Class -UxxU^^lL (x)[PglitN»__L9_La OJ- THE OVERLAPPING OF ATTAINMENTS IN CERTAIN SIXTH, SEVENTH, AND EIGHTH GRADES ' BY PAUL J. KRUSE, Ph.D. TEACHERS COLLEGE, COLUMBIA UNIVERSITY CONTRIBUTIONS TO' EDUCATION, No. 92 PUBLISHED BY ^eacf)er£( College. Columbia ®nibersiitp NEW YORK CITY 1918 iHonograpUii. ,^i\ ^'^Kl \^\l'>' Copyright, 1918 By Paul J. Kruse FEB 24 lt^l9 THE HERSHEY PRESS. HERSHEY. PA. >CI.Ar)|5332 'MO ( ACKNOWLEDGMENTS It is only through the co-operation of those interested in the promotion of the science of education that studies of this nature are possible. I am glad to acknowledge my indebtedness and record my thanks to those teachers in the Seattle Pubhc Schools whose inteUigent co-operation made the giving of the tests far less difficult than it would otherwise have been. To the genuine professional interest and ever courteous help of Supt. F. E. Wil- lard and Principals A. S. Gist, W. H. EUert, R. W. Moore and Charles W. Metsker I owe much. I am unable to express even a small part of my great obhgation and deep gratitude to Professor E. L. Thorndike for his helpful aclvice and encouragement through- out this investigation, and for his generous giving of valuable time and help on other occasions. Without the encouraging assistance given by my wife through the months of testing, scor- ing, and calculating, the completion of the study at this time would have been impossible. P J K CONTENTS Chapter P^ge I Introduction 5 1 Evidences of Overlapping 2 The Problems II The Data 3 1 The Subjects 2 Tests Used 3 Administration of the Tests 4 Scoring the Tests III Overlapping by Single Tests 30 1 The Measure of Overlapping 2 Elimination of Some Tests 3 Comparison of Different Tests as to Amount of Overlapping 4 Surfaces of Overlapping IV Overlapping in the Case of Composites 42 1 Results 2 Surfaces of Overlapping 3 Statistical Treatment V Reliability of the Tests Used as Determined by the Cor- relation OF Groups of Similar Tests 57 VI Comparison of Results in Overlapping With Those of Other Investigators 59 VII Conclusions 67 VIII Bibliography 69 IX Appendix 71 THE OVERLAPPING OF ATTAINMENTS IN CERTAIN SIXTH, SEVENTH, AND EIGHTH GRADES I INTRODUCTION 1. Evidences of Overlapping One of the most important findings resulting from the measure- ment of the achievements of pupils by means of standard tests is the great variabiUty in a given grade. This brings in question our whole system of grading, which proceeds upon the assump- tion that we have in a given grade pupils differing markedly in ability to do the work of the school from those in the grades above and below. There is a tacit understanding that the pres- ence of a pupil in the seventh grade, for example, means that his ability is greater than that representative of the sixth grade and less than that of the eighth grade. The results of much of the testing taken at face value seem to challenge very seriously the validity of these assumptions. Some of the findings of the Butte Survey Commission are in- dicated in the following extracts. (14) "A very wide range of ability in each grade is revealed (in com- position) . . . One eighth-grade pupil wrote a composition rated while two pupils wrote papers rated 7 on the scale of 10. The eighth-grade group range rather evenly over aU the steps of the scale from 1 to 6 ... We see that some fourth-grade pupils surpass the median ability of the eighth-grade pupils, while many eighth-grade pupils fall below the median ability of the fourth-grade pupils." (p. 74) "As in spelling and composi- tion, a very wide range of abifities in penmanship is found in each grade .... Some children in the second grade surpass the abiUty of the median eighth-grade pupil, while some in the eighth grade fall below the median second-grade pupil." (p. 81) 5 6 The Overlapping of Attainments in Certain Grades Dr. Daniel Starch in his "Educational Measurements" (16) makes repeated reference to this overlapping of grades. "There are pupils in the fourth grade whose attainment in reading is higher than that of the average eighth-grade pupil. Likewise, there are pupils in the fourth grade whose attainment in reading is inferior to that of the average first-grade pupil." (p. 40) "In speed and comprehension combined, 31.8 per cent of the pupils of any grade reached or exceeded the median of the next grade above, 20.1 per cent reached or exceeded the median of the second grade above, 13.2 per cent reached or exceeded the median of the third grade above, and 3.3 per cent reached or exceeded the median of the fourth grade above." (p. 42) "We find that in quaUty of writing on the average 37.1 per cent of the pupils of any given grade reach or exceed the median of the next grade above it, 24.0 per cent reach or exceed the median of the second grade above it, 14.6 per cent reach or exceed the median of the third grade above it, and 7.7 per cent reach or exceed the median of the fourth grade above it. Statements of the same sort apply to the speed of writing." (pp. 86, 88) "The spelling tests reveal identically the same facts concerning the distribution of the pupils and the over- lapping of the grades as were found in case of the reading and writing tests The overlapping among the various grades is enormous. There are two pupils in the second grade who can spell as well as two pupils in the eighth grade . . . On the average 20.8 per cent of the pupils of any grade reach or exceed the median of the next grade above it, and 6.2 per cent reach or exceed the median of the second grade above it." (p. 98) 2. The Problems These statements suggest many questions the answers to which should be highly significant for educational theory and school practice. a. Will further investigation under carefully controlled condi- tions support these findings? 6. Have we in the results from single tests in different traits a valid measure of the overlapping of general ability to do the work of a given grade? c. Is our grading system, in which we have had so much faith, really as bad as it appears to be as a means of select- ing on the basis of achievement? Introduction 7 d. There is a tacit feeling that the passing from one grade to the next — with all the tribulation which it entails on the part of pupil and teacher — means a very definite step in accompUshment. Is this feeling well founded? €. What constitutes a valid measure of overlapping? It was in the hope of presenting some facts that would have a direct bearing on these and similar questions that this study was undertaken. Primary Problem. To determine from data, adequate and re- liable, the amount and nature of the overlapping in certain sixth, seventh, and eighth grades. Related Problem. To get a measure of the reliability of certain tests as measures of attainment by school pupils in these grades. II THE DATA It was recognized at the beginning that whatever value might come from such a study as the one contemplated would depend very laigely upon (a) the number of pupils tested, (6) the number and worth of the tests used, (c) the uniformity of procedure in giving the tests, (d) the opportunity given the pupils to show their real capacity in each test, (e) the consistency in the scoring of the papers. 1. The Subjects The pupils tested were all the sixth-, seventh-, and eighth- grade pupils in four pubhc schools in Seattle, Washington. About 860 pupils in aU were tested, this being the number represented in the final composite: 254 in the sixth grade, 324 in the seventh, and 282 in the eighth. The discrepancy between these figures and the totals for certain tests is accounted for by the fact that missing scores for a certain minimum number of tests were inter- polated in making the composites, by a method that will be ex- plained in a later section. The schools were chosen so as to represent different types as to size and sociological conditions of environment. School No. 1 is a small, comparatively new school in the outskirts of the city, drawing some pupils from less well-graded districts. The com- munity is largely composed of people of the artisan classes. School No. 2 is also a small school, but in an old and rather unprogressive section of the city. No. 3 is a large cosmopolitan school with a varied school population. No. 4 is also a large school and is in one of the best residence sections of the city largely composed of business and professional classes. Table I shows the distribution of classes by schools, rooms, and grades. The Data TABLE 1 Distribution of Classes by Schools, Rooms, AND Grades No. School of Number of Classes Rooms Grade VI Grade VII Grade VIII Total No. 1 . . 5 2 3 2 7 No. 2.. 4 2 3 2 7 No. 3.. 7 3 3 2 8 No. 4.. 8 3 3 3 9 All 24 10 12 9 31 2. Tests Used The tests used were: Addition, Woody Scale, Series A Subtraction, W oody Scale, Series A Multiplication, Woody Scale, Series A Division, Woody Scale, Series A Problems I Problems II Problems III Trabue Completion-Test Language Scale B Trabue Completion-Test Language Scale C Trabue Completion-Test Language Scale D Trabue Completion-Test Language Scale E Composition I. A Letter ApplyingJor a Job. Composition II. On the topic, "The Study I Like Best and Why, and the Study I Like Least and Why." Paragraph Reading, Thorndike Reading Scale Alpha 2 SpeUing, last 20 words in column "U" of Ayres' Measuring Scale for Abil- ity in Spelling Opposites, Al Opposites, A2 Opposites, A3 Opposites, A4 Whole-Part, Bl Adjective-Substantive, B2 Verb-Object, CI Species-Genus, C2 Part-Whole, C3 Mixed Relations, Dl Mixed Relations, D2 Directions. X Directions, VI Directions, VII Visual Vocabulary, VIII Visual Vocabulary, XIII 10 The Overlapping of Attainments in Certain Grades Specific references to the sources of those tests which are readily available are given in the appended bibliography, as follows: Woody Arithmetic Scales (27) Trabue Language Scales (25) Thorndike Reading Scale Alpha 2 (20) Ayres' Measuring Scale for Ability in SpeUing (1) Directions, X (24) Directions, VI (24) The other tests are reproduced below.^ Write your name here Write your age here . . years . . months. Write the date here Write after each word on this page its opposite as shown in the first five. If you do not know the opposite of a word go on to the next word. good — bad day — night up — down friend — enemy true— false high summer long soft white far up smooth lost wet high dirty day yes wrong empty top north sour out weak good after above sick slow large rich dark front love . taU open summer new come male to reveal level past common foreign sane country rapid ^The writer is indebted to Professor Thorndike for these tests. The Data 11 A-2. Write your name here Write your age here, .years, .months. Write the date here Write after each word on this page its opposite as shown in the first five. If you do not know the opposite of a word go on to the next word. good — had day — night up — down friend — enemy true— false early extravagant dead obscure hot opaque convex heterodox collect prompt strong to lead courteous tragic diminish erroneous fact jtale genuine broken permit descend expensive divine all from serious grand to win to respect clean few alike deep hiss encourage coarse melt double dwarf plural valley to benefit future A-3. Write your name here Write your age here . . years . . months. Write the date here Write after each word on this page its opposite as shown in the first five. If you do not know the opposite of a word go on to the next word. good — bad day — night up — down friend — enemy true— false 12 The Overlapping of Attainments in Certain Grades frequently to lack apart stormy motion forcible straight to hold loquacious forbid sdways grief sickly negative soothing free superior separate city attractive stupid shallow savage different to marry every mascuhne obnoxious enlarge joy scowl sacred inhale aristocratic help foolish pay wild precise dwindle drunk fluctuating first horizontal A-4. Write your name here Write your age here . . years . . months. Write the date here Write after each word on this page its opposite as shown in the first five. If you do not know the opposite of a word go on to the next word. good — bad day — night up — down friend — enemy true — -false after to float rough to bless to take exciting clumsy unless tender public raise remember increase preserve debit if vertical ignorant rude simple deceitful stingy The Data 13 impoverish cruel generous ancient silly multiply desist survive proficient hindrance strength innocent permanent over to degrade weary to spend part together gradual victorious optimist laugh numerous B-1. B-2. Write your name here Write your name here Write your age here . . years . . months Write your age here . . years . . months Write the date here Write the date here Write words that fit the Write words that fit the words in this column in the words in this column in the way shown in the first three. way shown in the first three. face — nose tree — branch store — counter apple clock knife book hat pencil hand dog oyster church chair bird banana shoe train finger house coat cart face wet — water ted — rose sharp — knife sharp hot dusty raw deep ripe funny tall stormy new hiUy strong muddy pretty noisy white steep round smoky curly 11 The Overlapping of Attainments in Certain Grades C-1. Write your name here Write your age here . . years . . months. Write the date here After each word printed below you are to write some word, according to the further directions. Write plainly. If you cannot think of the right word, go ahead to the next. Write words that fit the words in this column, in the way shown in the first three. drink — water ask — questions subtract — numbers bake light spill sail kiss spin polish lock sweep wash fill pump sharpen learn write open chew eat drive climb read lend tear smoke throw singe paint dig mail sift C-2. Write your name here Write your age here, .years, .months. Write the date here After each word printed below you are to write some word, according to the further directions. Write plainly. If you cannot think of the right word, go ahead io the next. Write words that tell what sort of a thing each thing named is, as shown in the first three. lily — flower blue — color diamond— jewel penny iron dictionary green cabbage 42 Rhine parlor murder ruby dog orange The Data 15 sparrow cat London September foot-ball dime rose elm diphtheria France robin skirt Pacific a rattle cod Christmas baseball muslin C-3. Write your name here Write your age here . . years . . months. Write the date here After each word printed below you are to write some word, according to the further directions. Write plainly. If you cannot think of the right word, go ahead to the next. Write words that fit the words in this column, in the way shown in the first three. sleeve — coat nose— face roof — house elbow sleeve hinge brick page deck finger France wing pint morning fin Wade steeple mattress month chimney hub cent chin D-1. Write your name here Write your age here . . years . . months. Write the date here Write in each line a fourth word that fits the third word in that line in the way ttiat the second word fits the first, as shown in the first three fines. color— rerf name — John page— 600^ handle — knife fire — burns soldiers— fight good— bad long- deep — valley high- eagle— bird shark- growls— dog roars — eat— bread drink- brick— waU page— 16 The Overlapping of Attainments in Certain Grades fruit — orange sit — chair double — two England — London chew — teeth pen — write water — wet laborer — works come — came north — south mend — clothes lily — flower ton — pound elbow — arm pea — pod past — present bell — rings vegetable- sleep — triple — France — smell — knife — fire — soldier — go- far— bake — ■ oak — pound — chin — nut — present — clock — • lathe — machine hammer— pencil — lead book- high — ^low up— sheep — lamb dog- Thursday-Friday June — build — house paint- one — single two — eye — see ear — Monday-Tuesday April- do— did see — bird— sings dog- hour — minute minute — straw— hat leather- cloud— rain sun — hammer— tool dictionary- uncle — aunt brother — dog— puppy cat— D-2. Write your name here Write your age here . . years . . months. Write the date here Write in each line a fourth word that fits the third word in that line in the way that the second word fits the first, as shown in the first three lines. color — red name — John page — book handle — knife fire — burns soldiers— fight he — him she — dusty — dry muddy — boat — water train — floats— raft sails — crawl — snake swim — deck — ship mattress — horse— colt cow — iron — metal Latin- nose — face toe- cat — kitten hen— bad— worse good- dark— fight hot— hungry— food thirsty- month-December hoUday hat— head glove— spin — top sheupen — ship — captain army— second — minute minute — man — woman boy- pencil— lead book- axe — cuts pin— Uttle— less much — yard — foot foot- wash — face sweep— early — late new — house — room book- shoot — gun smoke — sky — blue grass— Atlantic— ocean Mississippi — swim — water fly- has— had is — once — one twice — month— year day — cat— fur bird- my — mine they— pan— tin table— room — ceiUng house- buy— sell come— baby — cries bird— oyster — shell banana— The Data 17 VII DO WHAT IT SAYS TO DO 21. Show by a cross which costs most: an orange a suit of clothes a pair of skates a pound of sugar 22. Show by a cross which tastes best: dirty paper coal dust roast beef sour milk 23. How many legs has a horse? Write the answer. 24. How many more legs has a horse than a boyP Write the answer. 23. Show by a cross which you would like best to own: a toy boat a gold ring a thoussmd dollars a sharp knife 26. Show by a cross which you would enjoy most: being hit with a club having your shoulder pinched receiving a gift of money losing many toys * 27. Show by a cross the word that means pleasant: entrance entreat entangle enjoyable 28. Show by a cross the words that tell something a boy ought to do: perform miracles overeat habitually study his lessons strain his eyes 2\>. Show by a cross which is hardest to do: To run a mile in a second To stay awake all night To pay attention in school To play out-doors 30. Show by a cross the least dangerous thing: To sit in front of a trolley car To sit in the snow all night To play with a loaded gun To sit still in front of a mirror 18 The Overlapping of Attainments in Certain Grades 31. Show by a cross each word that means unhappy. grotesque wretched sincere miserable sad notable sane joyless grateful uncomfortable 32. Show by a cross each word that means to make clear or something like to make clear: execrate elevate elucidate traduce explain clarify satisfy expound extort antipathy VIII Write a letter b under every word that means some part of the body. Write a letter t under every word that is the name of a tool. Write a letter s under every word that means something to do with the sea or ships. Write a letter h under every word that means some part of a house. Remember — b, for parts of the body t, for tools s, for words about the sea and ships h, for parts of a house arms, ear, wave, door, eye, ship, hall, saw, bone, sail, hammer, wall, ocean, face, deck, window, axe, float, mast, chamber, foam, file, canoe, billow, cellar, knife, harbor, elbow, coast, attic, brain, bosom, basement, breast, laimch, artery, chisel, shin, ceiling, brace, tongue, helm, garret, porch, gulf, kitchen, hatchet, fleet, balcony, gimlet, channel, casement, cargo, entry, cruise, embark, chimney, thigh, tiller, awl, alcove, ell, keel, piazza, cleaver, spine, screw-driver, dormer, bevel, cartilage, rudder, corridor, eaves, ratchet. The Data 19 kidney, hurricane, gable, lathe, cerebellum, lagoon, plane, maUet, leeward, hearth, vise, cranium, nauticeJ, wainscot, adze, ligament, cornea, augur, navigable, patella, starboard, lymph, mariner, peritoneum, hull, jetty. xm Look at each word. Think what it means. If it means a color like red or blue, write a letter c under it. If it means something about a number, like six, all, half, many, write n under it. If it means something about time, like now. Me, when, never, write t under it. If it means something about di- rection or location, if it is a word like east, north, up, down, above, behind, write d under it. Remember — c, for words about color, like red, blue, green n, for words sibout number, like more, five, many t, for words about time, like then, often, early d, for words about direction and location, like front, east, here last, west, here, half, early, brown, many, below, there, month, across, year, noon, back, forty, gray, first, right, pair, left, green, morning, whole, pink, while, where, afternoon, minute, gold, edge, soon, outside, some, more, evening, plum, inside, beneath, odd, even, every, yellow, monday, toward, dozen, June, group, future, buff, from, count, nine, forenoon, purple, middle, plural, tuesday, instant, couple, score, crimson, april, each, during, interior, forward, Wednesday, never, december, singular, center, hazel, outward, several, sceu-let, violet, double, frequent, chestnut, august, ruby, either, rarely, recent, over- head, million, tan, exterior, queutet, seldom, tawny, olive, January, margin, numerous, immediate, bounding, february, encircling, September, lilac, numeral, crowd, gradual, roan, quadruple, opposite, minority, november, russet, diagonal, October, faun, formerly, garnet, overhanging, multitudinous, lasting, meantime, duplicate, continue, intervening, perpendicular, henceforth, elevated, lavender, forthwith, sextet incessant, azure, surmounting, majority, 20 The Overlapping of Atlainmenis in Certain Grades previous, interim, thereabouts, plurality, narrow, perennial, frontal, treble, repeatedly, constantly, carmine, neighboring, parallel, contemporary, vertical, mauve, decade, emerald, magenta, octile, ecru, contiguous, qxiintet, ensuing, horizontal, evanescent, sepia, conterminous, multiple, integer, vermillion, eternal, turquoise, transverse, century, sexagesimal, ephemeral. Problems — I Begin with No. 1 and see if you can solve all ten correctly. 1. How much change should I expect from $5.00, after paying for 5 pounds of coffee at 38 cents a pound? 2. A baseball team played 160 games during the season and won 100 of them. What part of the whole number of games did the team win? 3. If $1,991 a day is paid to 724 men who each earn the same wages, how much does each man receive? 4. The children of a school made badges. Five hundred and fifty were needed. In 4 days grades 3 to 7 made 20, 25, 63, 132 and 144 badges. The eighth grade agreed to make the rest. How many did the eighth grade agree to make? 5. A man has a salary of $125 a month. He saves 20 per cent of his salary. How much will he save in a year? 6. A store takes in the following sums: $1,250.50, $300, $175, $16.25, $120.50, $32.75, $68.50. It pays out: $600, $360, $166.67, $44.33, $240. How much remeiins after payments are made? 7. Mr. Marshall receives a salary of $2500 a year. His rent costs him one-fifth of this and his other expenses are $1,500. He saves the rest. What per cent of his salary does he save? 8. A grocer had a tank holding 44% gallons of oil. One day he drew out 1534 gallons and the next day 9}i gallons. How many gallons were left in the tank? 9. How much will Mr. Fox receive for 8% dozen pencils at the rate of 6 for 25c? 10. John is 4 ft. 9 in. How tall will he be in two years if he grows 3J^ per cent of his present height the first year, and 3}^ P^r cent of his height a year from now the second year? Problems — II Find how long Meuy was allowed to play on each of these days. The Data 21 1. Monday. It is 4.10 P. M. Mother says to Mary, "You may play till quarter past five " 2. Tuesday. It is 4.20 P. M. Supper is at 6 o'clock. Mother says, "You may play half the time from now till supper time." 3. Wednesday. It is 4.05 P. M. Mother says, "If you will help me for half an hour now, and for 10 minutes before supper you may play the rest of the afternoon." 4. Thursday. Mother says, "You may play 20 minutes and 2}4 minutes more for every piece you iron." Mary irons 28 pieces. 5. Friday. Mother says, "You may play 2 minutes for every 3 problems you solve, and .5 minutes more for every problem you solve correctly." Mary solved 15 and had all but one right. 6. Saturday. Mother says, "You may play 2 hours for nothing. Also I will allow you 10 minutes for every problem you solve correctly, but wiU take off 10 minutes for every problem that is wrong. Also you may play IJ^ minutes for every minute you help me by minding your little brother." Mary did 10 problems, and had only one of them wrong. She minded the baby for an hour and a quarter. Problems — III See how many of these problems you can do correctly. All the problems ask the same question, "How many minutes is it from the time John begins to pump until the tank is filled?" The tank holds 120 gallons and is sup- posed always to be empty when John begins to work. 1. John pumps 2 minutes before any water reaches the tank. Then he pumps water into it at the rate of 3 gallons a minute until the tank is fuU. 2. John pumps IJ^ min. before any water reaches the tank. Then he pumps water into it at the rate of 3 gallons a minute, for 20 min. Then he pumps at the rate of 2 gallons a minute until it is fuU. 3. John pumps 1 min. before any water reaches the tank. Then he pumps water into it at the rate of 24 gallons in 10 minutes until the tank is full. 4. John pumps 2 min. before any water reaches the tank. Then he pumps for 15 min. at the rate of 3 gallons per minute. Then Dick helps him and they pump at the rate of 5 gallons per minute until the tank is full. 5. John pumps 1^ min. before any water reaches the tank. Then he pumps for 10 min. at the rate of 2.7 gallons per minute. Then the pump breaks and he spends 8 min. mending it. Then he pumps at the rate of 3.1 gallons per minute until the tank is fuU. 22 The Overlapping of Attainments in Certain Grades 6. John pumps for 1 min. 50 sec. before any water reaches the tank. Then he pumps at the rate of 3.6 gallons per minute for 10 min., then rests 5 min., then pumps 3.6 gal. per minute for 10 minutes, then rests 5 min., then pumps 3.6 gal. per min. for 10 minutes, then rests 5 min., and so on until the tank is filled. The selection of the tests used was made with the following aims in mind: a. To make as thorough a testing as time would permit. b. To have the two general fields of Arithmetic and English well represented. c. To use the best available tests in school subjects, suited to the purpose of this study. The Woody Arithmetic Scales furnish a real test for the sixth, seventh, and eighth grades, present a variety of types of opera- tions, and are a test of power rather than a speed test. The Problem Tests were known to be unfamihar to the pupils. The form of II and III is such as to test for a minimum of control of vernacular and a maximum of control of operations and math- ematical reasoning. The Trabue Completion Tests represent the scientifically de- rived results of much careful testing. Dr. Trabue says of them, "It will be found that ability to complete these sentences suc- cessfully is very closely related to what is usually called 'language abiUty'." (25, p. 1) Thorndike Reading Scale Alpha 2 is a scientifically sound scale for the measurement of paragraph reading. The only limitation is that there are not other comparable scales available. Composition. A letter was chosen as one form of composition because it is commonly taught in schools and is the most ordi- nary form of writing done outside of school. The topic, "The Study I Like Best and Why, and the Study I Like Least and Why,"^ was chosen as furnishing a theme on which every child had something to say and as giving scope for the more capable pupils. iThis topic is suggested to teachers as of particular value from the standpoint of throwing light on the "content of children's minds" with respect to their studies. There is, no doubt, an advantage in having the test given by an outsider as the pupils tested gave evidence of more frankness than might be expected from them if writing for the teacher to read. The Data 23 The Ayres Spelling Scale furnishes a comparatively large list from which to choose, and hence one less hkely to have been drilled upon. Column "U" was chosen as representing a degree of difficulty not altogether beyond reasonable attainment by the sixth grade and yet such as to test the eighth, d. To supplement these with a varied selection of psychological tests. Being tests involving knowledge of verbal relations, they serve to extend the scope of the English group. Opposites tests have been often shown to be very satisfactory tests as a measuie of the control of the vernacular. Visual Vocabulary tests supplement testing of paragraph read- ing. They are found to rank high as measures of "language power." Directions tests furnish a very real test of power to get meaning from a printed page. The other controlled association tests used had no particular claim except as supplementary to the others. 3. Administration of the Tests Uniformity of Procedure. The aim in this study being a comparison of achievements in the different grades tested, it was of primary importance that the procedure in giving the tests be uniform throughout. Consid- ering the number of tests used, the number of schools, rooms, and classes concerned, involving numerous repetitions, the neces- sity for rigid control of conditions during the testing is obvious. To this end the following procedure was maintained throughout. a. Every test exercise was started by the writer. b. The writer remained in the room throughout the first test. There- after, except in the case of the short tests, he went to another room leaving an assistant to receive the papers and record the time on each. c. Either the writer or an assistant was in the room throughout each test. d. The teacher was regularly present, but ordinarily engaged in other work at the rear or side of the room. The very best of cooperation on the part of the teachers made possible very rigid control of the con- ditions in the room so far as any possible influence on the work of the pupils by the teachers was concerned. It was thoroughly understood from the beginning that all directions were to come from the testers. On the other hand the presence of the teacher in the room was a real. 24 The Overlapping of Attainments in Certain Grades though unadvertised, help in insuring independent work on the part of each pupil. It is a pleasure, incidentally, to record the fact that the pupils were remarkably free from any tendency to depend on others for aid. e. At the very start of the testing in each room the mechanics of pro- cedure in the way of provision of pencils in good condition, clearing of desks, passing of papers, putting on of headings, signal for begin- ning and procedure on finishing the test, were fixed by rigid adherence to a pre-arranged plan, with modifications only to suit the needs of different tests. /. Preliminary to the beginning of each test specific directions to be given the pupils were worked out. These were adhered to carefully through- out. Any slight changes in general procedure were made at the be- ginning of a test, the aim being to maintain the very same procedure throughout all the rooms for a given test. g. The same order of tests was followed in all rooms with only a few slight changes made necessary by the time schedule. h. Special care was taken that those tests which by the nature of their content might easily become matter of common knowledge among the pupils of a school, such as spelling for example, were given during the same day's session. I. At the cost of considerable time no test was begun at such a time as would not insure ample opportunity for all to finish before an inter- mission as for recess or the noon hour, except in a very few cases where the custom of the room was well established to finish any given task even after other classes had been dismissed. Special Features. Spelling. Twenty sentences were framed, each containing one of the twenty words of the list, care being taken to use such a construction that the word could not be mistaken. These sen- tences were used throughout the testing. Previous to the test period papers with the regular heading and numbered from 1 to 20 at the left-hand margin, had been prepared by the pupils under the direction of the teacher. The following directions were given : "I shall pronounce a word; then use it in a sentence; then pro- nounce it once more. Don't write anything until I have done these three things. "When you are all through with one word I shall say, 'Next.' If anyone is not ready, raise your hand. When all are ready I shall give the number of the next word. Make sure that when I say 'two' you are writing the second word. If you can't write some word leave its place blank. "Write plainly. What I can't read is called wrong, and there is some writing I can't read." The Data 25 Reading. The Thomdike Reading Scale Alpha 2 is printed on two large sheets. The exercise headed Set IV, the last one on the first sheet, is repeated as the first exercise on the second sheet. In order to make sure that the pupils would not spend time on this second copy of Set IV, that part of the second sheet was cut off. Headings. Contrary to the usual practice the headings were put on the back side of the test papers. This insures that on the signal to begin all start work on the test material, and eliminates the variation in the length of time required to write the heading, as a source of error. Distribution of Papers. All test papers were distributed, al- ways with the printed side down and the top end away from the pupil, by the testers. This makes for a saving of time and regu- larity of procedure. Timing. As the time element is not made a feature of this study only a brief statement will be made regarding the timing of the tests. The following directions were given : "You will all begin at the same time. When you have done everything as well as you can, bring your paper to the desk." The time for each paper was recorded in minutes and seconds at the moment it was laid on the timer's desk — the teacher's desk in the front of the room. In case of the very short tests two timers were used in order to avoid delay in recording. In a group test without a time limit there is an unavoidable source of error in getting each individual's time recorded. It was decided this could be reduced to the minimum by taking the time at which the paper was put on the timer's desk. This method works a slight injustice to those in the rear of the room, but results in no constant error. The Tests as a Real Measure of Capacity. The statement was made at the beginning of this chapter that much of whatever value this study might have would hinge upon the extent to which each pupil was given an opportunity to show his real capacity in each test. To this end no test was given as a speed test; each pupil was given as much time as he wished. The assumption here is that in this way we get a much more ac- curate measure of what each pupil actually is capable of doing in his work from day to day. Our belief is that very Httle of the 26 The Overlapping of Attainments in Certain Grades work on which his grade standing is based counts speed as an important factor. Ordinarily there is ample time for the tasks re- quired. We do not raise the question as to whether this is as it should be. Our purpose is to compare the attainments of differ- ent school grades as they exist. However, while it was the intention to allow the pupils all the time they needed, some stimulus against the waste of time was necessary in the interest of economy. This was provided by the fact that the pupils knew their time was recorded. There cer- tainly was some tendency to undue haste especially in the short- er tests. Except for a few of these short tests (the B-C group), however, the writer is confident this was a negligible factor. Early in the testing the practice of turning to other work as soon as a test was finished was well established among the pupils. As a result little was known as to who was not through with the test and hence no particular social disapproval was attached to slow- ness. The further fact that all the pupils had been subjected to testing under controlled conditions made the experience not un- like regular school work. In the case of the longer tests, as for example Alpha 2, in order not to interfere too much with the regular schedule, pupils who had not finished within a reasonable time were taken to an un- occupied room and there permitted to finish. Another factor making for an adequate measure of the attain- ments of the pupils was the large number of occasions on which the testing was done. The number of half-days during which one or more tests were given was 10, 11, 15, and 17 respectively for the different schools. This fact tends to offset the error which comes from using a single day's results as the measure of a pupil's standing. None of us care to be judged on the work of a single day, and with good reason. The following statement by Pro- fessor HoUingworth (12) brings out the significance of this point: "The momentary ability revealed in initial trials, or even in the first half dozen trials in a given set of tests might well be expected to show only low degrees of correlation. These would not be measures of ultimate capacity, but would be highly determined by previous practice, chance variabiHty, momentary attitude and initial method of attack. They would, in short, be samplings only of momentary ability, not of final capacity." The Data 27 4. Scoring the Tests Consistency in the scoring of test papers is perhaps only second in importance to uniformity of procedure in giving the tests. This calls for particular care when the number of test papers in a given test is so large as to make impossible their scoring at one sitting, or indeed at a half dozen, but requires many sessions even though long. Furthermore, the large number of papers makes practically impossible the scoring by one person. Hence the need of careful supervision and accurate recording of method from the beginning to be followed throughout a given test. Of course the difficulty in scoring consistently is far less in the case of tests of the right or wrong sort, than in the case of those calling for partial credit and involving judgment; but even in the former the pos- sibility of error in a large number of tests is considerable. All the test papers in this study were scored by the writer or under his direct supervision. By far the larger part of the scoring was done by three readers, three others having done a small part. The purpose being to get results comparable with each other rather than with previously obtained data, no particular effort was made to use methods absolutely identical with those used elsewhere. This was at times sacrificed for the much more im- portant consideration of consistency throughout this study. However, for the benefit of any who may wish to compare other results with those here presented the following statement is made of the method of scoring. Woody Arithmetic Scales Score: Number of examples correct Problems Score: Number of points on basis of 3 for full credit " [ for partial credit for no credit The answer recorded in the place assigned was the criterion used. Trabue Score: Number of points on basis of 2 for full credit 1 for partied credit for no credit 28 The Overlapping of Atlainments in Certain Grades Composition a. The composition on the topic, "The Study I Like Best and Why and the Study I Like Least and Why" was scored with the use of the Thorndike Extension of the Hillegas Scale. (23) b. The letters were scored by means of an improvised 'scale.' Briefly the method of preparing this 'scale' was as follows. Forty papers were selected from the lot, representing in the judgment of the writer the whole range of ability from the poorest to the very best. It was not assumed that this number of different quahties could be detected. This large number was selected with the purpose of getting all grades of quahty. Five competent judges then ranked the papers according to the directions below: "Please rank the papers in eight groups of five each calUng Group I the poorest and Group VIII the best. "Enter the number of each paper (appearing in the upper right hand corner) on the appropriate line below. "Note that you are not asked to rank the papers within each group. "Assuming arbitrarily the value 10 for Group I and the value 80 for Group VIII, assign intervening values to the other groups. Enter these values at the left of the number of the group." From the results of these rankings a selection of eight papers was made, those being selected on which there was greatest agreement among the judges as to the group in which each belonged. Of two or three papers equally placed, that one was chosen which seemed to best Ct into the whole. The amount of agreement among the judges is indi- cated below: Number placed in same group by Number of judges 2 5 10 4 10 3 On the combined judgments of the five readers the values from 10 to 80 by equal intervals of 10 were given to the compositions in the eight groups from the poorest to the best. This is a rough evalua- tion and lays no claim to scientific accuracy. But the results justify the use of the 'scale' for the purpose of this study. The aim was to overcome the difficulty which all readers have who try to use the Hillegas Scale to judge compositions in the form of letters. The ar- bitrary method of assigning values to the letters composing the 'scale' was with the purpose of getting values roughly comparable with those on the Hillegas Scale. In all the work of scoring the compositions the reader was ignorant of the grade in which a given composition was written, this being very essential if a comparison of results by grades is to be made. The scoring was done by three readers, each paper being scored by each reader independently of the others. This gave three scores for each paper or six scores for each pupil. The final score for each pupil was obtained by taking the median of these six measures. ^ The Data 29 Thorndike Reading Scale Alpra 2 Score: Number of correct responses. The key published by the author (20) was used, with additions as necessary. Spelling Score: Number of words spelled correctly Opposites Score: 3 for full credit Mor partial credit for no credit All the available keys were used. However, numerous answers had never been passed upon in these keys and it became necessary to ex- tend them greatly. This was done by using the combined judgments of from two to eleven judges. Because of the very great number of answers occurring only a very few times it was necessary in these cases to rely on the judgments of only two persons, the writer and one other in every case. Bl, B2, CI, C2, C3 Score: 1 for full credit H for partial credit for no credit Dl, D2 Score: Number of correct responses X Score: A, B, D, E, F were allowed 2 each when correct. C and G were allowed 1 for each line correct. VI Score: Number of correct responses VII Score: One point was allowed for each step correct except 31 and 32. These two steps were scored as a unit. The number right minus the number wrong, counting omissions, being the score. VIII Score: Number of correct responses XIII Score: Number of correct responses Ill OVERLAPPING BY SINGLE TESTS 1. The Measure of Overlapping The measure of overlapping in this study is the per cent of the pupils in any grade who equal or exceed the median score of the next grade or the second grade above; or the per cent of the pupils in any grade who go below the median score of the next grade or the second grade below. The former will be referred to as over- lapping "upward" and the latter as overlapping "downward." The value of this measure of overlapping is shown in the follow- ing statements by Professor Thorndike, and by his accompany- ing diagrams. (21) "The great advantage gained by comparing groups by the per cent of one group reaching or exceeding the point on the scale that is reached or exceeded by a given per cent of the other group is that results are mutually comparable what- ever the traits may be . . . Another advantage lies in the fact that this percentile comparison reminds one constantly of the overlapping of the two groups, when such exists." The distributions on which the per cents of overlapping are calculated wiU be found in the Appendix. Table II shows the amount of overlapping in 21 tests, of the sixth grade on the eighth, the sixth on the seventh, and the seventh on the eighth, upward; and downward, the eighth on the sixth, the eighth on the seventh, and the seventh on the sixth. 30 Overlapping by Single Tests 31 TABLE II Overlapping of Grades by Single Tests Per cent in each grade who equal or exceed the median of the other grade; upward. Per cent in each grade who go below the median of the other grade; downward Upward Downward VI VI VII VIII VIII VII Average on on on on on on of the six Tests VIII VII VIII VI VII VI measures Addition 24.19 35.92 38.71 25.90 36.15 37.39 33.04 Subtraction 22.30 25.66 44.22 19.90 42.09 25.77 29.99 Multiplication 23.42 33.23 40.90 26.53 39.18 34.92 33.03 Division 28.27 35.36 43.57 26.76 38.73 37.43 35.02 Problemsl 7.14 13.31 32.77 9.63 33.49 18.45 19.13 Problems II 10.64 16.95 36.06 14.04 37.19 25.29 23.36 TrabueB 26.61 34.54 44.29 30.01 42.86 38.12 36.07 TrabueC 21.68 38.49 34.65 25.28 33.32 40.05 32.25 TrabueD 23.31 38.58 36.95 30.92 38.34 41.32 34.90 TrabueE 13.99 36.86 25.91 19.74 26.97 37.16 26.77 Alpha 2 11.61 28.02 31.99 12.73 33.00 26.49 23.97 Spelling 7.37 21.09 30.26 7.39 26.34 22.14 19.10 OppositesA2 8.76 19.37 36.57 14.08 35.44 23 65 22.98 OppositesA3 13.33 31.80 28.44 14.16 26.41 32.82 24.49 OppositesA4 10.05 22.18 3r.77 11.15 34.11 27.50 22.79 Part-Whole C3 35.18 45.59 35.42 34.85 37.82 45.77 39.11 Mixed Relations Dl... 42. 53 53.50 43.95 43.87 41.19 52.65 46.28 Mixed Relations D2... 38. 75 48.67 39.50 34.60 35.84 49.07 41.07 Directions VII 21.23 37.38 35.10 27.88 38.80 38.82 33.20 Visual Vocabulary VIII 10.64 24.22 24.06 9.20 28.34 24.69 20.19 VisualVocabularyXIII 11.06 26.53 29.94 11.41 28.29 23.40 21.77 Average 19.62 31.77 35.48 21.43 34.95 33.47 29.45 7.14 13.31 24.06 7.39 26.34 18.45 19.10 Range to to to to to to to 42.53 53.50 44.29 43.87 42.86 52.65 46.28 The table reads- In addition 24. 19 per cent of the pupils in the sixth grade equal or exceed the median of the eighth grade; 38.71 per cent of the pupils in the seventh grade equal or exceed the median of the eighth grade; 25.90 per cent of the pupils in the eighth grade go below the median of the sixth grade; and 36.15 per cent of the pupils in the eighth grade go below the median of the seventh grade. 32 The Overlapping of Attainments in Certain Grades 2. The Elimination of Some Tests The fact that there are figures for only 21 tests calls for explan- ation. It is the purpose to present the results for only those tests which proved to be fair measures for the three grades used, on the basis of the distributions of the scores for each grade. Some tests proved too easy with consequent piling up of undistributed scores at the upper end. The higher ranges of abihty were not measured. The way up was closed, so to speak. This naturally works to the greatest disadvantage of the eighth grade, to less for the seventh, and least for the sixth. The result is an unduly high per cent of overlapping upward. The closeness with which the grade med- ians approximate the total possible score is a rough measure of this exaggeration of the overlapping figures. For example, the median score of the sixth grade may be very near the maximum possible score, beyond which obviously none in the eighth grade can go. Table III has been prepared to show which tests are affected in this way. This table gives the median score for each grade in each test and the maximum possible score. With the purpose, then, of ruling out those tests which by their very nature, as shown in the distribution tables, make a high per cent of over- lapping inevitable, the tests Al, Bl, B2, CI, C2, X, VI are omitted from Table II. Problems III proved much too difficult with un- distributed scores at the lower end, and hence is not included.^ The two composition tests, the scores for which were not dis- tributed singly, do not appear in the table. Composition is treated only as a composite. It should be borne in mind in connection with the treatment in Chapter IV that the inclusion of the first eight tests mentioned above would have raised the average per cents of overlapping to higher figures. Hence any showing based upon the magnitude of the per cents of overlapping by single tests would have been magnified had these tests been included. There is no absolute line of demarcation between the tests that should and the tests that should not be used. The error, for the vafidity of the argument which follows in Chapter IV, could not have been in the direction of eliminating too many tests. ^A table of per cents of overlapping for these tests is given in the Appendix. Overlapping by Single Tests 33 Tests VI Addition 29.30 Subtraction 25 . 74 Multiplication 27.84 Division 24 . 54 Problems 1 9.11 Problems II 2.28 Trabue B 13.37 Trabue C 12.94 Trabue D 14.20 Trabue E 13 . 30 Composition 37 . 66 Alpha 2 22.61 Spelling 12.73 A 1 122.69 A 2 53.71 A 3 49.52 A 4 48.45 B 1 16.50 B2 * C 1 29.28 C 2 26.27 C 3 15.34 D 1 16.12 D2 17.19 X 14.69 VI 19.43 VII 10.17 VIII 70.12 XIII 75.38 TABLE III 'est for Each G RADE, WITH Maximum ssiBLE Scores Grades Maximum VII VIII possible score 30.89 32.35 38 28.37 28.87 35 29.65 30.69 39 26 06 26.85 36 15.36 18.35 30 5.06 6.54 18 14 13 14 57 20 13.57 14.65 20 14 61 15.32 20 13.93 15.50 20 42.89 49.27 87.5 25.37 27.08 38 17.15 18.97 20 126.19 128.77 138 66.11 73.44 138 56.50 68.07 138 65.08 73.37 138 18.29 17.57 20 * * 20 29.62 * 30 27.56 28.31 30 15.71 16.69 20 15.76 16.85 40 17.45 22.25 40 16.10 16 11 18 19.55 19.55 20 10.59 11.33 18 78.10 84.67 100 100.63 119.35 169 3. Comparison of Different Tests as to Amount of Overlapping We have been prepared to find large amounts of overlapping and we find our results no exception. Less to be expected, per- haps, is the great range among the tests. It is obvious that were one making deductions from overlapping figures from single tests *See distribution tables in Appendix. 34 The Overlapping of Attainments in Certain Grades much would depend upon the particular tests chosen. For ex- ample, any conclusions regarding overlapping in aiithmetic drawn from a per cent of 28.27 in division (VI on VIII) would be quite otherwise than if based on a per cent of 7.14 in Problems I. The possibility of error from inferences from such results as those presented in Table II is further illustrated by assuming that 13.99 per cent (Trabue E) measures the overlapping of VI on VIII in Completion-Test Language Scales rather than 26.61 per cent (Trabue B). Much would thus depend upon whether one used one or the other of these tests, though they are of demonstrated equal difficulty. "Language Scales D and E are practically of the same difficulty as Scales B and C." (25, p. 22) In the discussion of the relative amounts of overlapping in the different tests which follows let it be borne in mind that we make no claim that these figures derived from single tests are valid measures of overlapping in the traits measured. The great range in the per cents for tests counted of equal difficulty has raised a feeUng of doubt, to say the least. Later we shall consider this question specifically. But taking these figures of Table II for just what they are, per cents of overlapping in the different tests as indicated, we shall see what an analysis of them reveals. Tables IV, V, and VI have been prepared as an aid in answer- ing the question, How do the tests rank in amount of overlapping as determined by the six sets of comparisons, three upward and three downward? Table IV lists the tests in order of rank from the least to the most overlapping, on each basis. Overlapping by Single Tests 35 TABLE IV Lists of the Tests in Order from the Least to the Most Overlapping for Each of the Six Comparisons VI VI VII VIII VIII VII Rank on on on on on on VIII VII VIII VI VII VI 1 Problems I Problems I VIII Spelling Spelling Problems I 2 Spelling Problems I] [ Trabue E VIII A3 Spelling 3 A2 A2 A3 Problems I Trabue E XIII 4 A4 Spelling XIII A4 XIII A2 5 A4 Spelling XIII VIII VIII 5.5 Problems II VIII 6 VIII A4 Alpha 2 Alpha 2 Problems II 7 XIII Sub. Alpha 2 Problems II Trabue C Sub. 8 Alpha 2 XIII Problems I A2 Problems I Alpha 2 9 A3 Alpha 2 Trabue C A3 A4 A4 10 Trabue E A3 VII Trabue E A2 A3 11 VII Mult. C3 Sub. D2 Mult. 12 Trabue C Trabue B Problems II Trabue C Add. Trabue E 13 Sub. Div. A2 Add. Problems II Add. 14 Trabue D Add. Trabue D Mult. C3 Div. 15 Mult. Trabue E Add. Div. Trabue D Trabue B 16 Add. VII D2 VII Div. VII 17 Trabue B Trabue C Mult. Trabue B VII Trabue C 18 Div. Trabue D Div. Trabue D Mult. Trabue D 19 C3 C3 Dl D2 Dl C3 20 D2 D2 Sub. C3 Sub. D2 21 Dl Dl Trabue B Dl Trabue B Dl 36 The Overlapping of Attainments in Certain Grades Table V summarizes the facts of Table II on the basis of divi- sion into tertiles. It shows that Spelling and VIII fall in the lowest third in all six comparisons, no test fcJls in the middle and only one test, Dl, falls in the highest third in the six comparisons. One test, XIII, falls in the lowest third in five comparisons, and three tests, A4, Problems II, and Problems I, fall in the lowest third in four comparisons. At the other extreme we have one test, Dl, in the highest third in six comparisons, and so on. It appears therefore that there is most agreement among the differ- ent comparisons in placing Dl at the top of the list in amount of overlapping, and Spelling and YIII at the bottom of the list, that is, with least overlapping. TABLE V Showing What Tests are in the liOWEST, Middle and Highest Thirds AS TO Amount of Overlapping in Four or More of the Six Comparisons, VI on VIII, VI on VII, etc. In six In five In four comparisons comparisons comparisons In lowest third Spelling VIII In middle third None In highest third Dl XIII None Trabue B D2 A4 Problems II Problems I A3 Addition Division C3 VII Trabue D Table VI furnishes another basis for measuring the relative amount of overlapping for the different tests. A rank is given each test in each of the six comparisons. These ranks are added and another ranking made in column 8. A ranking in column 9 is made on the basis of the averages from the six measures. (See Table II, last column.) Finally these two rankings are added and the rankings of column 11 obtained, which may be taken as the order of the tests in amount of overlapping on this basis. From the combined results then, the following tests show the least amount of overlapping, roughly in the order named: Spelling, Problems I, VIII, XIII, A4, A2, Alpha 2. Overlapping by Single Tests 37 TABLE VI Rankings of the T 'est.s 1 [N Amount OF Overlapping 1 2 3 4 5 6 7 Sums 8 Rank- 9 Rank- ings 10 Sums 11 Final Rank- 1 Comp larisons of ings of of ings Rank- of Aver- Rank- of VI VI VII VIII VIII VII ings Totals ages ings Totals Tests on on on on on on of in of of in VIII VII VIII VI VII VI Columns Column Table Columns Column lto6 7 II 8 and 9 10 Problems I . . 1 1 8 3 8 1 22 2 2 4 2 Spelling.... . 2 4 5 1 1 2 15 1 1 2 1 VIII . 5.5 6 1 2 5 5 24.5 3 3 6 3 XIII . 7 8 4 5 4 3 31 4 4 8 4 A4 . 4 5 6 4 9 9 37 5 5 10 5 A2 . 3 3 13 8 10 4 41 6 6 12 6 Alpha 2.... . 8 9 7 5 6 8 43 7.5 8 15.5 7 Problems II . 5.5 2 12 7 13 6 45.5 9 7 16 8 A3 . 9 10 3 9 2 10 43 7.5 9 16.5 9 Trabue E. . .10 15 2 10 3 12 52 10 10 20 10 Trabue C. . .12 17 9 12 7 17 74 11 12 23 11.5 Sub .13 7 20 11 20 7 78 12 11 23 11.5 Mult .15 11 17 14 18 11 86 14.5 13 27.5 14 VII .11 16 10 16 17 16 86 14.5 15 29.5 15 Trabue D.. .14 18 14 18 15 18 97 17 16 33 16.5 Add .16 14 15 13 12 13 -83 13 14 27 13 Div .18 13 18 15 16 14 94 16 17 33 16.5 Trabue B . . .17 12 21 17 21 15 103 19 18 37 18.5 C3 .19 19 11 20 14 19 102 18 19 37 18.5 D2 .20 20 16 19 11 20 106 20 20 40 20 Dl .21 21 19 21 19 21 122 21 21 42 21 4. Surfaces of Overlapping Figs. 1 to 4 show in graphic form the overlapping of the three grades in tests Trabue B, Opposites-A2, Part-Whole^C3, and Division. These surfaces are all drawn of equal area and on the same base Une, so that they are directly comparable one with another. The vertical distance represents the per cent of each grade attaining the scores indicated on the horizontal. The surfaces are for the sixth, seventh, and eighth grades from the bottom up in every case. The medians are drawn for each grade through the surfaces of the other grades so as to faciUtate comparison. The measure of overlapping is represented by the portion of the surface of a given grade which extends beyond the median of the 38 The Overlapping of Attainments in Certain Grades grade with which comparison is made. For example, in Fig. 1 the area ABC, 34.54 per cent of the whole, represents the part of the sixth grade that reaches or exceeds the median of the seventh; and the area DBE, 26.61 per cent of the whole, represents the part of the sixth grade that reaches or exceeds the median of the eighth. Likewise, the area AiBiCi, 30.01 per cent, represents the part of the eighth grade that goes below the median of the sixth; and the area DiBiEi, 42.86 per cent, represents the part of the eighth grade that goes below the median of the seventh. These figures give us also a representation of the form of dis- tribution for these tests. Figs. 1, 2, and 4, for example, approxi- mate rather closely to the "normal" curve, in contrast with Fig. 3, which shows considerable skewness toward the upper end. It was this condition still more emphasized that made advisable the elim- ination of some of the tests. Grade Vlll g 12 A D 16 B 30 Fig. 1. Overlapping in Trabue B. Overlapping by Single Tests 39 Grade Viii 26 50 75 100 Fig. 2. Overlapping in A-2, Opposites. 40 The Overlapping of Attainments in Certain Grades Grode VIII 4 8 12 16 SO Fig. 3. Overlapping in C-3, Part-Whole. Overlapping by Single Tests 41 GraJe Vill Grade vii Grade vi n_r A]-! fl., n 10 16 22 28 34 Fig. 4. Overlapping in Division. The results presented in this chapter are in accord with those reported in the Introduction as showing great variability within the grades and consequent overlapping among the grades, when attainment is measured by single tests. Certainly measurements of this sort show large amounts of overlapping even when given under well-controlled conditions and to large numbers of pupils. We are now ready to consider the question raised at the beginning: Are measures obtained in this way valid measures of the over- lapping of ability of school pupils in different grades? The next chapter wiD consider this problem. IV OVERLAPPING IN THE CASE OF COMPOSITES 1. Results Suppose a single test be given to sixth- and eighth-grade pupils with a resulting overlapping of the sixth on the eighth of 25 per cent. Are we justified in using this figure as a measure of the overlapping of abilities in the trait measured? Only in so far, of course, as this single test is a measure of the trait. Suppose that in order to get a more valid measure of overlapping we give three more tests of the same sort and use the average of the four as the measure. This will tend to even up any extreme results from a particular test. Rut this average will retain any error resulting from the inadequacy of a single test as a measure of the trait in question. It does not take account of the fact that the pupil who is in the lowest quartile in one test may be in the high- est quartile in the second test, and still differently placed in the third and fourth tests. In proportion as the single test is an ade- quate measure of a trait, a pupil's place in the distribution wiU be the same in the different test exercises (except in so far as par- ticular conditions affecting the pupil at a given time influence the result). In order to give the individual pupil the benefit of more nearly adequate testing, a composite may be made of the differ- ent scores he makes in the different tests for a given trait and this used as his measure. Thus any error in his placing in the dis- tribution resulting from only a single test tends to be offset by the other measures. The principle of procedure here is simply that up to an undefined Hmit the larger the number of measures, the greater the accuracy of placing the individual where he be- longs in the distribution. Theoretically, then, the amount of overlapping as measured in single tests should be reduced when taken in terms of the composite. 42 Overlapping in the Case of Composites 43 TABLE VII Overlapping of Grades by Composites Per cent in each grade who equal or exceed the median of the other grade; upward. Per cent in each grade who go below the median of the other grade; downward Upward Downward No. VI VI VII VIII VIII VII Average of on on on on on on of the six Composites Tests VIII VII VIII VI VII VI measures Composition 2 13.64 27.86 24.09 8.02 21.26 24.95 19.97 Trabue 4 14.68 32.46 31.19 21.89 31.75 36.63 28.10 Opposites 3 8.57 24.87 30.49 13.41 29.58 26.97 22.32 B-C ■ 5 32.59 32.62 44.21 36.07 49.99 41.40 39.48 Mixed Relations 2 39.51 51.40 41.32 40.25 39.17 .50.69 43.72 Directions 3 27.38 31.67 45.30 29.64 45.42 31.96 35.23 Visual Vocabulary... 2 8.32 22.48 28.14 11.35 27.31 26.81 20.74 Arithmetic 6 7.08 18.13 33.21 12.00 32.68 24.28 21.23 English 22 6.97 24.86 25.83 9.68 23.18 26.12 19.44 Total 28 4.58 19.57 27.74 8.37 24.15 21.98 17.73 Table VII presents the facts of overlapping when the scores are thus combined for the diiferent groups of tests and redistributed. Table VIII brings together the data of Tables II and VII in such a vs^ay as to permit a comparisoi* of the results obtained by aver- aging the per cents of overlapping for the different tests of a group with those obtained from this method of composites. It will be seen, for example, that in the Vl-on-VIII comparison the average per cent of overlapping for the four Trabue tests, 21.40, is re- duced to 14.68 for the composite, a reduction of 6.72 points. Column "d" shows the difference between the two measures of overlapping, the " — " prefix indicating a reduction for the com- posite method. The totals for the "d" columns show that the few cases of the reverse order, an increase for the composite meth- od, represent a very, very small portion of the total difference. Further analysis of the table reveals some interesting facts. The greatest difference in the two methods is found in the Vl-on-VIII comparison, and the least in the Vll-on-VIII comparison, the two in which the per cent of overlapping was, on the other hand, the least and the greatest, respectively. (See Table II.) The difference is small in the case of those composites representing a small number of tests such as Mixed Relations and Visual Vocabulary, of only two tests each. The arithmetic composite of the results from six tests is consistently lower than the average, 44 The Overlapping of Attainments in Certain Grades o ^ o i2 Overlapping in the Case of Composites 45 2 § ■■3 -9 OS VO (N ,-^ ir> t- CO X 7 cj a c ! o a '0