Nature of the Physical World, A. S. Eddington, Macmillan Company, New York, 1928; The Logic of Modern Physics, P. W. Bridgman, Macmillan Company, New York, 1928; The Analysis of Matter, Bertrand Russell, Harcourt, Brace & Company, Inc., New York, 1927. At least, these three books should prove to be a tonic, if it is true that misery loves company. Certainly the serious difficulties involved in the interpretation of physical phenomena are common in all fields, and the discussions in these books show how much we must rely upon the application of probability theory even in an “exact” science. 482 ECONOMIC CONTROL OF QUALITY 2. Detecting Lack of Control The second fundamental statistical problem is that of determining whether or not a given set of data comes from a constant system of causes, or more generally it is the problem of dividing the universe of objective values into rational subgroups schematically represented in Fig. 144. In our discussion of ways and means for detecting lack of control, we have pointed out again and again the necessity of sub- dividing the data into rational subgroups. To do this requires the exercise of human judgment. In the last analysis we must depend upon the use of scientific method—that is, upon human intuition, imagination, reasoning, and knowledge. It is perhaps only through the application of this general method that we can hope to attain good data, one characteristic of which is that they be subdivided into rational subgroups. It may be of interest, therefore, to sketch briefly a course of reading which will be found helpful to the student in the application of scientific method to the further development of the theory of quality control. To do so necessarily takes us into the fields of psychology, philosophy, and logic; into the field of psychology because we must get some sort of picture of the way the mind works; into the field of philosophy be- cause we need some hypothesis as to the nature of reality and the function of laws, theories, and causal explanations; into the field of logic because it presents what we know about the formal methods available in the theory of deduction and induction. How do data depend upon the mind? What is the effect of factual experience and the effect of reasoning upon an observer? These are important questions. What we sense through any one of our senses depends partly upon previous use of these senses. Thus a child looking at a straight stick extending beneath the surface of a pool of water sees a bent stick. Similarly, the first time one sees what is shown in Fig. 131, he sees the length of the line (a) to be different from that of line (b), although they are of the same length. In this way, factual experience influences what we sense through any one of OUIT SCI1S6S. Perhaps more important, however, is that the mental experience involving reasoning influences to a marked extent what we sense. One looking at a line AB, Fig. I, and thinking of the points on the line, sees those points in an entirely different way after he has tried to place such points as V3 and iſ on that line. APPENDIX III 483 Almost every day one hears of some physical discovery which has been influenced by a conceptual theory. A trained experimentalist who is at the same time familiar with the current theory or theories having to do with the phenomena which he is investigating will, in many cases at least, be able to get better data for the particular purpose in hand than he would be if he did not know the theory. W3. Tr A O O FIG. I. In this same connection, it is important to note some of the applications of the theory of frequency curves in assisting one to break down an observed set of data into rational subgroups or to indicate in ways other than those described in the text whether or not this can be done. For example, the fact that an observed point in the 3132 plane is in the neighborhood of (o, I.8), Fig. 2, is consistent 4 rº 3 - A52 2H- © (0,1,8) | }=- O | I ſ ſ ſ | O O.5 I.O |.5 2.O 2.5 3.O ſ3) FIG. 2. with the hypothesis that the observed set of data came in approx- imately equal proportion from, let us say, m rational subgroups. In a similar way, an observed value of skewness may be consistent with some rational hypothesis in respect to the causes of variation. In other words, an observed set of statistics can be suggestive of a working hypothesis in much the same way that a rough plot of an observed frequency distribution may be suggestive in the sense in- dicated by E. B. Wilson in his article, “The Development of a Fre- 484 ECONOMIC CONTROL OF QUALITY quency Function and Some Comments on Curve Fitting,” Proceedings of the National Academy of Sciences, Vol. Io, 1924, pp. 79-84. Another very important use of the knowledge of the theory may be that of detecting mistakes in computation. For example, if one found a point (31, 32) below the line 82 – 31 — I = o, Fig. 3, he would know that a mistake had been made because, as was originally shown by Pearson, it is not possible for a frequency distribution function to have a point in this area. Broadly speaking, we see again why it is so necessary in the control of quality of manufactured product to have data accumulated by someone acquainted with the available factual and conceptual expe- 4 r- O I ſ ſ f 1—1 O O.5 I.O |.5 2.O 2.5 3.0 £31 FIG. 3. rience relating to the particular problem in hand. Books such as: Scientific Thought, C. D. Broad, Harcourt, Brace & Company, Inc., New York, 1927; The Function of Reason, A. N. Whitehead, Princeton University Press, Princeton, New Jersey, 1929; The Analysis of Mind, Bertrand Russell, George Allen and Unwin, Ltd., London, 1922; Conflicting Psychologies of Learning, H. B. Bode, D. C. Heath & Company, New York, 1929; The Principles of Psychology, William James, Henry Holt & Company, New York, 1890; The Revolt Against Dualism, A. L. Lovejoy, W. W. Norton & Company, Inc., New York, 1930; and Human Learning, E. L. Thorndike, The Century Co., 1931; contain much of interest in this connection. Having seen what an important part conceptual experience may play in taking data, one is likely to become more interested in formal logic. The meaning of the laws of thought and the application of APPENDIX III 485 syllogistic reasoning take on a new interest. For example, a funda- mental understanding of the theory of control tacitly involves such mathematical concepts as function, limit, continuity, and so on, developed to a degree of refinement which comes from the study of the discussion of these subjects in such a book as G. H. Hardy’s Pure Mathematics, Cambridge University Press, London, 1928. Perhaps of even greater interest, however, is the consideration of what we mean by judgment and common sense—two things which we find we must use so often in experimental work of all kinds. One soon finds that there is a considerable divergence of opinion in respect to such matters as will be evidenced by a more or less systematic browsing in the following treatises on logic. Elementary Logic, A. Sidwick, Cambridge University Press, London, 1914; Principles of Logic, H. W. Bradley, Vol. I and Vol. II, 2nd Edition, Oxford Univer- sity Press, London, 1922; An Introduction to Logic, H. W. B. Joseph, 2nd Edition, Oxford University Press, London, 1922; Formal Logic, J. N. Keynes, 4th Edition, Macmillan Company, Ltd., London, 1928; Logic, W. E. Johnson, Cambridge University Press, London, Vol. I, Logic, General, 1921; Vol. II, Logic Demonstrative Inference: Deduc- tive and Inductive, 1922; Vol. III, The Logical Foundation of Science, 1924; The Logic of Discovery, R. D. Carmichael, The Open Court Publishing Co., Chicago, IQ30; Rational Induction, H. H. Dubs, The Chicago University Press, Chicago, 1930; and Scientific Inference, Harold Jeffreys, Macmillan Co., New York, 1931. It will be noted that the application of the formal scientific method in discovery involves a human choice at every step. For example, in the discovery of a functional or statistical relationship, the follow- ing choices must be made: I. Choice of data. 2. Choice of functional form. 3. Choice of number of parameters, at least in certain cases. 4. Choice of method of estimating parameters. To a certain extent this field of choice is a kind of methodological No-Man's Land. History of science shows, however, that the discoverers of the past have, in general, been those broadly trained in the particular field of discovery of their choice. They have been those familiar with the status of experimental and theoretical results in their partic- ular field. The importance of theory in helping one to choose the 486 ECONOMIC CONTROL OF QUALITY right thing to be discovered is illustrated by the fact that several elements in the periodic table have been looked for and found because their existence was suggested by the blank spaces. So it is that many of the discoveries of science have been suggested by theory. Furthermore, it is of interest to note that important discoveries have usually come only after the investigator has surrounded himself for a considerable period of time with the facts bearing upon the subject and during this period has kept these more or less constantly in mind. It is true, however, history also indicates that many of these discoveries have only come after the investigator has dropped the search for a time more or less completely from his conscious con- sideration. In all cases, however, it appears that preliminary con- scious attention to the facts in hand is essential. Coming now to the more or less formal treatment of scientific method, the following books will be found helpful in something like the order listed: The Foundations of Science, H. Poincare, The Science Press, New York, 1929; The Principles of Science, W. S. Jevons, Macmillan Company, Ltd., London, 1924; Essentials of Scientific Method, A. Wolf, Macmillan Company, New York, 1927; Scientific Method, A. D. Ritchie, Harcourt, Brace & Company, New York, 1923; and Physics, The Elements, N. R. Campbell, Cambridge University Press, London, 1920, together with Vol. III of Johnson's Logic noted in the previous paragraph. Books such as the Quest for Certainly, John Dewey, Minton Balch Company, New York, 1929; and in particular, A. N. Whitehead's Process and Reality, Macmillan Company, New York, 1930, contain much of interest. Just as a simple example, it is necessary for us to think of a quality characteristic as an entity in the sense adopted by Whitehead if it is to be general enough to be of use in the many practical problems that arise in the interpretation of a sample. OTHER REFERENCES 1. Errors of Measurement It is assumed that the reader has available one or more of the following books on the discussion of the errors of measurement: The Combination of Observations, David Brunt, University Press, London, I917; The Calculus of Observations, E. T. Whittaker and G. Robinson, Blackie & Son, London, 1924; The Theory of Measurements, A. D. Palmer, McGraw-Hill Publishing Company, New York, 1930; and APPENDIX III 487 The Theory of Measurements, L. Tuttle and J. Satterly, Longmans, Green & Company, New York, 1925. g Brunt's book contains, in addition to the ordinary discussion of the theory of errors, an interesting introductory chapter indicating various ways of developing the normal law. The book by Tuttle and Satterly gives a particularly good elementary discussion of many things which must be considered in correcting data for errors of meas- urement. Palmer's treatise is of particular value in outlining things which must be considered in planning physical measurements so as to reduce the errors of measurement to a minimum. 2. Tables Of course, every one needs a table of squares, reciprocals, and square roots such as that of Barlow published in revised form by E. and F. N. Spon, Ltd., London, 1930, and a table of logarithms such as those of Vega published by D. Van Nostrand Company, New York, 1916. In addition to these, any one interested in the theory of quality control will find much use for Pearson’s Tables for Statisticians and Biometricians, published by the Cambridge University Press, London, 1924. The second volume of these tables which is now in the process of preparation is supposed to contain the tables which have appeared in Biometrika since the publication of the first volume in 1924. In a way, the promised second volume will be even more helpful than the first. The books by Fry, Arne Fisher, and R. A. Fisher contain many useful tables. For a more complete bibliography, the reader is referred again to that of Yule. 3. Magazines Without question, one magazine which has been found most useful in our study of quality control has been Biometrika, edited by Karl Pearson and his son Egon Pearson, and published by the Cambridge University Press, London. It has carried many of the important papers of “Student,” R. A. Fisher, L. H. C. Tippett, J. Neyman, J. O. Irwin, Karl Pearson, E. S. Pearson, J. Wishart, and their associates. The Skandinavisk Aktuarietidskrift, Stockholm, contains many important articles in English as well as in foreign languages. The same is true of Metron, an international review of statistics published in Rome, Italy. 488 ECONOMIC CONTROL OF QUALITY The journal of the American Statistical Association, New York, contains many discussions of the applications of the more elementary theory of statistics in the field of economics. The same can be said of the journal of the Royal Statistical Society, London, although this Journal has also published several important articles on the theory of statistics. Both of these Journals are of value because of their reviews of current literature. The Annals of Mathematical Statistics is a Journal recently started in cooperation with the American Statis- tical Association. It is devoted to both theory and application of mathematical statistics. A glance at any of the complete bibliographies previously referred to will show that important articles have appeared in many other journals than those listed here, 4. Mathematics It is assumed, of course, that the student of the theory of control is equipped with elementary texts up to and including differential and integral calculus. For a more complete treatment than is ordi- narily given in any elementary text, the following books are suggested. In questions involving purely algebraical manipulation as in the dis- cussion of the multinomial theorem, the student will find Algebra, G. Chrystal, Vol. I and Vol. II, 5th Edition, A. and C. Black, Ltd., London, 1920, of great help. For a discussion of the subject of sym- metric functions and related topics of interest in the application of the method of moments and the use of semi-invariants, M. Bôcher's Introduction to Higher Algebra, Macmillan Company, New York, 1921, will be helpful. Advanced Calculus, W. F. Osgood, Macmillan Com- pany, New York, 1925, treats in sufficient detail for most purposes the analytical methods required for an understanding of the math- ematical theory found in most articles on the subjects of specification, distribution, and estimation. Two excellent books on the mathemat- ical theory of statistics are: Statistique Mathématique, G. Darmois, Gaston Doin et Cie., Paris, 1928, and Statistique Mathématique, Charles Jordan, Gauthier-Villars et Cie., Paris, 1927. 5. Graphical Methods We have neglected to consider in any great detail the important problem of presenting the results of quality control studies in a way to be of greatest service even though so much depends upon a thought- APPENDIX III 489 O. O O | tº º O. O O2 +- i + O.OOO3 r + O.OOO4 * * O.OOO5 -H. O.OOO6 -- O.OOO7 -H. O.OOO8 —H O.OO O * * O,OO2O —- O.OO3O —H. O.O.O.4O - O.OO5O + 0.0060 T O.OO 70 —H O.O.O.80 -- O.OO 90 —H O,O || OO * * O,O2OO * * O.O3OO -H. O.O4OO —H. O.O6OO * =e O.O 700 -H O.O.800 - O,O900 + O. 1000 - O. 12 OO I A L I G N M E N T C H A RT = \|p' (1 - p") ©p. |ege, FIG. 4. + O.O500 - 10,000 – 90 OO – 8 OOO -- 7OOO —H 6000 : -H 5 OOO -- 4 OOO -H 3OOO +- 2000 -H |OOO + 900 + Bog -H 7OO -- 6OO -H 5 OO -H 4OO + 3OO •+- 200 +-IOO -- 9 O -- 8 O -- 60 -- 5 O 49C ECONOMIC CONTROL OF QUALITY - T O.6 NOMOGRAPHIC RERRESENTATION ~ of PROBABILITY (Pn) THAT AT ERROR (X-X") |N AVERAGE ESTIMATED FROM THE MEAN OF O.7 || O O + A sampl E (X) of SIZE n AND MEASURED IN TERMS OF ITS STANDARD DEVIATION LIES O.72OO + BET WEEN - CO AND + t FOR VARIOUS n (FROM n = 4 To n=25) O.73OO -- + 0.7 O.74OO + O.75 OO -- 4 -t- O.76OO -- j O.8 o.7700 + O.78OO -- wº- .9 O.7900 + + O ,8000 + 5 -H. O O.8 |OO + | + 1. O tº 0.8200 + 3. o,83ool * 6 + § 0.8400+ + | . I § | 0.8500+ LL) Uſ) C —l Cl. O.86OO + * + 1.2 | 7+ O C O,87OO + f 8 -- o,8800 + + + 1.3 O.8900 + 9 -- O,900 O + + 1.4 JO -- O.9 |O O - + O.92OO - + 1.5 12+ O.93OO + - 14 + O.94OO –– |## O,9500 + #3 + O.96OO + 25 + O.97OO + O.98OO + O.985O + o,9900+ PROBABILITY Pt of HAVING THE SAME O.995 O + ERROR IN AVERAGE BET WEEN -t, AND + t ojof CAN BE CALCULATED FROM: O.998O + Pt = 2 (Pn-o.5) O.999C) + O.9999 + NUMERICAL VALUES OF PROBABILITY Pn ARE TAKEN FROM R. A. FISHER's TABLES GIVEN IN HIS PAPER: EXPANSION - OF "STUDENT'S" INTEGRAL IN POWERS of n-1 ("METRoN' volumE v. No. 3. 1925) FIG. 5. APPENDIX III 491 ful and artistic layout of the graphical presentation. In this con- nection, Layout in Advertising, W. A. Dwiggins, Harper & Bros., New York, 1928, should prove to be suggestive. In closing, we should note that in the application of the method of control, it is sometimes advisable to substitute nomograms for tables in shop practice. For example, Fig. 4 gives a nomogram which enables one to read off the standard deviation a in terms of a given sample size n and probability p’. In a similar way, Fig. 5 presents in graphical form the very complicated table of “Student’s” integral. For a discussion of this nomogram and of the application of nomog- raphy in this way, see the paper by V. A. Nekrassoff, “Nomography in Applications of Statistics,” published in Metron, Vol. VIII, 1930, pp. 95-99. INDEX OF NAMES [The numbers refer to pages] ADYANTHAYA, N. K., 282, 287 ANDREW, S. L., 475 BARLow, 487 BAYES, R., 48o BecKER, 4, 476 BERNOULLI, 318 BETTs, H. S., 387 BLOCH, E., 474 BöcHER, M., 488 BoDE, H. B., 484 Bowley, A. L., 243, 429, 478 BRADLEY, H. W., 485 BRIDGMAN, P. W., I, 481 BROAD, C. D., 484 BRow N, Io BRUNT, D., 486, 487 BURGEss, R. W., 478 CAMP, B. H., 177, 179, 185, 187,208, 478, 479 CAMPBELL, N. R., 486 CARMICHAEL, R. D., 485 CAve, B. M., 217 CHADDock, R. E., 478 CHARLIER, I 58, I 59, 267, 475, 476 CHRYSTAL, G., 488 CHURCH, A. E. R., I80, 188,241, 244, 245, 479 CoGGINs, P. P., 343 CoolTDGE, J. L., 124,476 CRAIG, C. C., 234, 241 DAEves, K. H., 4 DARMOIs, G., 488 DARRow, K. K., 474 DEBROGLIE, L., 35I DEw EY, J., 486 DoDD, E. L., 168, 232, 479 DoDGE, H. F., 48, 341 DoNHAM, W. B., 475 DUBs, H. H., 385, 485 DUSHMAN, S., 378, 379 DwigGINs, W. A., 491 EDDINGTON, A. S., I, 481 EDGEworth, F. Y., 243, 476 EDwARDs, G. D., 477 ELDERTON, P., 207, 270, 271, 475, 479 Evans, G. C., 477 - EwALD, P. P., 474 FERMAT, I22 FISHER, ARNE, 475, 480, 487 FISHER, R. A., 184, 188, 189, 207, 217, 221, 240, 24I, 279, 479, 48o, 481,487 Forsyth, A. R., 386 FostER, W. T., 141 Four IER, 234 FRY, T. C., 124, 476, 487 GAUss, 135, 240 GEIGER, H., 127 GRAM, I 58, I 59, 267, 475, 476 GRoNDAHL, L. O., 352 HAAs, A., 351, 473 HARDY, G. H., 485 HAYES, A., 391 HETTINGER, A. J., 141 Holzing ER, K. J., 245 HoTELLING, H., 241, 479 IRWIN, J. O., I80, 241,487 Isser LIS, L., 197, 479 JAMES, W., 484 JEFFREYs, H., 485 493 494 INDEX OF NAMES JEvoNs, W. S., 38,486 JoHNson, W. E., I31, 485, 486 Jones, C., 407 Jones, R. L., 19 JoRDAN, C., 480 Joseph, H. W. B., 485 KELLEY, T. L., 228, 475, 479 KEYNES, J. M., 481 KEYNES, J. N., 485 LAGRANGE, 259 LAPLACE, 135, 367, 368 LEE, A., 217 LEXIs, 318 Loeb, L. B., 474 LovEJOY, A. L., 484 LUCAS, F. F., 357 MARKOFF, A. A., 479 MAxwell, 122, 133 MEIDELL, M. B., 177, 179, 185, 187, 208 MEYER, O. E., 134 MILLIKAN, R. A., 21, 62, 163, 377 MILLs, F. C., 476, 478 MITCHELL, W. C., 474 MoLINA, E. C., 370, 480 MooRE, H. F., 410 MoULTON, F. R., 367 NEKRAssoFF, W. A., 491 NEw LIN, J. A., 23 NEWTON, 122, 142, 353 NEYMAN, J., 193, 195, 292, 369, 47", 481, 487 OsgooD, W. F., 488 PAGE, L. L., 473 PALMER, A. D., 486, 487 PAssANo, R. F. 391 PEARL, R., I4I, I42 PEARSON, E. S., 282,287,369,479,481,487 PEARSON, K., 138, 184, 206, 207, 217, 241, 329, 475, 476, 479, 484, 487 PERRIN, II PERsons, W. M., 141 PLAUT, H. C., 4, 476 PoſNCARE, H., 385, 486 Poisson, 318, 407 Pope, A., 3 Pöschl, TH., 474 PRANDTL, L., 474 RADFORD, G. S., 39 RICE, J., 474 RICHTMEYER, F. K., 474 RIDER, P. R., 198, 241, 479 RIETz, H. L., IO4, II 5, I 16, 124, 478,479 RITCHIE, A. D., 486 - t RoBINSON, G., I43, 234, 388,479, 486 ROMANovsky, V. J., I 55, 234, 235, 241, 479 ROMIG, H. G., 341 RUARK, A. E., 474 RUNGE, I., 4, 476 RUNNING, T. R., 479 RUSSELL, B., 481,484 RUTHERFORD, E., 127 SATTERLY, J., 487 SHEPPARD, W. F., 78, 79, 329 SHEwHART, W. A., 190, 382, 383 SIDWICK, A., 485 SLUTSKY, E., 479 SoPER, H. E., 217, 218 STEFFENSEN, J. F., 476 STUDENT, 184, 188, 189, 196, 217,479, 487, 49I TAYLOR, 393 TCHEBYCHEFF, 88, 95, IO7, I 17, 176, 179, 181, 189, 193, 202, 221, 245, 257, 265, 266, 267, 269, 277, 370, 397, 479 TCHOUPROFF, 24I, 244, 479 THIELE, T. N., 476 THORNDIKE, E. L., 484 TIPPETT, L. H. C., 202, 282, 287, 479, 487 ToLMAN, R. C., 474 TUTTLE, L., 487 UREY, H. C., 474 WALKER, H. M., 48o WALSH, C. M., 53 t WATANABE, M., 479 WEBER, P. J., 132 INDEX OF NAMES 495 WHETHAM, C. D., 473 WHITEHEAD, A. N., 369, 484, 486 WHITTAKER, E. T., 143,234,388,479, 486 WILKINson, R.I., 370, 48o Wilson, E. B., 483 WILSON, H. A., 474 WINTERs, F. W., 190, 315 WISHART, J., 241, 479, 487 Wolf, A., 486 Young, A. W., 217 YULE, G. U., 126, 229, 407, 478,479 INDEX OF SUBJECTS [The numbers refer to pages] Arithmetic mean as measure of central tendency, 71 distribution of, 179-183; 191-196 efficiency of, 279-284 measurement of, 369-375 method of calculation of, 72 significance of, 86–88 use in general, 94-96 use when quality is controlled, 89-94 Assignable cause definition of, I4 effect of removing, 26-34 eliminating (see Criteria) Type I, I46 Type II, 321 Bibliography, 473-491 Cause assignable (see Assignable cause) causal explanation, 364-369 chance constant system of definition of, I2, 130; errors of, 381-385; evidence for existence of, 12, 130; resultant effect of, 425-436 definition of, 7 variable system of, I31 macroscopic, 322 meaning of, I31 ultimate, 321 Cells boundaries of, 66-70 number of, 66, 69 Chance cause, 7 (see Cause) how chance effects have been portrayed, I36–137 - variations that should not be left to chance, 4, 17, 21, 22, 54, 275-422 Chi square distribution of, 205-2Io mean, 207 mode, 2O7 use of for detecting change in universe of effects, 297 use of in Criterion V, 328-338 Consistent estimate, 284 Consumer’s risk, 341-347 Control characteristics of, 6 chart (see Criteria) conditions for, I45-I48 definition of, 6 detection of lack of, 301-349 choice of criteria, 303-304 in respect to standard quality, 275-3CO economic advantages of, 26-34; 430-436 graphical methods in, 488-491 importance of in setting tolerances, 250- 259 laws basic to, 12 I-I44 limits, basis forestablishment of,275-277 maximum characteristics of, I 52-158 definition of, I 50-I 52 nature of, 5 need for, 354-356 objective state of, 34 postulates basic to, 8-14 problem of, 3-5 program, 418-42I report, 421-422 sampling in relation to, 341-347 statistical, I 45-I49 Correlation coefficient as measure of relationship, 71 calculation of, 81 commonness of causation measured by, 223-228 definition of, 8o 497 498 INDEX OF SUBJECTS Correlation coefficient distribution of, 217-223 importance of in specification of quality, 228 -- interpretation of, IoA-II 5; 228-229 Correlation ratio as measure of relationship, 71 definition of, II 5 use as measure of relationship; II 5-II.7 Criteria for detecting trouble or assignable C421/J&S character of, I 5 choice of for detecting lack of control, 3O3–3O4 Criterion I, 304–318 Criterion II, 318-320 Criterion III, 322–325 Criterion IV, 325–328 Criterion V, 328-338 evidence that they work, 18–22 rôle of judgment in choice of, 338-34o Data analysis of, 385-388 correction of for errors, 379-385 frequency distribution of, 63, 66 permuted series of, 63 prerequisites for good, 379 presentation of graphical, 65 grouped, 66-70 problem of, 57-61 ungrouped, 63 reasons for taking, 55-56 sampling, 437-471 Design, 249-272 control in, 419 for minimum variability, 259-261 limits on variability, 248-26 I Dispersion, measures of, 7I Distribution frequency, 63-68 cumulative, 68 references, 475-476 relative, 66 function, 41-44 Gram-Charlier, I4o of arithmetic mean X, 179-183 Distribution - of correlation coefficient r, 217-223 conditions under which significant, 22I-223 of flatness or kurtosis 32, 197 of fraction defective p, 178-179 of other measures of central tendency, I97-I99 of other measures of dispersion, 199-205 of quality characteristics, 44-46 of skewness k, 197 ofstandard deviation a, I84-188; 191-196 of statistic in relation to control, I75 of throws of dice, I25 of variance a *, 188 Pearson types of, 138-139 statistical, 22 “Student's,” 189-191 theory importance of 232 mathematical, 233-245 phases of 230-232 simple illustration of, 169, 171 Efficiency definition of, 172 of arithmetic mean, 279-284 of Criterion I, 315 of measures of dispersion, 287-289 max. + min. of median and , 279-284 Errors correction of data for, 379-385 different kinds of, 378-379 law of propagation of, 392–394 of grouping, 78 of measurement, 61; 379-385; 486-487 Flatness or kurtosis distribution of, I71, 197 importance of, 96-98 measure of, 7I method of calculation of, 75 significance of, 75 Fluctuations (see Sampling fluctuations) Fraction defective definition of, 71 distribution of, I78-179 INDEX OF SUBJECTS 499 Frequency definition of, 66 distributions, 63-68 histogram, 67 polygon, 67 relative, 66 Histogram cumulative, 68 frequency, 67 surface, 68 Index of quality, 48 Information essential, definition of, 58 total, definition of, 85 Inspection reduction in cost of, 26-27 sampling in relation to control, 341-347 to detect lack of control, 275-422 Judgment references on use of, 482-486 rôle of in choice of criteria, 338-34o Law comparison of exact and statistical, I40- I44 exact, I2I; I4O-I44 normal (see Normal law) of large numbers evidence for existence of, I25-130 statement of, I22-125 physical, 360-364; 473–474 statistical, I33-I44; 473–474 Limits basis for establishing control limits, 275-277 for detecting trouble (see Criteria) statistical, 361-364; 437-439 tolerance (see Tolerance limits) Maximum + Minimum r) - as measure of central tendency, 71 distribution of, 171; 197-199 efficiency of, 279-284 Mean Deviation as measure of dispersion, 7I Mean deviation distribution of, 199-205 efficiency of, 287-289 Measurement - a sampling process, 378-379 how many, 390-392 minimizing cost of, 388-390 of average X and standard deviation or, 369-375 place of in control, 376-378 through statistical relationship, 394-403 Median - as measure of central tendency, 71 definition of, 63 distribution of, I71; 197-199 efficiency of, 279-284 Method graphical, IoS of least squares, Io9 of moments, Iog Mode - as measure of central tendency, 71 definition of, 63 Molecular motion, Io-I2; 129-131; 133-135 Normal law definition of, I2 graphical representation of, 94 table of integral of, 90 Normal surface analytical expression for, Ioo graphical representation of, IoI. Parameters definition of, 99 four methods of estimating, Iog Point binomial in relation to control, 125 Polygon cumulative, 68 frequency, 67 surface, 68 Probability a posteriori, 342-347; 370-37I a priori, I23, 168 Constant, I2 mathematical, I23, 168 objective, 123, references, 476 5OO INDEX OF SUBJECTS Quality allowable variability in, 173; 273-347 as an attribute, 40–41 as a point in space, 4o control in practice, 351-422 object of, 356-357 report, 42 I-422 distribution of, 44-46 importance of correlation coefficient in specification of, 228 index, 48 objective, 53 of a number of the same kind of things, 4I-44 of product, 44-47 popular conception of, 37 rate, 47-49 relationship between several qualities, II3-II 5 - representation in m space, 39, 49 set of characteristics, 38 standard detection of failure to maintain, 298– 3OO specification of, 262-272 statistics, 46 statistics to be used when quality is controlled, 89-94 subjective, 53 tolerance on controlled quality, 256- 259 true versus observed, 61 uniform, 31 Range, 63 arithmetic mean of, 203 as measure of dispersion, 71 distribution of, 201-205 efficiency of, 287-289 standard deviation of, 203 Rate, quality, 47-49 Rational subgroups, 299, 3O4, 305, 309, 312, 313; 409–412 Regression line of definition of, Io; standard deviation from, Ioë Regression plane of graphical representation of, 33 standard deviation from, I I4 Rejections, reduction in cost of, 27–29 Relationship between several qualities, II3-II 5 functional, 99 general comments, II6-II 7 measured by correlation coefficient, IO4-II 5 measured by correlation ratio, II 5. II6 measures of statistical, 71, 80 observed, IoI-IO4 statistical, IOO Sample definition of, 163 random, 406-407 relation of to universe, I70 representative, 4IO-4II size of, 313-314; 4II-417 Sampling, 404-417 for protection, 408-4IO in relation to specification, 417 theory, 405-406 Sampling fluctuations, 69, 163-245 experimental results, 164-167 in correlation coefficient, 2I4–217 in simple statistics, 167 problem of predicting, 167-170 Scatter diagram, 68 Second approximation definition of, 94 in relation to controlled quality, I 59 table of integral of, 90-91 use in Criterion V, 329 Significant figures, 79 Skewness distribution of, 171, 197 importance of, 96-98 measures of, 7I method of calculation of, 74–75 significance of, 75; 86–88 Specification, 249-272 of standard quality, 262-272 Sampling in relation to, 417 INDEX OF SUBJECTS 5OI Standard deviation as measure of dispersion, 71 distribution of, 184-188; 191-196 efficiency of, 287-289 measurement of, 369-375 method of calculation of, 73 significance of, 86–88 use in general, 94-96 use when quality is controlled, 89-94 Standard quality detection of failure to maintain, 298–300 specification of, 262-272 Statistical comparison of statistical and exact laws, I4O-I44 indeterminateness, 362 laws, 133-140; 473–474 limit, 361-364; 437-439 magazines, 487-488 measurement through statistical rela- tionship, 394-403 nature of macroscopic properties of matter, I28 nature of modulus of rupture, 23 relationship, Ioo tables, 487 theory rôle of, 22-24 texts on, 478-48 I Statistics calculation of, 72-81 choice of to detect change in average quality, 278-284; in correlation coeffi- cient, 290; in standard deviation, 284-289; in universe of effects, 297 choice of method of using, 290-300 classes of, 80-84 consistent, 284 expected and modal values of, 212 quality, 46 simple, 71 sufficient, 279 to be used when quality is controlled, 89-94 why average and standard deviation are always useful, 94-96 Tolerance for quality of finished product, 252- 256 importance of controlin setting, 250-252 limits definition of, 249 reduction in, 32 on controlled quality, 256-2.59 range, 249 where Ioo per cent inspection cannot be made, 250 Value, use, cost, esteem, exchange, 53 P'ariability, design limits on, 249-272 Variance as measure of dispersion, 71 distribution of, 188 ||||||||| §juſ& 4 | | | . y - , . . . . . . * * * * - - - º * . . . . . . . . . . . . . . . 2 W = 1 . . . . . . . . . . . . * * * * * * * * " " . . . . . . * * * * * * : * * * * * . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * * * * * * * * * * . . . . . . . . * * * * * * ſ . . . . . . . . . . . . . . . . . . . . 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Wºº'ſ sº sº -i-ste § 2; . . . . . . . . . . . . . . . . . . . .eſºt - e, is -- . . . . . .tºs Wººtºſ e º see Wºº C C C. ſº a tº-º-º-º-º-º-º-º: º !. ... * * * * * ºf:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: º.º. º. 3 & - - º * | * El a 2. a et sº * º º […] º C ºſ-ºs-i: - . . . . . . . . . . . . . .” -- - - - - - - - ºne - sº-º. . . . . . . . †::::::::::::::::::::::::: * I aſ wºeſ & ) ºi.º.º.º.º.º.º.º.º.º.º. - * * * ***** tº . Jºa ſº- let '~' ºs- ºr 12teſe * * * * * * * * ~ 1:. . . . . . e. . . . . . & 1-1, ſº "? ſet. * : **** - : * . . . . . . . . . . . . . . . . . . . . . . ºffeels tº 1-tºn. ... e.g.,. renº -ſcrew. - . . . . . . fººt-tº-1- ºne-ºf-ſets safe-----. ---------nºi...º.º.º.º.º.w....nº.º.º.º.º.º isºtº-tº-1: - - - . . . . . . . . . * * * * * * * * * * •ºº e-ºe ſet “ . . . He is ete v we we . . . . . . .*.*.*********** -- e.les, tº sº; e - even-º-º-º-º: ... . . . . . . . . sº ºr • We were *"Sººts ºne ... ..., ºft-º-º-º-º: - º:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: . . . ... etes sketc - “ºre eters, eſs isºere a neº- a ...A.'s tººsteºst-º-º-º-º- ºr-. . . .". -k, -ia ...t. tºº ſets tº Wewalsº ſelºº." - º º sº ******** -- i.... • *s---|-- Sº...º.º.º.º.º.º.º.º.º.º.º.º.º.º.º.º. º !. •l-ºr-º- 'º. -- . . . . . . . . . . . . . . ;-ºne tº seſsiºn. Jº...ºrº, ºw--- . . . . . . . . . . . . . ...,...; tº "... ſºlº tº ſº. • Wese effe - $º ºf jº } Wº ºil. i. neº-t ********teis.e., . . . . . . . . . . . [...º.º is is is iſ ºº:::::::::::...º.º.º.º.º.º.º.º.º.º.º. º lik.º.º.º.º. º ºº::::::::= -- tº sº...º.º.º.º. -º-º-º-º-º-º-º-telenecºlºnsºliºsºsº.º.º.º. ºº:::::::::::::::::::ſºli. * * : * ~ a ſal ** * * * * ******* - - is . . . . .eps. - esºtetsie's sis !º.º.º.º.º.º.º.º.º.º.º.º.º.º. º: |-te. , tº “H. alsº a tele sº-ſºº tº: ºriº.º.º.º.º.º.º.º.1:1:...]...ſºlºisſºlº sºlº-rºº º: . . . . . . . ;--1, its tº nº *** We ... ....:*-ºn-tº- SH-1--, . . . . . º.º.º.º.º.º.º.º.º.º.º.º.º.º.º.º.º.º.º.º.º. º.º.º. 1-stºlºi-º-º. 11... ... ---, -ºº:::::: !... . . . . . . . . . ; ; ; ; ; ; ; ºf º º ... . º ºlº. º reº a a we as we . • is “ --wºº is is sº stwº sºls -- tºte tº she rºws. . . . . . . .- ... . . . . . . . . . . . . . . tººl. J., . . . . . . . . . . . . . .'; 1:1 tº 'Sºlº ; : 1-14 a taſ, as sº is tº º'ºts º rººtsiºſº ºf *** * * * * * : * ~ *...*.*.* let-reſs ºilº.º.º.º.º.º.º.º.º.º.º.º.º.º.º.º.º.º.º.º. º º º ***** * * * * * * * * . . - - - be is teleºsis tº is . . . . . . . . v.- : * ~ * is jºi. s - - - , , . . . . * * *s ºfºrºſ.ºr-tºº-ºº: | " ": sº-º-tº at * 1sh; is . . . ; | * * ~ * ºw | * * * * * * * * ºfºº tes, is sº-º-ºw........ • * * * * ****** - - - - - -, -sisteſ. . . ejºſ.t. --, -, ---ſººk: ***** * * *. . . . . . . ; ********* * * * * * * * * * * * * * * * - - - - - - - - - ºº:::::: ; ; ; ; ; ; sº tº tº: ...tº *** *.*.* to R- ºt-lºs ... •egrº . ſº...º.º.º.º.º.º.º...!!! !...iº.º.º.º.º.º.º.º.º.º isºlº.º. ººººººººººº... º º:::::::::::tºn. Sºtº, -º-º-º-º-ºi.........jºi..jºnºiºsºsºlºi-º-º-º-º-ºº-ººººººº-º-º::::::::::ſº. • * * * * * : * ~ * º ºº:::::::::::::i-lº arsºnsists tºº. is tº . . . … --res left-l.,...º.º.º.º.º.º.º.º.º.º.º. - . . . . . ; isºlº Sº...ºu...t-tsºº-ºº ºl-" tºº. *** - ſº :::::::::::::::::::::::::: iº.º.º.º.º isºtº-º-º:*::::::::::: ſº a lº & --..º.º.º.º.º.º.º.º.º.º.ºr-tºº--ºf- ºtels. . . . . .1st. . . . . . . . .'; ' ' ' ' ' ' ' ' 'º'; '• ºº:::::::::::::::::::::::::::::::::::::::::::::::::::::::: º sº ºº::--------fºr-º-º: sters cº-ºre, irºº-ºººººººººººººººº-ººrsists;º sº... fºſsºstºlers ºf ººººº. lººk it ſºil; | º Sºº-º-º-º-º-º-º-º-º-º-º-ºil...º.º.º.º.º.º.º.º.º.º.º...i- •lsſºn-ºf-f. * Teiº º lººr; ºr ºlºr-º-º-º-º-º: stºº: º . sº *** **** * * * * *. . • * ~ *feterstate eleº- *** * * * * * * * * * * * ~ * º, e - i. º.ºte a systems tº cº- sº Iº sº-º-º-º: sºletºstals tº sºlº sºlºr-º-º: •º-º-º-º-º-, * ~ ** Wety" ******* . . . tº sº º ºpeſº 4. º ºº:: *...*.*.*fººt-º-º-º-º-º-º:...i.º.º.º.º.º. scºre-º-n -->|- sº sº. -t-. [...] ºust...ºlsº tºº." - * - - - a jºjºſº.º.º.º. º ºl; * * * * * : * ~ * ******* -º-º-º-º-º-º-º-ººººººººººº-º-º-º-º-ººººººslatsiºs::snerººrººººººº-º-º-º-º-º-º-º-ºººººººººº; reſºrt.rº ºfºº tº ºr cºſt º º º: jº-ºº:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: º::::::::::::::::::::::::::::::::::::::::::::::::: º: M. i. ºnestºr... a lºwer...º.º.º.º.º.º.º.º. 3 – ests sº-, - " - ºr “; ; ; * : "... 'ſ Nº. " * * * º - Tº ſº §:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: sº stºne - eſs, we eteº º *> *-** ** - - - - - - - -, ºr - tº ſei-ts'ºe weſt ſets where is is . . . . * . . . . ~ * r * : * ~ * * * * * ******* a s tº e.e. is stwºstº, tº wººl--tº “*” "F": - - - - - -ºº ºr º;ſº º t º . ...tº'º. ºf-º-º-tººt-i-º-º-º-º-º-º-º-º-º-º-º-º-º-º-º-º-º-º-ten-tºº-ºº-ºººººººº-º-º-º-º-º-º-º-º-º: ſº-ºº-ººººººººººº-ººsiºs-ºs-Hºlsº al . t *:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ...! ††: ºf . º.º. ºº::::::::::::::::::::::::: a Wºº • * * * * * * * * * * : * > . . . . a . . . . . . . . . . . . . . is e - e s - I - - , = . . . . . . . . . . . . . . . . . . . . . sºme - $2 a eſs º f * * * * * * * * * is . . . . . . . . . . . . . . . . . . [- - - - - - - | * * * * * * * * * * * ****** a e i P. t - tº " ** *. ºº::::::::::::::::::: ſº e º s ..º.º.º.º.º.º.º.º. º:::::::::::::::::::::::::::::: I. º lº.º. • * !. º lºſſº ºº:::::::::::::::::::::::::: is ,-4s tº Fºº ºº::::::::::::"...º.º.º.º. ..i...º.º.º.ºttisºlºis-tº-ºn-º-º-º-º::iºt-ºf-flººssº: ºf * - - ? . . jº". *** *********** * ***** cºsts * Wis * * * * * * * ~ *** * : * ~ *-**** *** - - - - - - - - - - - - stºr. ...ºneta ºn... asſeſseſſee's 2 tº ºtº-º-º-º: º,...tº state tººl: ; “tº ** **-->ſe is º at: * * **** ºrs stºº stºl-ter: sº sºvº.4 • *relekºest, sº-º-º-º-º-º-1- lºsis º.º.º. ººººººººº-ºº:::::::::::::::::::::::::::::::::::::::::::::tº |- sº ſensiseſ, ſcies. ..º.º.º.º.ºtsºlelºlº ºneſ: º ºte: * * * * ºº:: º º º * * * * * * : * : * * * * * * * ... ... ... -: , -º-, - ... tº #4 ºf ºst-ºr-nº sº lºſe effºrt.ust... . . . . tº ºf aſ-isºn stee i...º.º.º. ººse--- - - - , , ; ;st-ſels, ºr cºsts --- ºften a 1, •,• . . . . . . . . ºf-2 tº t- ºr- is . . . . a.º.º.º. º. ſºlºsſe sºletºn-i- i.e. --- - , , , , , , etc. sie, a ye • , , , , , -º- a b > * * * * * , s as . . . . . . . . . . * ~ * * - d * nº s , * * * * * : * : * : * : * : * : * * * . . . . . . . . is is . . . . . . . . . - - º - . •le - tº tºº - . . . . . ;- * * : * * * * * * * * * * * * - tº . . . . . . . . . . . . . . . . . . . is a tº be sº * * * * * * * * * * * * * * * * * * º: . º º :::::::::::::::::::::::::::::: * sºlets; tº !" • • * * * *** * * *. lºsſº is lº * 1: * *::veraldi: ºn-º-º-º-º-º-º-º-º-º-º-º- |-i-Haº. ...l.º.º.º.º.º.º.º.º.º.º.º.º.º.º. i*isterºtº-ºº::::: ºt-tº-geºtets is lºoselºsº ivº tº vºl. 3 º' º *** * * * * : * ~ *-ºs isºtºniºte's sº tºº isi. . . . . . . . . . . . . . . . . . . . . . . . . -- 'Yº -- “tº ºº: ºne fººt.ºr. ... ----ºº-ºº: tº . . ºlºts > * * * : *t ºr sºlº ºr ºf tº tº st-ſeº tº is . . ...; tº-º-º-º-º-º-º-º-º-º-º-º-sº-tº-º-º-º-º-º-º-º: -i.iº islaº-º-º-º-º-º-º-º...º.º.º.º.º.º-º-º-º-º: * - sº a lºt.-- ºlºſ; - ; sº a a cus, i. p. ... *** -i-tº-he twº. was reº ºweſs, is bººm. -- i.e.-- is ºx-ºiki.º.º.º.º. ºnºsºlºiskeletºniºn....jºsº. ºf-ºf-ºbst:sºlº tººl--tº ſº a s º º:************* - ... -----------tº-sº tººl: ºssºstºl-fºr-º-º-º-º-º-º-º-i-º-º-º-º-º-º-º-º-º-º-º-te. ...º.º.ºrºssrºº ſ **:::::::::"º wº ºf-ºf-tºº-º-º: sheet. .º.º.º. ºººººº-ºººººº...º.º.º.º. ººº-ºººººººº-º-º: * -*** - • “1” is tº shºe he sº ºrsters sists she sº ºt.; sºciº-i- ºr-ſºlº Hº jºº ºf -º-º: ººlººf-ºf- - *** * * * rº. ſe ****** *** * * * : * ~ *ſ-----|--|-- - - - - •º-tºº"...ºne stºle], . . . . ;- 1st-tº-ſº º:- - - - -, -ºº::....." "...º.º.º.º-º-º-tºº" ſº # * : * * *::::::::::::::::::::::::::::::::: **** -º-º-º-º-º-º-º-º-º-º. -i-...-,-,-,-,-ºº-ºººº-j.n.d.º.º.º.º.º.º.º.º.º.º.º. ºrieneſ. ...) ..º.º.º.º.º. lºſs ºbst: " - x -: , ; , , , , ºesº ºff-tº ºne issus is rºº le ****** -., -º-º-º-º-ºp . *****, *s--be-reſsiete weſel-ele-ſense reſºleta i. 1-4-1-4. 1-1s-ºf-ºf-ºf-i- sº-ſºº ºn. Nºrst-ºf-ºf-ºf- ºf" is ºet, tºº *::"Nº - " - . . . . ºlºr:l-tº-nºt-fººt - w at - - tº *** a visiºns tº ºne tºº ſº tºº tºº - . ,-, -jee's as ºne-º-º: |- . . . . ." it is, “…! ºº::::::::::::::::::::::::::::A; ſº º * * * * * * I e i = 0 = 0 ° a s = - • , sº * * * * * *t → • * * * * * * * * * - e º ºº:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: º',','º','!'}; "º, ** * * * * * * * * ſ -º- > -e-,-,-,-, . . . . . . . . . ...º.º.º.º.º-º-º-º-º-º-º-º-º-º-º-º-º-º.º.º.º.º.º.º.º. is ... ---iſº... .º.º.º.º.º.º.º. 1-1, -ºº º ºſº.º.º.º.º.º.º.º. ºººººººººººººº. six is tº ſº men. . . ...º.º.º.º.º.º. ºil lºſſº. ... - º glºſſ. º iſ:-i: -; ; ; ; *ſ::::Hºte:: “. Neºs º tº * * * * * * * * * tºº - 'stºº - ºften stºrale-º-º-º-º-º-º- ºne is shººtsºeisers tº "rººtst-tests i.i.d. ºiliºt-rººf;::::::::::::::::::::::::::::::::::::::::::::::::::::::: * * e º ºſ w tº tº ſet- tº . ~ * * * * * * * * * * Hººle ºffers is a twº me is ºw. . . sº. He is is tº ete left- - , º 'º tº “” is ſet ºne tº . . . e. is to seizelº ºtsiºlº, *** *** **** * }; i. 1; º; ;" | Fººtº: "º ******** ill; *... a as wººl-tºº-ºº-ºº-ººººººººººººº-º-º-º-º-º-º-º-º-º-º-º-º-º-º-º-º-º-º-º-º: sºlºissºis...º. [. ºº:::::::: ...tº 1-ft. ººº-º-tºisºtº ºrºk: Sºº-º-º: 1-º-º-º-º-º: sº tº sº-ºes tº . . . . . . . . . . . . . . . . . . . . . . . . . is ºne ºfferen-º-º- 'º'; tº:*:::... º. . . . . . . | | - a sº tº zºº º ...lººkººlºsſº ºfºsſeſs tº fººtº -i-º-º: ſº ºf alsº stºº tº ºr ºfºlºſs. 1. -º-º-º-º-º-º-º-º-º: •-ºf-tºº-º-º-teleſsi-et-ºu-ºo-ºº-ººps ºt-ºf-ºf-ſºlº •ºst, i. 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