AN ANTHROPOMETRICAL STUDY OF COLLEGE 
 
 FRESHMEN 
 
 
 BY 
 
 MILDRED LEE ECKI 
 B. A., Northwestern College, 1920 
 
 THESIS 
 
 Submitted in Partial Fulfillment of the Requirements for the 
 
 Degree of 
 
 MASTER OF ARTS 
 IN MATHEMATICS 
 
 !N 
 
 THE GRADUATE SCHOOL 
 
 OF THE 
 
 UNIVERSITY OF ILLINOIS 
 1921 
 
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 UNIVERSITY OF ILLINOIS 
 
 THE GRADUATE SCHOOL 
 
 c 
 
 I I92X 
 
 I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY 
 SUPERVISION BY- 
 
 
 BE ACCEPTED AS FULFILLING THIS PART OF THE REQUIREMENTS FOR 
 
 THE DEGREE OF 
 
 
 V\ 
 
 J 
 
 In Charge of Thesis 
 
 Head of Department 
 
 Recommendation concurred in* 
 
 Committee 
 
 on 
 
 Final Examination* 
 
 ^Required for doctor’s degree but not for master’s 
 
 
Digitized by the Internet Archive 
 in 2015 
 
 https://archive.org/detaiis/anthropometricalOOecki 
 
TABLE OF COi^iTENTS 
 
 I Introduction 1 
 
 II Description of Data 1 
 
 Table I- University of Illinois Health Service '6 
 
 III Tne Frequency Distributions 4 
 
 IV Calculation of Averages 4 
 
 Table Ila- Frequency Distribution of Heights 5 
 
 Table lib- Frequency Distribution of Heights 6 
 
 Table Ilia- Frequency Distributions of Weignts 7 
 
 Table Illb- Frequency Distributions of Weignts 8 
 
 Table IV- Summation .Method for Heignts 11 
 
 Table V- Summation Method for Weights . . 13 
 
 V Calculation of the Standard Deviation 14 
 
 VI Explanation of k 15 
 
 VII Classification of Distributions According to Types .... 17 
 
 VIII Calculation of Curve of Type IV 19 
 
 Table VI- Calculation of the Normal Curve. . . 20 
 
 IX Calculation of the Curve of Type VI 32 
 
 Table VII- Calculation of Yq fof Type IV 23 
 
 Table VIll- Calculation of the Curve for Type IV .... 24 
 
 X Calculation of the Curve of Type V 35 
 
 Table IX- Calculation of Constants for Type VI 36 
 
 Table X- Calculation of the Curve of Type VI ..... . -27 
 
 XI Meaning of Correlation 38 
 
 Table XI- Calculation of Constants for Type V 39 
 
 Table XII- Calculation of the Curve for Type V 30 
 
 XII Calculation of r 31 
 
 Table XIII- Calculation of Correlation Coefficient ... 34 
 XIII Summary 35 
 
9 
 
 LIST OF FIGURES 
 
 Figure 
 
 1. 
 
 Frequency 
 
 Distribution 
 
 of 
 
 Heights 
 
 of 
 
 1182 Freshmen 
 
 Figure 
 
 2. 
 
 Frequency 
 
 Distribution 
 
 of 
 
 Heights 
 
 of 
 
 638 Freshmen 
 
 Figure 
 
 3. 
 
 Frequency 
 
 Distribution 
 
 of 
 
 He ight s 
 
 of 
 
 1004 Freshmen 
 
 Figure 
 
 4. 
 
 Frequency 
 
 Distribution 
 
 of 
 
 Heights 
 
 of 
 
 991 Freshmen 
 
 Figure 
 
 5. 
 
 Frequency 
 
 Distribution 
 
 of 
 
 Heights 
 
 of 
 
 877 Freshmien 
 
 Figure 
 
 6. 
 
 Frequency 
 
 Dist r ibut ion 
 
 of 
 
 Heights 
 
 of 
 
 971 Freshman Women 
 
 Figure 
 
 7. 
 
 Frequency 
 
 Distr ibut ion 
 
 of 
 
 Weights 
 
 of 
 
 1182 Freshmen 
 
 Figure 
 
 8. 
 
 Frequency 
 
 Distribution 
 
 of 
 
 Weights 
 
 of 
 
 638 Freshmen 
 
 Figure 
 
 9. 
 
 Fr equency 
 
 Distr ibut ion 
 
 of 
 
 Weights 
 
 of 
 
 1004 Freshmen 
 
 Figure 
 
 10. 
 
 Frequency 
 
 Distr ibut ion 
 
 of 
 
 Weights 
 
 of 
 
 991 Freshmen 
 
 Figure 
 
 11. 
 
 Frequency 
 
 Distribution 
 
 of 
 
 Weights 
 
 of 
 
 877 Freshmen 
 
 Fig’ur e 
 
 12. 
 
 Frequency 
 
 Distribut ion 
 
 of 
 
 Weights 
 
 of 
 
 971 Freshman Women 
 
 Figure 13. 
 Figijre 14. 
 Figure 15. 
 Figure 16. 
 Figure 17. 
 
 Correlation Between Heignt and Weight for 638 Freshmen 
 
 Correlation Between Height and Weight for 1004 Freshmen 
 
 Correlation Between Height and Weight for 9S1 Freshmen 
 
 Correlation Between Height and Weight for S77 Freshmen 
 
 Correlation Between Height and Weight for 971 Freshman 
 
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I, INTRODUCTION 
 
 The object of tnis study nas been to investigate the anthro- 
 pometric measurements of students, in particular the height and 
 weight of college freshmen. In connection with anthropometric meas- 
 uraaients the statement is often made by well-known statisticians 
 that such measurements follow the normal probability curve - in fact, 
 that statement is almost an axiom in such studies. It has been my 
 purpose to investigate whether or not that is the case, and if not, 
 What curve will fit those measuremients more closely. 
 
 In connection with such an investigation there are certain con- 
 stants which are always of interest; these include several averages, 
 and a measure of the spread of the data. The three common formiS of 
 average are the arithmetic mean, the median, and the mode, while the 
 standard deviation is the usual mieasure of the spread of the data. 
 
 The calculation of these constants was the first step in this study. 
 
 The question of the dependence of weight upon height is one 
 that often arises. This too has been investigated, and the usual 
 m.easure of the relationship (the coefficient of correlation) calcu- 
 Idt ed. 
 
 In particular, one object has been to set down on paper a sta- 
 tistical description of certain classes of students at the University 
 of Illinois in the years just following the war, this description to 
 be used later, for comiparison purposes. 
 
 II. DESCRIPTION OP DATA 
 
 Every year the University of Illinois requires that the studeni 
 entering that Institution for the first time take a physical examin- 
 ation. The informxation obtained from this examination, which include^ 
 

 
 
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2 
 
 various physical measurements, is recorded on individual cards for 
 each person and is then filed in the offices of the University Health 
 Service Department (See Table I). The date of exaniinat ion and the 
 classification and age of the student are also recorded on this card, 
 so that it is possible to get the heights and weights of men and 
 women, taken when they were freshmen. As a result of this system, 
 
 I was able to get sufficient data to separate the men into classes 
 according to age, so that I have the measurements of 1004 men who 
 were 18 years old when they entered the university as freshmen, of 
 991 Who were 19 at the same time, 6b8 who were 17, and 877 who were 
 20. The age recorded is the one given by the student himself and 
 means that that was his age on his last birthday, rather than his 
 age to the nearest birthday. 
 
 Due to the fact that there are fewer women in the university, 
 
 I was unable to get enough individuals to divide into classes as I 
 did the men, ana instead, i have the heights and weights of 991 
 women of ages 16-26 (the range of the ages of a very large proport ic: , 
 of the 991 women is 17-20 inclusive). This does not give as satis- 
 factory results as one could desire, for it leaves nothing for com- 
 parison at the present time, and the more or less heterogeneity of 
 the group makes the results of less worth. 
 
 Then as a matter of interest and comparison, I made a frequen- 
 cy distribution of the mieasurements of the men, of ages 17 to 20 
 inclusive, in the present freshman class (1920-1921), 
 
 It is understood that all of the mieasuremient s recorded are 
 measurements of American born students. 
 
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TABLE I 
 
 University of Illinois Health Service 
 
 FAMILY HISTORY 
 
 {jJate 
 
 
 f Racial '] 
 
 
 
 
 ! extraction J 
 
 
 j in the vear 
 
 
 f Racial I 
 
 
 J Wellj has 
 
 
 Lextraction I 
 
 
 , in the year 
 
 of. 
 
 uaU of Exam.) 
 
 What relatives have had Tb? Cancer?.,.. 
 
 Neurasthenia? Epilepsy?.. 
 
 Other possibly inherited disease? 
 
 PERSONAL HISTORY 
 
 Birthplace 
 
 What injuries? {Give age) 
 
 What operations? (G/ait age) 
 
 Age of last vaccination scar; under 10 yrs. 
 
 ., 10 to 20 yrs over 20 yrs... 
 
 Have done ; also 
 
 (Mental work other than schooling) (physical tc'ork) 
 
 Are you a waiter, dish-washer or cook? Where? 
 
 Present general health Appetite Sleep 
 
 Tea and Cof. Tob Ale Drugs., 
 
 .hrs. 
 
 Disease 
 
 Measles 
 
 Rubella 
 
 Mumps 
 
 Chicken-pox 
 
 Whooping Cgh- 
 
 Scarlet Fever 
 
 Typhoid Fever- 
 
 Typhoid Vac. 
 
 Diphtheria 
 
 Meningitis 
 
 Malaria 
 
 Smallpox 
 
 Smallpox Vac._ 
 
 Pneumonia 
 
 Asthma 
 
 Pleurisy 
 
 Rheumatism 
 
 Amygdalitis 
 
 Chorea 
 
 Influenza 
 
 Otitis Media 
 
 Gonorrhea 
 
 Syphilis 
 
 Constipation 
 
 Dysentery 
 
 Appendicitis 
 
 Neurasthenia 
 
 Poliomyelitis 
 
 Tuberculosis 
 
 Glasses 
 
 Age 
 
 PHYSICAL EXAMINATION 
 
 ^Gen. DevL; exc., good, fair, poor. W.; thin, av., obese. lbs ins. 
 
 Build: stocky, medium, slender. Head: length ^.'9. cms., width (rT".. ems 
 
 Eyes: blue, gray, dark gray, greenish, hazel, dark. Hair: fair (flaxen, reddish, light brown), brown, dark 
 brown, black. 
 
 ‘Skin: type. 
 
 T . . funder 15 mm. 
 
 Hcne\ ^Vac.: R. L. arm, leg; pitted, keloidal, smooth; ! 15-20 mm. 
 
 Temp.. 
 
 Lover 20 mm. 
 
 ‘■Teeth: 876543211234567 8) Remarks., 
 87654321 1234567 8 j 
 
 "^Thyroid F. H.. 
 
 I Present 
 Res.-^ Past 
 
 I Adolescent 
 
 Epi., 
 
 .Exp. 
 
 '■"Lymph N.:C Ax Ing 
 
 "Chest: norm "Lungs: norm In.sp 
 
 "Heart: rate recumb , erect , norm V. C cc. 
 
 “B. P. (max.), recumb mm., erect mm. B. P. (min.) 
 
 "Abdomen: norm., rigid, relax. "Hernia: 
 
 "Palpable: Liv , Spl , R. Kid , L. Kid _. , Other 
 
 "Knee jerk: R , L "Penis: norm., circum. ^''Testes: R L "Varic: R L 
 
 ^^Urine: Col Sp. gr , R , Alb , S 
 
 (Kyphosis 
 
 "Vertebral column-J^ Lordosis "Feet: Long arches ) R Anterior arches ) R 
 
 ^Scoliosis ) L ) L 
 
 "Other joints: 
 
 "Nose: Nor., Spur., Div., C.C.Rh., Turg. Rh., Hyp. Rh., Atrop. Rh. "Adenoids: L., S. ^Chr. Pharyn. 
 
 "Tonsils: Nor., Ab., Bur., Proj. Path "Larynx: 
 
 "Ears: Nor., Cer., T. T. Chr. S., Wch Speh Whisp 
 
 "Eyes: Lids: nor Muscles; nor Fundus; nor ;Col. vis.: nor 
 
 Refraction : O.D O.S 
 
 Defects 
 Treated (t) 
 Corrected (c) 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 Wl9 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 32 
 
 Examiner. 
 
 
4 
 
 III, THE EREQUEHCY DISTRIBUTIONS 
 
 When the height and weighs of tnese 4481 students had been re- 
 corded according to the classifications mentioned above, I was able 
 to make twelve frequency distributions, that is, distributions of 
 height and of weight for 17 year , 18 year , 19 year , 2u year ,men^ 
 for women, and for the men of one freshman class (Tables II , III). 
 For heights I have used a class interval of one inch, the frequencies 
 as given in the tables representing the number of students whose 
 height lies in the interval of .5 Inch on each side of the class as 
 given; for example, a man is recorded as 68 inches tall, if he is in 
 reality between the limits 67.5 and 68.5 inches. The central points 
 such as 68 are called the mid-ordinates, since ail the frequencies 
 are assumed to be concentrated at these mid-points of the intervals 
 Of the base. When heights were given as the limits of intervals 
 (e.g. ,67.5), the unit was aiviaed, with one half given to each mid- 
 ordinate just above and below tne real value. The weights were dis- 
 tributed in class intervals of 5 pounds, in order to make fewer 
 classes to compute, and because with intervals of 1, tne dlstributioi 
 would nave been m.ore or less discontinuous and by no means smooth. 
 
 As it is, in several of the distributions the frequencies do not run 
 smoothly, but show frequent jutting points in tne frequency polygons 
 (See Figs, 7,8,9,10,11). The multiples of 5 were taken as the mid- 
 ordinates in the case of weights and when the latter were given as 
 the limits Of intervals (e. g. , 152%5), the unit was divided as in 
 heights. 
 
 IV. CALCULATION OF AVERAGES 
 
 Although frequency distributions help to make any series of 
 
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5 
 
 TABLE Ila 
 
 Frequency Distributions of Heights of College Freshmen 
 
 
 Freshman Class 
 
 1920-21 
 
 
 Women 
 
 
 18 
 
 years 
 
 Height 
 
 Actual 
 
 Grad. by 
 
 Grad. by 
 
 Actual 
 
 Grad. by 
 
 Grad. by 
 
 Actual 
 
 Grad. by 
 
 in 
 
 Freq. 
 
 Normal 
 
 Type IV 
 
 Freq. 
 
 Normal 
 
 Type IV 
 
 Freq. 
 
 Normal 
 
 inches 
 
 
 Curve 
 
 
 
 Curve 
 
 
 
 Curve 
 
 55 
 
 
 
 
 1.0 
 
 0.1 
 
 0.3 
 
 
 
 56 
 
 
 
 
 
 1.0 
 
 1.0 
 
 
 
 57 
 
 
 
 
 1.0 
 
 3.0 
 
 3.0 
 
 
 
 58 
 
 
 
 
 11.0 
 
 9.5 
 
 9.0 
 
 
 
 59 
 
 0.5 
 
 0. 2 
 
 0. 3 
 
 23.0 
 
 26.0 
 
 34.0 
 
 1.5 
 
 0.2 
 
 60 
 
 1.5 
 
 1.0 
 
 1.0 
 
 58. 0 
 
 57.0 
 
 54.0 
 
 1.5 
 
 0.8 
 
 61 
 
 4.0 
 
 4.0 
 
 4.0 
 
 94.5 
 
 101.0 
 
 101.5 
 
 3.0 
 
 3.0 
 
 63 
 
 b. 5 
 
 11.0 
 
 10.0 
 
 168.0 
 
 147.5 
 
 153.0 
 
 4.0 
 
 9. 0 
 
 63 
 
 32 . 0 
 
 28.0 
 
 26.0 
 
 168.0 
 
 174.5 
 
 180.0 
 
 21.5 
 
 23. 0 
 
 64 
 
 54,0 
 
 59.0 
 
 57.0 
 
 155.0 
 
 168.0 
 
 170.0 
 
 50.0 
 
 48.0 
 
 65 
 
 107.5 
 
 104.0 
 
 104.0 
 
 147.5 
 
 131.0 
 
 138.0 
 
 88.5 
 
 87,0 
 
 66 
 
 163.5 
 
 154.0 
 
 157.5 
 
 73.5 
 
 83.0 
 
 79.0 
 
 123.5 
 
 130.0 
 
 67 
 
 190.5 
 
 190.0 
 
 195.5 
 
 40.5 
 
 43.0 
 
 41.0 
 
 165.5 
 
 162.0 
 
 68 
 
 187.0 
 
 197.0 
 
 200.0 
 
 22,5 
 
 18.0 
 
 18.0 
 
 172.5 
 
 169,0 
 
 69 
 
 175.5 
 
 171.0 
 
 169.0 
 
 5.5 
 
 6 . 0 
 
 7.0 
 
 160.0 
 
 147.5 
 
 70 
 
 125.0 
 
 124.0 
 
 119.5 
 
 0.5 
 
 2.0 
 
 3.0 ■ 
 
 98.0 
 
 107.0 
 
 71 
 
 74.0 
 
 76.0 
 
 72.0 
 
 0.5 
 
 0.4 
 
 1.0 
 
 53.5 
 
 65. 0 
 
 72 
 
 34. 5 
 
 39,0 
 
 37.0 
 
 
 0.1 
 
 0,3 
 
 41.0 
 
 33.0 
 
 73 
 
 18.0 
 
 16.5 
 
 17.0 
 
 1.0 
 
 
 
 13.5 
 
 14.0 
 
 74 
 
 3.5 
 
 6 . 0 
 
 7.0 
 
 
 
 0.1 
 
 4.5 
 
 5. 0 
 
 75 
 
 3. 0 
 
 2.0 
 
 3.0 
 
 
 
 
 0.5 
 
 1.0 
 
 76 
 
 2.0 
 
 0.4 
 
 1,0 
 
 
 
 
 1.5 
 
 0.4 
 
 77 
 
 0.5 
 
 0.1 
 
 0.3 
 
 
 
 
 
 
 
 1182.0 
 
 1183.2 
 
 1181.1 
 
 971.0 
 
 971. 1 
 
 “972. 1 
 
 100470 
 
 1003.9 
 
V 
 
6 
 
 TABLE Ilb 
 
 Frequency Distributions of Heignts of College Freshmen 
 
 
 17 
 
 year 
 
 19 
 
 year 
 
 20 
 
 year 
 
 Height 
 
 Actual 
 
 Graduated 
 
 Actual 
 
 Graduat ed 
 
 Actual 
 
 Graduat ed 
 
 in 
 
 Freq. 
 
 by Normal 
 
 Freq. 
 
 by Normal 
 
 Freq. 
 
 by Normal 
 
 inches 
 
 
 Curve 
 
 
 Curve 
 
 
 Curve 
 
 56 
 
 
 
 2.0 
 
 ■ 
 
 
 
 57 
 
 
 
 
 
 
 
 58 
 
 
 
 
 
 
 
 59 
 
 
 
 
 
 
 
 60 
 
 1.0 
 
 0.5 
 
 
 1.0 
 
 1.5 
 
 0.4 
 
 61 
 
 4.0 
 
 2.0 
 
 1.0 
 
 3.0 
 
 1.5 
 
 2.0 
 
 62 
 
 6. 0 
 
 5.5 
 
 D. 0 
 
 8.0 
 
 6.5 
 
 6. 0 
 
 65 
 
 13.0 
 
 14.0 
 
 26.0 
 
 21.0 
 
 11.5 
 
 15.0 
 
 64 
 
 33. 0 
 
 30.0 
 
 40.0 
 
 45.0 
 
 33 . 5 
 
 35.0 
 
 65 
 
 40.5 
 
 53. 0 
 
 84.5 
 
 81.0 
 
 61.0 
 
 66. 0 
 
 b6 
 
 77.5 
 
 79.5 
 
 122.0 
 
 122.0 
 
 112.0 
 
 104.5 
 
 67 
 
 109.0 
 
 100.0 
 
 158.5 
 
 155.0 
 
 145.5 
 
 137.0 
 
 68 
 
 106.5 
 
 106.0 
 
 155.0 
 
 165.0 
 
 152.5 
 
 150.0 
 
 69 
 
 89.5 
 
 94.0 
 
 136.0 
 
 147.0 
 
 129.5 
 
 136. 5 
 
 70 
 
 71.0 
 
 70.0 
 
 138.5 
 
 111.0 
 
 91.0 
 
 103.0 
 
 71 
 
 51.5 
 
 44. 0 
 
 71.5 
 
 70.0 
 
 75.5 
 
 65. 0 
 
 72 
 
 24.0 
 
 23.0 
 
 28.0 
 
 37.0 
 
 35.0 
 
 34.0 
 
 73 
 
 8. 5 
 
 10.0 
 
 13.0 
 
 17.0 
 
 16.0 
 
 15.0 
 
 74 
 
 2.0 
 
 4. 0 
 
 7.0 
 
 6 . 0 
 
 1.0 
 
 5. 0 
 
 75 
 
 1.0 
 
 1.2 
 
 3.0 
 
 2.0 
 
 0.5 
 
 2.0 
 
 76 
 
 
 
 
 
 2.5 
 
 0.4 
 
 77 
 
 
 
 
 
 0.5 
 
 0.1 
 
 
 638. 0 
 
 636 . 7 
 
 991.0 
 
 991.0 
 
 877.0 
 
 876.8 
 
1 
 
 
 
 
 

 TABLE Ilia 
 
 Frequency Distributions of Weights of 
 Fresnman Class 1920-21 Women 
 
 7 
 
 College Freshmen 
 
 18 year 
 
 Weight 
 
 Actual 
 
 Grad, by 
 
 Grad. by 
 
 Actual 
 
 Grad. by 
 
 Grad. by 
 
 Actual 
 
 Grad. by 
 
 in 
 
 Freq. 
 
 Normal 
 
 Type VI 
 
 Freq. 
 
 Normal 
 
 Type V 
 
 Freq. 
 
 Norm^al 
 
 pounds 
 
 
 Curve 
 
 
 
 Curve 
 
 
 
 Curve 
 
 85 
 
 N 
 
 
 
 5.0 
 
 19.5 
 
 5.5 
 
 
 
 90 
 
 1.0 
 
 4.0 
 
 0.1 
 
 30. 5 
 
 52. 0 
 
 35,0 
 
 
 
 95 
 
 4.0 
 
 8.0 
 
 1.0 
 
 55. 5 
 
 47.0 
 
 o6. 0 
 
 4.0 
 
 6, 5 
 
 100 
 
 10.0 
 
 15,5 
 
 7.0 
 
 85.0 
 
 65. 5 
 
 92.0 
 
 7.0 
 
 12.5 
 
 105 
 
 15.0 
 
 27.5 
 
 21.0 
 
 119.0 
 
 84.0 
 
 118.0 
 
 16.0 
 
 22. 5 
 
 110 
 
 56. 5 
 
 44.0 
 
 47.0 
 
 ia2,5 
 
 99.0 
 
 127.0 
 
 52.0 
 
 57.6 
 
 115 
 
 71.5 
 
 b5. 5 
 
 80.0 
 
 135,0 
 
 108.0 
 
 121.0 
 
 58.0 
 
 o6. 5 
 
 130 
 
 115.5 
 
 89,0 
 
 112.0 
 
 98.0 
 
 108.5 
 
 105.0 
 
 85.5 
 
 78.0 
 
 125 
 
 155. 5 
 
 111.5 
 
 154. 5 
 
 85. 5 
 
 101.0 
 
 65. 0 
 
 145.5 
 
 98.5 
 
 IdO 
 
 150. 5 
 
 128. 5 
 
 145.0 
 
 66.0 
 
 87.0 
 
 6b. 0 
 
 120.5 
 
 114.0 
 
 155 
 
 128.5 
 
 156. 5 
 
 159.0 
 
 58. 5 
 
 69.0 
 
 50.0 
 
 126.0 
 
 120.0 
 
 140 
 
 155.5 
 
 135.0 
 
 135.5 
 
 42.0 
 
 50. 0 
 
 56 . 0 
 
 129.5 
 
 115.5 
 
 145 
 
 101.5 
 
 119.0 
 
 103.0 
 
 25. 5 
 
 54.0 
 
 26. 0 
 
 78.5 
 
 102.0 
 
 150 
 
 bb. 0 
 
 98.0 
 
 81.0 
 
 14.5 
 
 21.0 
 
 18.0 
 
 60.5 
 
 83.0 
 
 155 
 
 51.5 
 
 74.0 
 
 61.0 
 
 14.5 
 
 13.0 
 
 15.0 
 
 45. 5 
 
 60.5 
 
 160 
 
 54.5 
 
 51.5 
 
 44.0 
 
 5. 0 
 
 7.0 
 
 9.0 
 
 58.0 
 
 40.5 
 
 165 
 
 54.5 
 
 55 . 0 
 
 50.0 
 
 5.0 
 
 5. 0 
 
 6. 0 
 
 21.0 
 
 25.0 
 
 170 
 
 20.5 
 
 19.5 
 
 20.5 
 
 6.0 
 
 1.5 
 
 4.0 
 
 12.6 
 
 14. 0 
 
 175 
 
 11.5 
 
 10.5 
 
 13.5 
 
 2.0 
 
 1.0 
 
 5. 0 
 
 7.0 
 
 7.0 
 
 180 
 
 7.5 
 
 5.5 
 
 9.0 
 
 2.0 
 
 0.2 
 
 3.0 
 
 11.0 
 
 3.5 
 
 185 
 
 b.O 
 
 2.5 
 
 5.0 
 
 0.0 
 
 0.1 
 
 1.5 
 
 1.0 
 
 1.5 
 
 190 
 
 1.0 
 
 1.0 
 
 5.0 , 
 
 1.0 
 
 
 1.0 
 
 1.5 
 
 0.5 
 
 195 
 
 4. 0 
 
 0.4 
 
 2.0 
 
 1,0 
 
 
 0.7 
 
 6. 5 
 
 0.2 
 
 200 
 
 5. 0 
 
 0.1 
 
 1.0 
 
 2.0 
 
 
 0.5 
 
 
 
 205 
 
 1.0 
 
 
 0.7 
 
 1,0 
 
 
 0. 5 
 
 
 
 210 
 
 
 
 
 1.0 
 
 
 
 
 
 215 
 
 
 
 
 
 
 
 1.0 
 
 
 255 
 
 
 
 
 
 
 
 
 
 240 
 
 
 
 
 
 
 
 
 
 285 
 
 
 
 
 
 
 
 
 
 1182.0 
 
 1178.0 
 
 1181.8 
 
 971.0 
 
 950. 5 
 
 969. 5 
 
 1004.0 
 
 998.2 
 
alt 
 
 . X 
 
 
8 
 
 TABLE Illb 
 
 Frequency Distribucion ci nyeignts of College Fresiimen 
 
 
 17 
 
 year 
 
 19 
 
 year 
 
 20 
 
 year 
 
 Weight 
 
 Actual 
 
 GracLuat ed 
 
 Actual 
 
 Graauat ed 
 
 Actual 
 
 Graauat ed 
 
 in 
 
 Freq. 
 
 by Normal 
 
 Freq. 
 
 by Normal 
 
 Freq. 
 
 by Normal 
 
 pounas 
 
 
 Curve 
 
 
 Curve 
 
 
 Curve 
 
 90 
 
 2.0 
 
 3.0 
 
 
 
 
 
 95 
 
 2,0 
 
 6.0 
 
 
 
 1.0 
 
 5,0 
 
 100 
 
 2.5 
 
 11.0 
 
 6 , 5 
 
 10,0 
 
 4.0 
 
 9.0 
 
 105 
 
 11.5 
 
 18.0 
 
 9.5 
 
 18.0 
 
 8. 5 
 
 16.0 
 
 110 
 
 20,0 
 
 27.0 
 
 31.5 
 
 31.0 
 
 23. 5 
 
 26.0 
 
 115 
 
 39.5 
 
 39. 0 
 
 53.0 
 
 48 . 0 
 
 30.5 
 
 39. 5 
 
 120 
 
 64.0 
 
 51.0 
 
 96. 5 
 
 68.0 
 
 63. 0 
 
 55.5 
 
 125 
 
 84.0 
 
 62. 0 
 
 85.0 
 
 89.0 
 
 88.5 
 
 72.0 
 
 130 
 
 82.5 
 
 69.0 
 
 117.0 
 
 106.0 
 
 122,5 
 
 87.0 
 
 135 
 
 89.0 
 
 71.5 
 
 128.0 
 
 116.0 
 
 107.0 
 
 96.0 
 
 140 
 
 70.5 
 
 68. 5 
 
 123.5 
 
 116.0 
 
 111.0 
 
 98.5 
 
 145 
 
 53.0 
 
 61.0 
 
 101.0 
 
 106.0 
 
 80.5 
 
 93.0 
 
 150 
 
 48.5 
 
 49.5 
 
 64.5 
 
 89.0 
 
 55.0 
 
 81.0 
 
 155 
 
 19.0 
 
 37.0 
 
 54.5 
 
 68. 0 
 
 65. 5 
 
 66. 0 
 
 160 
 
 17.0 
 
 26.0 
 
 34.5 
 
 48.0 
 
 45.0 
 
 49.0 
 
 165 
 
 8.0 
 
 17.0 
 
 40.0 
 
 31.0 
 
 29.5 
 
 34.0 
 
 170 
 
 8.0 
 
 10.0 
 
 21.0 
 
 18.0 
 
 15.0 
 
 21.0 
 
 175 
 
 5.5 
 
 5. 5 
 
 6,0 
 
 10.0 
 
 4.5 
 
 13.0 
 
 180 
 
 4.5 
 
 3.0 
 
 3.0 
 
 5.0 
 
 6.5 
 
 7.0 
 
 185 
 
 0.0 
 
 1.0 
 
 10.0 
 
 2.0 
 
 2.0 
 
 3.0 
 
 190 
 
 2,5 
 
 0.6 
 
 2.0 
 
 1.0 
 
 5.0 
 
 2.0 
 
 195 
 
 0.5 
 
 0.2 
 
 1.0 
 
 0.3 
 
 1.0 
 
 1.0 
 
 200 
 
 1.0 
 
 0.1 
 
 2.5 
 
 0.1 
 
 2.0 
 
 0.3 
 
 205 
 
 
 
 0.5 
 
 
 1.5 
 
 0.1 
 
 210 
 
 
 
 
 
 2.5 
 
 
 235 
 
 1.0 
 
 
 
 
 
 
 240 
 
 1.0 
 
 
 
 
 
 
 245 
 
 
 
 
 
 1,0 
 
 
 255 
 
 
 
 
 
 1.0 
 
 
 2^5 
 
 1.0 
 
 
 
 
 
 
 
 638.0 
 
 636.9 
 
 991.0 
 
 980.4 
 
 877.0 
 
 874.9 
 
i 
 
 
9 
 
 ODservaD ions conprenensible, tnere is a need for quantitative ex- 
 pressions to cnaracterize the histributions. Two ways in which ordi- 
 nary distributions may differ are: (1) in position, that is, in tne 
 values ot the variable around wnicn they center, and (2) in the rang? 
 of variation. Expressions which measure position are usually called 
 averages, of which three common ones are the arithmetic mean, the 
 m.edian, and the mode. The more closely a distribution can be fit by 
 the normal curve the more nearly will these three measures approach 
 coincidence. Measures of the range of variation are termed mieasures 
 of dispersion, of wnlch the mxst useful is the standard deviation. 
 
 The values of these expressions, for the distributions in the present 
 study, have been calculated. 
 
 Tne mode of a frequency distribution is the class which has the 
 greatest frequency. since the division into classes is arbitrary, 
 the value of the mode thus obtained can only be approximate. A m.ore 
 accurate definition, as given oy Gr.U . lule^ is: ’’The mode is the 
 value Of tne variable corresponding to the maximum of the ideal fre- 
 quency curve wnicn gives tne closest possible fit to the actual dis- 
 tribution. ” Tnis theoretical mode is obtained in several cases 
 under tne discussions of the types of frequency curves. The emiplri- 
 cal or approximate value of this average can be easily obtained from 
 tne frequency polygons. 
 
 The median is tnat value of tne variable of an ordinary fre- 
 quency distr lout ion, on eacn side of wnich tnere are an equal number 
 of observations. The m^edian for each of the given distributions was 
 obtained by simple interpolation, which gives the value only approx- 
 imately, since sucn a mietnod assumes that the values in each class 
 
 1. ”An Introduction to the Tneory of Statistics”, p 120. 
 

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 >aoQ r i<f %j^ jic^a cQuAJ 9i>om lo 
 
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 are uniformly distrilDuted throughout tne interval. Using the figures 
 of the freshman group to illustrate, tne total number of observations 
 is 1182, he-if of which is byi. Fromi tne table of distribution for 
 this group (Taole ila), we see tnat there are 5b9 students wnose 
 heignt is not greater than 67.5 incnes, and 187 more whose height 
 comes in tne interval 67.5 to 68.5. however, only 62 are required 
 to ria,ke up the total of 591, so tnat the value of the median is 
 
 67.5 + T^.l = 67.5 + .171 
 187 
 
 = 67.671 inches. 
 
 The fraction added must be multiplied by the class interval, which ii 
 this case is unity. 
 
 The arithmetic mean is the particular form of average which is 
 commonly termed the mean or the average value. It is defined by the 
 formula 
 
 in which X represents the various values of a variable, f is the fre- 
 quency of each variable, and N is the total number of observations, 
 ■To calculate the arithmietic means of the various distributions, a 
 
 modification of Mr. Hardy’s summation method, as given by W. P. Elder- 
 
 1 2 
 ton , was used. By m.eans of this mjethod the monients are calculated 
 
 and it is these which lead to the criterion k, which shows the type 
 
 of frequency curve (as they have been classified by Karl Pearson) to 
 
 which the data belong. The sumimation method is shown by Table IV in 
 
 which I calculated the momients for the distribution of heignts of 
 
 the freshman group. Using a central term as the starting point for 
 
 the summiation, we get S 3 whicn is the difference between the sums on 
 
 1. Frequency Curves and Correlation, 1917 Edition, pp 22-33. 
 
 2. For a discussion of the method of moments see Elderton, "Frequen- 
 cy Curves and Correlation", Chap. III. 
 

TABLE IV 
 
 Summation Method 
 for 
 
 The Calculation of Moments and Other Constants 
 Example : Distribution of Heights of 1183 Freshmen 
 
 11 
 
 ■equency 
 
 1st Sum 
 
 3nd Sum 
 
 3rd Sum 
 
 4th Sum 
 
 5th Sum 
 
 .5 
 
 ,5 
 
 .5 
 
 .5 
 
 .5 
 
 .5 
 
 1.5 
 
 3.0 
 
 3.5 
 
 3.0 
 
 3. 5 
 
 4.0 
 
 4.0 
 
 6.0 
 
 8.5 
 
 11.5 
 
 15.0 
 
 19.0 
 
 5.5 
 
 11,5 
 
 30.0 
 
 3l. 5 
 
 46,5 
 
 65. 5 
 
 33.0 
 
 54.0 
 
 107.5 
 
 163.5 
 
 43.5 
 
 97.5 
 305.0 
 368. 5 
 
 63.5 
 161.0 
 366. 0 
 734.5 
 
 95. 0 
 
 356.0 
 
 633.0 
 
 141.5 
 
 397.5 
 
 307.0 
 
 190.5 
 
 187.0 
 
 633.0 
 
 1531.0 
 
 3433 . 5 
 
 6938.5 
 
 13385.0 
 
 175.5 
 
 436. 0 
 
 938.0 
 
 1861,5 
 
 3506. 0 
 
 6356. 5 
 
 135.0 
 
 360,5 
 
 503.0 
 
 933,5 
 
 1644.5 
 
 3850.5 
 
 74.0 
 
 13b. 5 
 
 341.5 
 
 431,5 
 
 731.0 
 
 1306.0 
 
 34 . 5 
 
 61.5 
 
 106.0 
 
 180.0 
 
 399 , 5 
 
 485.0 
 
 18,0 
 
 37.0 
 
 44.5 
 
 74.0 
 
 119.5 
 
 185.5 
 
 3.5 
 
 9.0 
 
 17.5 
 
 39,5 
 
 45.5 
 
 66.0 
 
 3.0 
 
 5.5 
 
 8.5 
 
 13.0 
 
 16.0 
 
 30.5 
 
 3.0 
 
 3.5 
 
 3.0 
 
 3.5 
 
 4.0 
 
 4.5 
 
 0.5 
 
 0.5 
 
 0.5 
 
 0. 5 
 
 0.5 
 
 0.5 
 
 S 3 
 
 1561 734.5 
 
 1183 1183 
 
 o _ ^433.5 . 633 
 
 1183 1183 
 
 .69933858 
 
 3.43174380 
 
 6938 5 397.5 
 
 S 4 = ^ItIs - -TleZ = 5. 53538071 
 
 = -^1455160 
 
 ^3 = SS 3 - d(l+d) = 5.65531343 
 
 ^3 = SS^ - 3^(l+d) - d(l+d)(3+d) = 1.11594631 
 
 ^4 = S 4 S 5 - 3/^[3(l+d)+lj -/< 2 [ 6 (l+d) (3+d)-l] -d(l+d) (3+d) (3+d) 
 
 ^1 = .00688530 /^g = 3.19830317 103.38991868 
 
 m: 
 
 /^i ) 
 
 A. 
 
 ^ -m (^:3“^6Tf4^-i'0iT = • °1277189 
 
 _ 8 ( 03 -%-!) = i4. 97401766 m = i(r+2) = 18.48700883 
 
 P./^ —A (O' ' 
 
 203 -^ 01-6 
 
 V _ r(r-3)./6i 
 
 VTSU-lj-^tr-S)' 
 
 = (-)4. 13388736, negative because 5 is posi 
 
 tiV€ 
 
 a = y-^16(r-l)-/3^(r-3)2^ = 13,76545077 
 
1 
 
 1 
 
 « 
 
 t 
 
 i 
 
 i 
 
 ! 
 
 ''I 
 
 I 
 
 I 
 
 
 'i' 
 
 
 
 
 
 I 
 
 t 
 
 V 
 
 I 
 
12 
 
 each side of the central point; in this work it is an advantage to 
 consider tne total frequency as unity, hence the sums of the columns 
 must be divided by M, the total number of observations, in order to 
 keep the work consistent. Sg is d, the difference between the act- 
 ual mean of the distribution and the arbitrary starting point. Since 
 the starting point in the example was 190.5 , which corresponds to 
 class 67, and since d is .699, the actual mean of the data is 67.699 
 inches. Using the same group (freshman class) as an illustration of 
 the calculation of the mean for weight, we find 150,5 taken as the 
 central point, which corresponds to class 130; summing just as in 
 the case of heights, d was found equal to 1.187, However, instead 
 of adding this directly to tne arbitrary point 130, it was necessary 
 first to multiply it by 5 in order to get it in terms of the class 
 interval wnich is 5. Thus the mean weight of the men of one fresh- 
 man class is 135.9 pounds (Table V), 
 
 The 3.rithmetic means of the various distributions for the 
 height of men are very close together, varying between 67.7 and 68 
 inches. The 20 year group averaged the tallest at 67.98 inches, 
 with the 19 year next at 67.36, then the 17 year at 67.83, and the 
 
 18 year at 67,74; the group of freshmen men averaged 67.70. The 
 range on each side of the mean is 8 or 9, with an exception in the 
 
 19 year group in which two men were only 56 inches. The women 
 averaged 63.31 inches. 
 
 There was a very great range in several of tne weight distri- 
 butions, for example, one of the 17 year men weighed 285 pounds, 
 although tne distribution was by no means continuous that far. The 
 range of continuity was usually 90 to 210 pounds. The 20 year group 
 averaged not only tallest but heaviest with an average weight of 139 
 
TABLE V 
 
 13 
 
 Summation Method 
 for 
 
 The Calculation of Moments and Other Constants 
 
 Example : Distribution of Weights of 1183 Freshmen 
 
 Frequency 
 
 1st Sum 
 
 2nd Sum 
 
 3rd Sum 
 
 4th Sum 
 
 5th Sum 
 
 1.0 
 
 1.0 
 
 1.0 
 
 1.0 
 
 1.0 
 
 1.0 
 
 4.0 
 
 5.0 
 
 6. 0 
 
 7.0 
 
 8.0 
 
 9.0 
 
 10.0 
 
 15.0 ■ 
 
 31,0 
 
 28.0 
 
 36. 0 
 
 45.0 
 
 15.0 
 
 30.0 
 
 51. 0 
 
 79. 0 
 
 115.0 
 
 160.0 
 
 36, 5 
 
 66 . 5 
 
 117.5 
 
 196.5 
 
 311.5 
 
 471. 5 
 
 71.5 
 
 138.0 
 
 256. 5 
 
 452.0 
 
 763.5 
 
 
 115.5 
 
 253. 5 
 
 509,0 
 
 961. 0 
 
 
 
 153.5 
 
 407.0 
 
 916.0 
 
 
 
 
 150.5 
 
 128.5 
 
 624. 5 
 
 2319.5 
 
 7622.0 
 
 33017.5 
 
 65016. 5 
 
 13o. 5 
 
 496.0 
 
 1695 . 0 
 
 5302.5 
 
 15395.5 
 
 41999. 0 
 
 101.5 
 
 362, 5 
 
 1199.0 
 
 3607.5 
 
 10093,0 
 
 26603. 5 
 
 66. 0 
 
 261.0 
 
 836.5 
 
 2408.5 
 
 6485.5 
 
 16510.5 
 
 51.5 
 
 195.0 
 
 575.5 
 
 1572.0 
 
 4077.0 
 
 10025.0 
 
 54.5 
 
 143. 5 
 
 380. 5 
 
 996.5 
 
 2505.0 
 
 5948.0 
 
 34. 5 
 
 89.0 
 
 237.0 
 
 616,0 
 
 1508.5 
 
 3443. 0 
 
 20.5 
 
 54.5 
 
 148.0 
 
 379.0 
 
 892. 5 
 
 1934.5 
 
 11.5 
 
 34,0 
 
 93.5 
 
 231.0 
 
 513.5 
 
 1042.0 
 
 7.5 
 
 22.5 
 
 59. 5 
 
 137.5 
 
 282.5 
 
 528.5 
 
 6.0 
 
 15.0 
 
 37.0 
 
 78.0 
 
 145. 0 
 
 246.0 
 
 1.0 
 
 9.0 
 
 22.0 
 
 41.0 
 
 67.0 
 
 101.0 
 
 4.0 
 
 8.0 
 
 13.0 
 
 19. 0 
 
 26.0 
 
 34.0 
 
 3,0 
 
 4,0 
 
 5.0 
 
 6.0 
 
 7.0 
 
 8. 0 
 
 1.0 
 
 1.0 
 
 1.0 
 
 1.0 
 
 1.0 
 
 1.0 
 
 Q- _ 3319 . 5 _ 916 _ 1 107 'zq/jot:; q _ 33017,5 _ ^ 63. 5 _ tq 007 / t tic 
 
 ^2 - 1182 “ 1.18739435 S^ 1183“ HB3 “ 18.83741116 
 
 o _ 7633 . 961 
 
 ^3 TT^ TT^ = 
 
 = 11.93554339 
 
 7.36143133 
 
 65016.5 . 
 — ri8~ 
 
 471.5 
 
 116^ 
 
 =55. 40439933 
 
 = 36,ft3835548 
 
 ^ = 523.99711354 
 ^ 2 
 
 j AA^ '4. 
 
 1 = -f = . 411815 
 
 -^3 
 
 A^A 
 
 — =- = 3. 684<i:49 
 
 ^2 
 
 k = 2.553all 
 
1'4 
 
 pounds; the other groups follow in good order, the 19 year averaging 
 137,5, the 18 year 135.4, ana the youngest group weighing lightest 
 with 134.7 pounds as an average. The mean weight (117.9) of the 
 women is considerably less than those of the men, although the range 
 is practically the same. 
 
 V. CALCULATION OF THE STANDARD DEVIATION 
 To measure the spread of the data about the mean, the stand- 
 ard deviation, O'y was calculated. This measure which is usually used! 
 to show the dispersion of a distribution is given by the formula 
 
 / where f is the frequency of a given interval, x is the dsvi- 
 
 ^ n 
 
 at ion of that interval from the mean, and n is the total number of 
 observations. But it, too, can be obtained by the summation m.ethod 
 for its square is the same as the second moment, that is, the stand- 
 ard deviation is equal to (Tables IV, V). The spread of the data 
 from the mean was least in the distribution of women’s heights, the 
 standard deviation being 2.30 inches; the dispersion for the other 
 distributions of heights was very nearly the same for all groups: 
 
 30 year- 2.33, 18- 3.35, freshman class- 3.38, 17 year- 2.39, 19 year- 
 2.40. 
 
 To find the standard deviation for weights it must be remem- 
 Dered that in order to get it in terr/.s of pounds, the class interval 
 of 5 must be taken into account; thus, is the symbol for 
 
 second moment) equals the standard deviation of the distribution of 
 (heights. As one would expect, the deviation is quite large due to 
 bhe' range, especially virhen thought of in terms of one pound. The 
 smallest standard deviation is 16, S8 for the 18 year group, and the 19 
 /■ear group is next with (T as 16.82. This can be seen from the dis- 
 tribution <5f the data, for in neither case is the range very great 
 
■a:,-. 
 
 V- . . V 
 
 t*w 
 
 
 
 4 
 
 fii :« . . 
 
 '.'•' .tf i‘ Y’j 
 
 
 I '^ Oi!u 
 
 
 
 At: 
 
 Y 
 
 i'r ( 
 
 i). 
 
 !»■ 
 
 
 9 
 
 IC/ 
 
 w*i-w 
 
 . " V • . J 
 
 ■ ■. \ "z\ 
 
 ■ W 
 
 ' ■ .' 
 
 I 
 
 .-. w it» 
 
 i. u 
 
 vt er.*‘ 
 
 :Vfc^£ . 
 
 • *,'<■ 
 
 
 I ti^O.^. pt* '’r.i, t 
 
 
 .p 
 
 .'fL'pe 
 
 0; . ■■ ji;'.’ 
 
 n 
 
 r. 
 
 A 
 
 ka M 
 
 ' J ■ • .' . 
 
 U: 
 
 i 
 
 «*« ^ ‘ 
 
 
 . V . . T\ , 
 
 '': '“■■ •:;^r 
 
 ', £• 
 
 TxA^' 
 
15 
 
 and in the 18 year group especially there is a noticeable clustering 
 about the mean. The 20 y^ear and 17 year groups with their longer 
 ranges have deviation of 17.74 and 17.79 respectively. The distri- 
 bution of the weights of women is spread over quite a large range 
 and consequently it, too, has a highCT: 17.74. 
 
 VI. EXPLANATION OF k. 
 
 After the moments are calculated, k, the criterion for Pearsonl 
 frequency curves, is immediately calculable. As is well known, the 
 normal curve does not fit all types of data, in fact, actually it 
 fits very few. In many physical, economic, and biologic investi- 
 gations, we find the tendency to deviation on one side of the mean 
 unequal to the tendency on the other side. It becomes necessary, 
 therefore, to find equations for curves which would fit these cases, 
 and Karl Pearson miet this need in his articles on "Skew Variations 
 in Homogeneous Material",^ "Supplement to a Memoir on Skew Variation^ 
 and "Second Supplement to a Memoir on Skew Variation"^. In the firBr 
 article he deduces a generalized probability curve which fits 'Sur- 
 prisingly accurately observations of an asymmetrical character." 
 
 Then he goes further in order to meet the problem of limited range; 
 he says, "We have, then, reached this point: that to deal effectively 
 with statistics we require generalized probability curves which in- 
 clude the factors of skewness and range. The generalized curve we 
 have already reached, possesses skewness, but its range is limited 
 in one direction only. " 
 
 As we know, the normal probability curve is obtained 
 
 1. Phil. Trans, of Royal Soc. of London, (A) Vol. 186,Pt. I,p344,et se'l, 
 • 2. Phil. Trans. Royal Soc. London, (A),Vo 1.197, p245, et seq. 
 
 3. Phil. Trans. Royal Soc. London, (A), Vol.216, p 429, et seq. 
 
5 . 
 
 £1 
 
 
 ■^i 43 V oX ® bn-- 
 
 1 
 
 "V -a /3 • 
 
 .fv.:o' 
 
 .' i. ?c .. tiv^ . . >< -: 
 
 
 
 , - i X f ii:.i . : . . ■ f! :- c :r j 
 
 V 
 
 
 nj; 
 
 1 -.: 
 
 
 f . 
 
 •• ^ ^■ 
 
 
 
 ■-i ,*vii ^0 « £ '■ 
 
 v.- t -fcc ^ 
 
 • i V- 
 
 --! *i : , * xo/ 5 ? • '*■ 
 
 ‘ -. ^ 
 
 • ‘ ■ '-. I.i .:" . 
 
 
 i *i .. ''■? 
 
 ' -j- 
 
 nox^. ;,v.n .: •- i.'-A;: 
 
 rv • • *. ♦ ; , 
 
 4 ^ Vj V* , 
 
 ’ "iLX^y _v*; • 
 
 '.’ i *■ I,’ *' '?<»• v*. •v.-'i i . • , . f _ •"'” 
 
 fc-',' V A . . , 
 
 .- ^ ■' ;<• 
 
 ■~' •Tf" -* - t . '••*■ 
 
 1 ■ ^ 
 
 * * - "I'j. V 
 
 ... t / ‘7 *; \-’ ■«■ ' 
 
 • i - ‘' 5 ' .. 
 
 »t ■'■;•• 
 
 / : . 
 
 ^ uu •’ ■ X, .' 
 
 ftriti'. ♦ ''V* ', . y__T . J'v ^ 
 
 / ■' - 
 
 Td-'lC If-./t ' • tl 
 
 ;.:r. 
 
 • ■ <’■ * 
 
 iv^ a X ^ -V ^ oc>a^ xX|tcic , • 
 
 '1 
 
 *. A » 
 
 •r 
 
 \x •.•.'■ 
 
 »4 is?»£iX i 
 
 '■'n 
 
 -‘ V£.'. . v'v”, 'Y *'9 
 
 
 •J . "i 
 
 ’■ 
 
 
 
 
 ♦- r ‘ 
 
 .-' li>r. ,-T. 
 ' i 
 
 ; • : X6 1 ? y, 1 o;^ 0,5 T ij n-r ptt j 
 
 
 ’ . ' : 70 7 ;.r 
 
 ' •' >? 
 
 -t J ^ - 7 1 U 
 
 
 * I, 
 
 ri 
 
 1. .. -JHfl 
 
 ( 
 
 t 
 
 ft 
 
 ^"\n*.. ...- J ; , 
 
 • -• f * > 
 
 
 . • . 1 ,-^ 
 
 ^ : 
 
 f < • I 
 
 '.-'i 
 
 
 • -' 
 
 ■ 7 y ,jK lpir ■ > ' , ' . ■■ .vuu.i 
 
 '■ v; 
 
16 
 
 approximately by running a continuous curve tnrougn tne point bi- 
 nomial + 3) . By allowing n to increase indefinitely and letting 
 the interval between ordinates approach zero the limiting curve is 
 the theoretical probability curve. We usually get this binomial ex- 
 pansion by tossing coins, for we have the probability of throwing 
 heads equal to the probability of throwing tails'^, that is, the proba- 
 bility of the success and failure of an event is one half. The gen- 
 eralized form of the probability curve may be obtained from the gen- 
 eral binomial (p+q)^, where p+q=l, but p^q. The example usually 
 used for this form is the throwing of dice, where the probability of 
 throwing an ace, for instance, is one sixth, and the proba-bility of 
 not throwing it is five sixths. The normal curve may be considered 
 as a special case of this generalized form. The still more general 
 form worked out by Pearson includes both the normal probability and 
 the generalized probability curves as special cases. To illustrate 
 how a frequency curve with limiited range, ana skewness, may arise, 
 he uses the example of drawing balls from a bag. "Take n balls in 
 a bag, of which pn are black, and qn are white, ana let r balls be 
 drawn and the number of black be recorded. If r> pn, the range of 
 black balls will lie between 0 and pn; the resulting frequency poly- 
 gon will be skew and limited in range."* This polygon is given by 
 a hypergeometric series from which he obtains the differential equa- 
 tion 
 
 1 ^ = a+x , 
 
 y dx b4cx+dx^ 
 
 In oraer to get this in the form. y=f(x), it is necessary to integrate 
 
 3/^X 
 
 , and the form the integral takes depends upon the values 
 
 b+cx+dx® 
 
 of the coefficients of x in the denominator. Thus the criterion for 
 the form in a particular case is the same as that for the nature of 
 
' * Vt:. -7 I'j 
 
 o a »: 7 
 
 ..«--l;ftf- rv: ;r. f^CLMUaOr j; t-,;. 
 
 : -ri j.r n .. i v.?.‘Xi. 
 ■ - •••--■ ■ 
 
 T . r 'i V f ' . • *« . 
 
 *‘ ■ • - . . _ ' .f. 
 
 i£ia. 
 
 T- • - i.<:U .. . -x-i 
 
 ^ • * 0- ^ Am 
 
 
 "<■ 
 
 •• N *• 
 
 
 tn:' i. 
 
 
 \ - 
 
 V - '-v. 
 
 V ^ 
 
 
 - -> ^;ji? p 5^-52, , 
 
 “ vr- .rn 
 
 /w 
 
 - . 1 
 
 - — L 
 
 M.*' ox *•. 
 
 fc ’^Tt; 
 
 ^ .. 
 
 
 ■ • .. -I 
 
 
 - ^-- ^ -X A 
 
 : 
 
 . -V-;: _ ,:, 
 
 . V . 
 
 * w ( • - •«* 
 
 • » 4 
 
 - « I J . 
 
 •‘ jC 
 
 - I'A, ' 
 
 ' ^ xOS-lL’ 
 
 *. r ;. 7 7 
 
 - •“ '• '•■ , 
 
 • •■ * 
 
 1*^ 
 
 
 ^ - i . .’ ' 
 
 • 
 
 “ - . i-Z 
 
 •1. • i.1' .. 
 
 • -: . 
 
 . ; A:z: ■ 
 
 ~ ' ' n n.-i* ^ 
 
 :o 
 
 i ;: d. 
 
 : '-/vv 
 
 £[ ^ 
 
 ■.. ' .:) Il: 
 
 J . -> - t 
 
 ? f ^ 
 
 10. . 
 
 f - 
 
 - m *ir 
 
 
 •>..^•L »;i . - . . .• , , ^ . 
 
 '4. 
 
 *. V ^*V>-T StO' 
 
 l-.J, 
 
 * '*■■■>' --.ii T -1. iT 
 
 /Z;r!-r 
 
 - • 
 
 ‘ * • 
 
 r,^ 
 
 • > ^ tS' J'^ 
 
 ‘* ' - - 
 
 * - ’'i.. 7 . 5j||^, , 
 
 
 
 i. z 
 
 ' ' . '■ ^'' ' 
 
 ' - -ii -^lO . {■ rrii 
 
 
 
 . Ai , 
 
 ,; •: xilTi iV - X 
 
 
 
 '■C'-" L: 
 
 
 V ^ 
 
 -*• ■. 
 
 ^.vr,. H?* 
 
 i: •■ . 
 
 >J. V 
 
 
 t '/ / '!' ' 4lo :: 
 
 ^ .. ^ - ‘ ? 'wi n. t , 
 
 t .ff 
 
 u 4: 4>4i(4"' (|'T :.' X i': r ,. j„ .v ^ 
 
 ^ J 'Xii 3^ /*i£, X re 
 
 
 *. * ** rw , 
 
 Lj.'v ::3 
 
 -' j. 'Mb'. 
 
 
 
17 
 
 the roots, nar/.ely, the discriminant: yc^-4hd, which Pearson puts in 
 
 the form / t, allowing to equal k. Then, according as k=0, 
 
 ^4bd “ ^ 
 
 k^l, or k<l, certain equations result and fit various cases of 
 skewness and range. 
 
 If the differentia^ equation is integrated with respect to x, 
 after multiplying each side by x^, according to the method used by 
 Elderton,^ the coefficients of the powers of x can be found in the 
 terms of the moments. Letting /3 = ^ and 02 ~ ^ 
 
 
 
 pressed in terms of the/3’s as given in Table IV. We have then, the 
 following table: 
 
 k<0 
 
 Type 
 
 I 
 
 k = 0 02 ~^ 
 
 Type 
 
 VII, 
 
 k = 0 
 
 Type 
 
 II 
 
 0<k<l 
 
 Type 
 
 IV 
 
 1 — 1 
 II 
 
 Type 
 
 V 
 
 k >1 
 
 Type 
 
 VI 
 
 k = 00 
 
 Type 
 
 III. 
 
 VII. CLASSIFICATION OF DISTRIBUTIONS ACCORDING TO TYPES 
 Using k, then, according to the above table, I found that the 
 distributions of heights always approached very nearly tne normal 
 curve, in fact to all appearances as near that as to any curve, al- 
 though according to k ana 0 ^ the distributions of the lb) year men, 
 the v;omen, ana the freshman class groups were fit more closely by 
 type IV. The 17 year group with its neirative k is classed v/ith Type 
 I. To compare the fit of the two curves (the curve of Type IV 
 and the normal curve) I calculated the equations of the two, for 
 the heights of women and of the freshman class, ana plotted them- 
 
 1. Frequency Curves and ’ Cor r el at i 
 
 on, p 3S. 
 
t . 
 
 ‘ •- 
 
 
 -,-^V i*>nt: ' iX'l XcUi -^y^od 
 
 I -5 
 
 
 
 ' »*:a 
 
 .3 
 
 - 
 
 », . j^'uV&4|6 ,4^^' 
 
 it * ■ ' •• j ■ y" ■ '•' 4 ■’^ ‘ ■ '■'JS- t ■ ■■1®' IT 
 
 ^1 1:-“’'* ®' ’ ■^'•^;"^l!0 
 
 .'ft v£df.r‘ ».*^v«i^<i§.'8isWfcn£4i^^ 
 
 ■- X ■ -• ^ ■ I5®IW 
 
 i: .ner-t,. 
 
 ‘^ 3^Ti 
 
 ^ , ,1 Y'' »-,y 
 
 •* 
 
 
 . -vvrO iU.V lyritt ,{ 5]>'- ' • 
 
 ,,, >9 
 
 I ” • - f • _ L.’ * • ' -w — ^ -M ^ . 
 
 
 
 i' 
 
 ir 
 
 St. "^ix.' .y 
 
 • -V V I) 
 
 t>.a>: 
 
 ■h 
 
 • *$ '\ - * 
 
 .y' ’.<' *. -'vV ■; • • ^ 
 
 kh: 
 
 fei.' ’ ■■ 1 ■■"■!■'■ ■ ^■■. ■ , 16,^' ^ 
 
 {} ■ <■: i I ,^ Ip'^-,l itTw3 (Jj^i ;■. 5 m.!,,^. 5»4« - x^i. .4^4, 
 
 
 ■ ,i 
 
 r* . *i ^ Ei W IGpijL^ ..^^, 
 
 '»• •. %?■ 
 
 . y ■ .-'T ' • y* IT 
 
 '■ ' -1 \ ■•. -.rf- 
 
 ■#«flS' »■'.*•,'»«> -- VW'f v.A 
 
18 
 
 following the method of Elderton, pp 6y-73. The calculation of the 
 constants for the equation of Type IV, 
 
 is shown in Tables TV and VII, as worked out for the freshman group. 
 From the graphs of the two curves for both the freshman and women, 
 it seems that the normal curve is as close, if not a better fit than 
 that of Type IV (Figs.l and 6). However, it may be that the e 3 '’e can 
 not judge accurately from, figures the closeness of fit, and it miay 
 be tnat k has a relatively large probable error such that the normal 
 curve is really the better fit. 
 
 The types represented by the distribution of weights are three* 
 Types IV, V and VI. The 17 year, 18 year, and 20 year men are class-' 
 ified under Type IV, wnile the 19 year and freshman groups are rep- 
 resented by Type VI, and the w'eights of women are fit best by a 
 curve of Type V. From* the figures (7--12) showing the distribution 
 of weights, it is easily seen that they are not fit by the normal 
 curve. In two cases I have drawn the curves which fromi k snould 
 better represent the data, and from figures 7 and 12 we see that 
 they do. The constants of the equation of Type V: 
 
 were calculated and the ordinates obtained as shown in Tables XI an 
 
 and the calculation of these constants and the curve is shcwmi in 
 
 For each of the six groups, the frequency polygon of both 
 
 1. See Elderton , pp 78-80. 
 
 2. See Elaerton, pp 8o-85. 
 
 y = 7o X 
 
 XII?’ The equation of Type Vi is 
 
 y = yQ(x-a )^2 
 
 Tables IX and X 
 
 .2 
 
19 
 
 height and. weight was drawn, and on the polygon was graphed the nor- 
 mal curve for the respective data. To get the ordinates for the 
 normal curve I used Sheppard's tables, wnich require that the devi- 
 ations be expressed in terms of the standard deviation of the given 
 
 data, tnat is, as The z which is given in the tables is equal to 
 1 _ 1 x2 
 
 e ^ ^ for corresponding ^ 's. Since tne equation for the 
 
 N -1 
 
 normal curve is y = — ;■ ■ ■ ' ■ , it is necessary to multiply the z 
 
 crysTT ® 
 
 given in the tables by ^ in order to have the proper ordinates for 
 
 the given distribution. An example is given by Table VI, Since for 
 the normal curve the origin is at the mode, and the mode and the mear 
 are the same, we have yo(= — — ) a-s the maximum and central ordinate, 
 corresponding to the abscissa represented by the mean. 
 
 VIII. CALCULATION OF CURVE OF TYPE IV. 
 
 It was shown above that the distribution of heights of the 
 freshman group was among those which were better fit by type IV. 
 The equation of this curve is 
 
 y 
 
 yod 
 
 -V tan 
 e 
 
 and in order to graph it, it is necessary to evaluate the constants 
 a, m, V, and y^. These were calculated by means of the formulae on 
 p 69 of Elderton, Each of these constants can be expressed in terms 
 of another constant r, the value of which in terms of /3]_ and /J 3 is 
 derived in Elderton (pp 74-76). The calculation is simple until the 
 integral G(r,v) is met in the formula 
 
 V — N e 
 
 o aG(r , v) * (a) 
 
 In deriving the value of y^ (See Elderton, p 74), the integral 
 
20 
 
 
 
 TABLE 
 
 VI 
 
 
 
 
 
 Calculation of 
 
 the Normal 
 
 Curve 
 
 
 Example: 
 
 Distribution of Heights 
 
 of 1004 Freshmen Men, 
 
 Age 18. 
 
 y = 
 
 x^ 
 
 2(T3 
 
 ^0 ® 
 
 
 V J.. 
 
 - = 170.4923 
 
 
 
 
 
 ^ -v/2tt jla 
 
 2 
 
 
 
 
 Origin = Mean 
 
 = 67.73556 
 
 
 
 ^ = 2.3493 
 
 ^ = .42566 
 
 N 
 
 O' 
 
 = 427.3613 
 
 
 
 
 Normal ^ 
 
 Curve 
 
 
 
 He ignt 
 
 Frequency ^ 
 
 z 
 
 N 
 
 2 • cr 
 
 y 
 
 59 
 
 1.5 
 
 3.71838 
 
 . 0003968 
 
 0.170 
 
 0.2 
 
 60 
 
 1.5 
 
 3.29272 
 
 .0017646 
 
 0.754 
 
 0.8 
 
 61 
 
 3.0 
 
 2.86706 
 
 . 0065461 
 
 3.798 
 
 5.0 
 
 62 
 
 4.0 
 
 2.44140 
 
 . 0202597 
 
 8.658 
 
 9.0 
 
 63 
 
 21.5 
 
 2.01574 
 
 .0523132 
 
 32.357 
 
 22.0 
 
 64 
 
 50.0 
 
 1.59008 
 
 .1136899 
 
 48.16 
 
 48. 0 
 
 65 
 
 88.5 
 
 1.16442 
 
 . 2035293 
 
 86.55 
 
 87.0 
 
 66 
 
 123.5 
 
 0.73876 
 
 . 3036668 
 
 129.78 
 
 130.0 
 
 67 
 
 165.5 
 
 0.31310 
 
 . 3798557 
 
 163.34 
 
 162.0 
 
 68 
 
 172.5 
 
 0.11356 
 
 . 3964193 
 
 169.41 
 
 169.0 
 
 69 
 
 160.0 
 
 0. 53823 
 
 .3451472 
 
 147.50 
 
 147.5 
 
 70 
 
 98.0 
 
 0.96388 
 
 . 2507066 
 
 107.14 
 
 107.0 
 
 71 
 
 53.5 
 
 1.38954 
 
 .1519282 
 
 64.93 
 
 65. 0 
 
 72 
 
 o 
 
 • 
 
 1 — 1 
 
 1.81520 
 
 .0768127 
 
 32. 83 
 
 33. 0 
 
 73 
 
 13.5 
 
 2. 24086 
 
 .0323983 
 
 13.80 
 
 14.0 
 
 74 
 
 4.5 
 
 3.66652 
 
 .0114012 
 
 4.87 
 
 5.0 
 
 75 
 
 0.5 
 
 3.09218 
 
 .0035471 
 
 1.43 
 
 1.0 
 
 76 
 
 1.5 
 
 5.51784 
 
 .0008198 
 
 0.35 
 
 0.4 
 
 1003.837 
 
 1003.9 
 
22 
 
 4> desired; the difference between Au^ and its succeeding value 
 
 as given in tne table. The value of log H(r,v) known, F(r,v) was ob- 
 tained from equation (c) ana substituted in the equation 
 
 a ■F(r,v) 
 
 obtained from equations (a) and (b) . 
 
 After the constants had been evaluated. Table VIII was made 
 
 and the ordinates calculated. The origin for Type IV is equal to 
 
 the mean - and for the freshman group is 66.07. The mode is 
 1 /H'i> r-2 
 
 the mean - — which in the example cited is 67.6, and the cor- 
 
 responding ordinate 202.77. 
 
 Vk. CALCULATION OF THE CURVE OF TYPE VI 
 The distribution of the xveights of the freshman group was 
 found to approach most nearly the curve represented by the equation 
 of Type VI: 
 
 ^2 
 
 y = y^Cx-a) x ^ . 
 
 The three constants, a, qg, q^ can be expressed, too, in terms of /3;j_, 
 and r, which is derived in the discussion of Type I, pages 59-60, of 
 Elderton. However, in getting the value for y^, it is necessary to 
 
 evaluate the so-called T functions. in deriving the formula for y^, 
 
 X T "p—i q— 1 
 
 e^ (1-x) dx appears, and is another one 
 which can not be expressed in terms of elementary functions. This 
 integral is called the fi function and is represented by the symbols 
 
 /3(p,q) . There is another so-called non-integrable expression: 
 
 p-1 -n 
 
 I x^ dx which is terrr.ed the / function of (o) and it has beer 
 
 Vo ‘ 
 
 found that the l3 function can be expressed in terms of the / function 
 
 This relation is expressed by the following equation:^ 
 
 /3 (p Q ) »Ti£lZM 
 
 ’ T (p+q) 
 
 1. Elderton, Appendix Ii, p 152-156. 
 
23 
 
 TABLE VII 
 
 Calculation of for Type IV 
 Exaimple: Distribution of Heights of 1183 Freshmen. 
 
 ^ VTT 
 
 aG(r^v) 
 
 f,f- _ F(r,vl 
 
 e ^ G(r, v) 
 
 
 r+1 
 
 N = 1182 
 r = 34.97402 
 
 a = 13.76545 
 V = -4.13389 
 
 F(r,v) = e_l££|^)_H(r.v) 
 
 tan cj> = “ = -.118170 
 
 (f> = 6°. 74738 = -.1176256 (circular measure) 
 
 For log H(r, v) : 
 
 r = 34 r = 35 
 
 cj? = 6° .3894611 .3897389 
 
 4> = 7° .3894778 .3897551 
 
 cf> = 67. 74738 r = 34 
 log H(r,v) = .3894611 + (. 74738) [l67j - -|( . 74738) (. 25262) [25] 
 = .3894733 
 
 4>= 6®. 74738, r = 35 
 
 log H(r,v) = .3897389 + (. 74738) fl62] - •!(. 74738) (. 25262) [2^ 
 
 = .3897508 
 
 (p = 6°. 74738, r = 36 
 
 log H(r.v) = .3900011 + (. 74738) [l58] - -g(. 74738) (. 25262) [ 24 ] 
 
 = .3900127 
 
 Hence 6®. 74738, r = 34.97402 
 
 log Jl(r,v) = .3894733 + (. 77402) [2775;i - . 97402) (. 02598) [rl56] 
 
 = .3897438 
 
 log F(r,v) = V cj> log e + (r+1) log cos^ --3 log(r-l)+ lQ©H(r,v) 
 
 = -.27303500 
 _ N 1 
 
 y© = ^ 
 
 ^ F(rQ^v) 
 
 = 161.013 
 
 log y^ = 2.20686204 
 
TABLE VIII 
 
 Calculation of tne Curve for Type IV 
 
 Example : Distribution of ueignts of 1182 Eresnmen. 
 
 . -V tan''^ 
 
 y = yo(l+ — ) e 
 
 Origin = Mean + = b7, 69925858 ~ 1.62666655 = 66.07257223 
 
 r 
 
 Mode = Mean - ^ = 67.61124887 
 
 24 
 
 c 
 
 
 * 
 
 
 t 
 
 
 
 
 JL 
 
 a 
 
 
 i 
 
 / X 
 a 
 
 Co / (4) cn, 
 C-LTe., mess. 
 
 Co/.toX Co/.{5)x 
 
 Y 
 
 -.5158 
 
 1.2640 .;017 
 
 27° 11 
 
 •37”^ 
 
 -.4746 
 
 -.8519 -1.8809 
 
 1.4741 
 
 0.3 
 
 -.4412 
 
 1.1946 .0772 
 
 25°48 
 
 •17« 
 
 -.4155 
 
 -. 745? -1.4277 
 
 . 0365 
 
 1.0 
 
 -. 5686 
 
 1.1558 .0553 
 
 20^15 
 
 •44” 
 
 -. 65o1 
 
 — . *4.. 022o 
 
 . OOO'd 
 
 4. 0 
 
 — . 2959 
 
 1.0875 .0564 
 
 16° 28 
 
 •51" 
 
 -.2876 
 
 -.5163 -.6?67 
 
 1. 0169 
 
 10.0 
 
 -.2252 
 
 1.0498 .0211 
 
 12° 34 
 
 •58 ” 
 
 -.2196 
 
 -.3942 -.6904 
 
 1.4226 
 
 26.0 
 
 -. 1506 
 
 1.0227 .0097 
 
 8°35 
 
 •44 ” 
 
 -.1494 
 
 -.2686 -.1800 
 
 1.7587 
 
 o7,0 
 
 -.0779 
 
 1,0061 .0026 
 
 4° 27 
 
 • 20 ” 
 
 -.0778 
 
 -.1396 -.0486 
 
 2. 0187 
 
 104.0 
 
 -. 0055 
 
 1.0000 ,0000 
 
 0°18 
 
 1 711 
 
 -. 0055 
 
 -.0095 -.0002 
 
 2, 1971 
 
 157.5 
 
 . 0674 
 
 1.0045 .0020 
 
 6 ° 51 
 
 •15" 
 
 .0673 
 
 .1207 -.0364 
 
 2.2912 
 
 195.5 
 
 .1400 
 
 1.0196 .0084 
 
 7° 58 
 
 •15” 
 
 .1391 
 
 .2497 -.1559 
 
 2. 5006 
 
 200.0 
 
 .2127 
 
 1.0452 .0192 
 
 12 ° 0 
 
 • 22 ” 
 
 .2095 
 
 . 5761 — . 5552 
 
 2.2278 
 
 169.0 
 
 .2853 
 
 1.0814 .0340 
 
 15° 55 
 
 •26” 
 
 .2779 
 
 .4988 -.6283 
 
 2.0774 
 
 119.5 
 
 .3580 
 
 1.1281 .0524 
 
 19°41 
 
 •43” 
 
 .5437 
 
 .6170 -.9680 
 
 1.8558 
 
 72.0 
 
 .4306 
 
 1.1854 .0739 
 
 23° 17 
 
 •48” 
 
 .4066 
 
 .7298 -15655 
 
 1.5711 
 
 57.0 
 
 . 5053 
 
 1. 255P . 0980 
 
 26°42 
 
 •50"’ 
 
 , 4662 
 
 .8369-1.8125 
 
 1. 2312 
 
 17.0 
 
 .5759 
 
 1.3517 .1244 
 
 29° 56 
 
 •14” 
 
 . 5225 
 
 .9378 -2.2996 
 
 .8451 
 
 7.0 
 
 .6485 
 
 1.4206 .1525 
 
 52°57 
 
 •54” 
 
 ,5755 
 
 1.0327-2,8187 
 
 .4208 
 
 3.0 
 
 .7212 
 
 1.5201 .1819 
 
 o5°47 
 
 •54” 
 
 .6248 
 
 1.1214 45.5623 
 
 1.9660 
 
 1.0 
 
 .7958 
 
 1.6502 .2122 
 
 68 ° 26 
 
 •37" 
 
 .6710 
 
 1 . 2043 -3 . 9256 
 
 1.8876 
 
 0.3 
 
 ie. 
 
 .1118 
 
 
 
 
 
 
 
 1181.1 
 
 1.0125 .0054 
 
 6° 22 
 
 •41” 
 
 ,1113 
 
 .1198 -.0997 
 
 2.5070 
 
 202. 77 
 
 An eight-place logarithmic table was used in the actual calculation 
 of this table. 
 
 t 
 
 The values in circular measures were obtained to seven places. 
 
 These values were worked out to five places. 
 
35 
 
 The reason for expressing in terms of T rather than /3functicns 
 is that tables of r functions have been made. Due to the charac- 
 teristic property of T functions, namely, T(p+1) = pT(p), it is 
 necessary to tabulate only those values of the function from 1 to 2 
 
 since all others can be reduced to a value in that interval, How- 
 
 1 
 
 ever, for very large values of p, there is a close approximiat ion 
 of the form: 
 
 logT(x+l) = logy'^ + (x + ^)log X - (x - 1 ^) ® 
 
 This approximation was used in evaluating y^ (Table IX), 
 
 After the value of log y^ v/as obtained, a table for the calcu- 
 lation of tne curve was made, and the ordinates calculated (Table X) . 
 The origin of this type is equal to the mean - — ; instead of 
 using the mean expressed in pounds, it is necessary to use it in 
 termjs of unit intervals which in the example is 27.187; this puts 
 the origin at -71.6. The miOde is equal to the miean - 
 which in tnis case equals 26.12. This, in terms of pounds, is 130.6, 
 and the corresponding ordinate is 143.5, 
 
 X. CALCULATION OF THE CURVE OF TYPE V 
 The curve of Type V, expressed by the equation 
 
 -p 4 
 
 y = yjj X e ^ 
 
 proved to fit very closely the distribution of the weights of women 
 (See Fig. 12). The formulae for the three constants y^, p and y are 
 given in Elderxon ( p 78) and the proof for these formulae on page 
 82. TheT’ function is again met in tne calculation of Yq (for ex- 
 planation of the T' function see tne discussion of the Curve of Type 
 VI). In the example given (Table XI) this function was easily 
 
 1. Proof of this approximation is given in Elaerton, p 15b. 
 
 
26 
 
 TABLE IX 
 
 Calculation of Constants for Type VI 
 Example : Distribution of Weignts of 11S2 Freshmen 
 
 Q2 ' ~^1 
 y = yo(x-a) x 
 
 g^i3p-A-l) 
 
 = 6 r Ji - 202 
 
 Q 2 and -q]_ are given by 
 
 log r(117. 592096) = .59908995 + (117. 092096) (2. 06666911) - 
 
 (116.59158126) (.43429448) 
 
 = 191.75471442 
 
 log T(q 3 _-q 2 -l) = log T(103. 176826) 
 
 = 162.33854039 
 log T(q2+1^' = log T(1441527) 
 
 = 10.26646036 
 
 13.415270 
 
 -117.592096 
 
 Using the approximation: 
 
 log yo = 222.11534746 
 
27 
 
 TABLE X 
 
 Calculation of the Curve of Type VI 
 
 Example : Distribution of Weights of 1182 Freshmen 
 
 y = yo(x-a) ^ X ^ 
 
 Origin = Mean - = 27.187394 - 98.786951 = 71.599557 
 
 T *T»X Q 
 
 Mode = Mean - ~ ZT = 26.121887 
 
 2/^3 
 
 X 
 
 X 
 
 lo^X* 
 
 1.9523 
 
 f ^ 
 
 .4809 -229.5758 
 
 6.4519 
 
 4: 
 
 y 
 
 y 
 
 y 
 
 18 
 
 89.5996 
 
 2.9914 
 
 .098 
 
 0.1 
 
 19 
 
 90. 5996 
 
 1.9571 
 
 .6049 -230.1425 
 
 8.1152 
 
 .0880 
 
 1.225 
 
 1.0 
 
 20 
 
 91.5996 
 
 1.9619 
 
 .7013 -230.7031 
 
 9.4076 
 
 .8198 
 
 6. 605 
 
 7.0 
 
 21 
 
 92.5996 
 
 1.9666 
 
 .7801 -231.2577 
 
 10.4648 
 
 1.3224 
 
 21.010 
 
 31.0 
 
 22 
 
 93. 5996 
 
 1.9713 
 
 .8467 -231.8062 
 
 11.3592 
 
 1.6683 
 
 46.594 
 
 47.0 
 
 23 
 
 94. 5996 
 
 1.9759 
 
 .9045 -232.3489 
 
 12. 1344 
 
 1,9008 
 
 79.591 
 
 80.0 
 
 24 
 
 96.5996 
 
 1.9805 
 
 .9555 -332.8860 
 
 12.8185 
 
 2.0479 
 
 111.662 
 
 112.0 
 
 25 
 
 96.5996 
 
 1.9850 
 
 1.0011 -233.4174 
 
 13.4307 
 
 2. 1286 
 
 134.477 
 
 134.5 
 
 26 
 
 97.5996 
 
 1.9894 
 
 1.0424 -233.9434 
 
 13.9846 
 
 2.1565 
 
 143.404 
 
 143. 0 
 
 27 
 
 98.5996 
 
 1.9939 
 
 1.0801 -234.4639 
 
 14.4903 
 
 2.1417 
 
 138.596 
 
 139. 0 
 
 28 
 
 99 . 5996 
 
 1.9983 
 
 1.1148 -234.9792 
 
 14.9557 
 
 2.0918 
 
 123.544 
 
 123.5 
 
 29 
 
 100.5996 
 
 2.0026 
 
 1.1469 -235.4895 
 
 15.3866 
 
 2.0125 
 
 102.924 
 
 103.0 
 
 30 
 
 101.5996 
 
 2. 0069 
 
 1.1769 -235.9946 
 
 15.7879 
 
 1.9085 
 
 81.015 
 
 81.0 
 
 31 
 
 102.5996 
 
 2.0111 
 
 1.2048 -236.4948 
 
 16 . 1632 
 
 1.7838 
 
 60.788 
 
 61.0 
 
 32 
 
 103.5996 
 
 2.0154 
 
 1.2311 -236.9902 
 
 16.5159 
 
 1.6410 
 
 43.757 
 
 44.0 
 
 33 
 
 104.5996 
 
 2.0195 
 
 1.2559 -237.4808 
 
 16.8484 
 
 1.4829 
 
 30.407 
 
 30.0 
 
 34 
 
 105.5996 
 
 2.0237 
 
 1.2794 -237.9667 
 
 17. 1629 
 
 1.3116 
 
 20.494 
 
 20.5 
 
 35 
 
 106.5996 
 
 2.0278 
 
 1.3016 -2o8.4480 
 
 17.4614 
 
 1.1287 
 
 13.452 
 
 13.5 
 
 36 
 
 107.5996 
 
 2.0318 
 
 1.3228 -238.9248 
 
 17.7453 
 
 .9358 
 
 8.626 
 
 9.0 
 
 37 
 
 108.5996 
 
 2. 0358 
 
 1.3429 -239.3973 
 
 18.0160 
 
 .7340 
 
 5.420 
 
 5.0 
 
 38 
 
 109.5996 
 
 2. 0398 
 
 1 . 3622 —239 . 8654 
 
 18.2746 
 
 .5245 
 
 3. 346 
 
 3.0 
 
 39 
 
 110.5996 
 
 2. 0438 
 
 1.3807 -240.3292 
 
 18.5223 
 
 .3084 
 
 2.035 
 
 2.0 
 
 40 
 
 111.5996 
 
 2.0477 
 
 1.3984 -240.7889 
 
 18. 7599 
 
 . 0863 
 
 1.220 
 
 1.0 
 
 41 
 
 112.5996 
 
 2.0515 
 
 1.4154 -241.2445 
 
 18.9882 
 
 1.8589 
 
 .723 
 
 0.7 
 
 1181.8 
 
 ^oJe. 
 
 97.7214 
 
 1.9900 
 
 1.0472 -234.0071 
 
 14.0486 
 
 2, 1568 
 
 143.506 
 
 
 * Six decimal places were used in the original calculation. 
 
28 
 
 evaluated both by the approximation method used for Type VI and by 
 making use of the cnaracter ist ic property of the T function and the 
 table of log T' (p) . The latter method is shown in the table. 
 
 For this type the origin is expressed as mean - and using 
 
 unit intervals, the mean for the example is 2b. 59 and the origin at 
 
 10.003, The mode is equal to rne mean which in the ex- 
 
 p(p-3) 
 
 ample is 32.05, and its corresponding ordinate is 125.9 (Table XII.) 
 
 XI. MEANiIJU OF CORRELATION 
 
 Without going into technicalities, tne meaning of correlation 
 is easily explained. If we have two series of paired numbers, e.g. , 
 Heights and arm lengths of each individual of a group, or prices of 
 flour and of cotton on certain dates, or marks in two school subjects 
 of individual pupils of a school, we are interested in the connec- 
 tion, if any, between the two sets of figures. There may be little 
 or no connection between two such sets of numbers as in 'cne case of 
 ages and heights of a group of adults. Again, there may be a very 
 close relation, as would be shown in the case where the pairs of num- 
 bers are the radii and corresponding circumferences of a set of cir- 
 cles. It is clear that the set of pairs of numbers representing 
 height and arm length is different from these two extreme cases. We 
 express the connection between arm length and height roughly by say- 
 ing that in general a tall man has a long arm, but given the height 
 of a man we cannot compute the length of his arm. We need something 
 besides the words "in general" to tell us how the two sets of numiber£ 
 are related or correlat ed . we seek then some number which will 
 measure for us the degree of this relationship and of this correla- 
 t ion between paired numbers. Many measures have been devised but the 
 one in widest use is tne measure often called Pearson's coefficient 
 

 i ' * ^ 
 
 _____ , . . y ■ ? ^ ^ ■' 
 
 J* - ■ ' ■ ■ ' ■ 0* ,-. 'C;‘>'.. *'.- V .v "I ■ ■ '*'*'vi''''tj|^*‘ «! 
 
 
 
 .•y>. 
 
 
 !;I ff'vV ^ ’Sf^'UiJXPf'Ca .«.■■ )*r W ^'i 
 
 f i t . - Kp ,n^' 
 
 ..^^. rjo% a I ai’xri. 'iO 
 
 ‘■^'irv:/ 'Qof <e^ k* ' 3 ;^ 
 
 w' ;, ■■., ^ \-._,r "r-' 
 
 p ^,*jtsir,p.<if iit Fts £i-^>yiAun ?;iip it '■ 
 
 ■r' 6 - • • ■ •■ ^ •■ ■ 
 
 L- - -. • • ‘ ■ ' '.'^ 'V '■ ■ :m 
 
 ^.rf^ i ‘‘-j T*f - f* A .1 A.>. # ♦ A I • - 4 . -^ • ik . ^ « ««>/ ■ 
 
 ‘ • * ). .»>- . -L t . ‘"iM ■ ‘ ■ ; 
 
29 
 
 TABLE XI 
 
 Calculation of Constants for Type V 
 Example : Distribution of Weights of 971 College Women. 
 
 y = y, 
 
 *“P X 
 X e 
 
 p = 4 + 8 = 17.652903 
 
 Pi 
 
 Y = (p-2)^A^Tp-^ ~ 212.601063, the sign of y is the same as tnat of>- 
 
 . p-1 
 
 T(ip-i) 
 
 Since T(p) = (p-l) r(p-l) , 
 log T(16. 6529) = log 15.6529 = 1.19459481 
 
 14.6529 = 1.16592359 
 
 13.6529 = 1.13522491 
 
 12.6529 = 1.10319008 
 
 11.6529 = 1.06643402 
 
 10.6529 = 1.02746785 
 
 9.6529 = .98465781 
 
 8.6529 = .93716168 
 
 7.6529 = .88382604 
 
 6.6529 = .82301100 
 
 5.6529 = .75227130 
 
 4.6529 = .66772372 
 
 3.6529 = .56263778 
 
 2.6529 = .42372088 
 
 1.6529 = .21824658 
 
 log T(1.6529)= 1.9545121 
 
 12.89960415 
 
 log Vq = 28.848329 
 
K^flXv.rrDO 76 
 
 
 / 
 
 • j* %• W# - 
 
 •ca 
 
 r«>< 
 
 cv'^- t-ir 
 
 . 
 
 -^v.r,v\ 
 
 r"'*... -"• 
 
 •Li 
 \ 
 
 W-''‘0:' 
 
 y_ 
 
 5. 
 
 '0- 
 
 
 •\ 
 
 • TT et - 
 
 It d.:j < 
 
 > I 
 
 •‘*;-nr 
 
 
 
 » i, {:r\^ 
 
 • • ■ :*i - - •• *- . GioX 
 
 ■3 
 
 L^v.f.vKO'* 
 
 ' ^ 
 1 
 
 ^ . (, 
 
 » '.r* . . .■ 
 
 f . Jxijoi 
 
 * .V --; n. ^ j!;^ 
 
 .rj n* .rr^ 
 
30 
 
 TASLE XII 
 
 Calculation of the Curve of Type V 
 
 Example : Distribution of Weights of 971 College Women. 
 
 Y 
 
 y = X 
 
 Y 
 
 “P e X 
 
 Origin = Mean - — r- = 23.585479 - 13.582313 = 10.003266 
 
 p-3 
 
 Mode = Mean - 
 
 2V 
 
 p(p-2) 
 
 22.046671 
 
 X 
 
 X 
 
 log X -p log X 
 
 1 
 
 X 
 
 (-yiog e) 
 
 log y 
 
 y 
 
 y 
 
 17 
 
 7.003266 
 
 .84530062 -14.922010 
 
 
 13. 184058 
 
 . 742261 
 
 5.5241 
 
 5 . £ 
 
 18 
 
 8. 003266 
 
 .90326725 -15.945289 
 
 
 11.536724 
 
 1. 366316 
 
 23. 244 
 
 23. C 
 
 19 
 
 y.00o266 
 
 .95440009 -16.847932 
 
 
 10.255330 
 
 1.745067 
 
 55.599 
 
 56. C 
 
 20 
 
 10.003266 
 
 1.00014181 -17.655406 
 
 — 
 
 9, 230132 
 
 1.962791 
 
 91.789 
 
 92. C 
 
 21 
 
 11.003266 
 
 1.04152161 -18.385880 
 
 
 8.391278 
 
 2.071171 
 
 117.807 
 
 118. C 
 
 22 
 
 12.003266 
 
 1.07929943 -19.052768 
 
 — 
 
 7.692195 
 
 2. 103366 
 
 126.872 
 
 127. ( 
 
 23 
 
 lo . 003266 
 
 1.11405244 -19.666260 
 
 — 
 
 7. 100637 
 
 2.081432 
 
 120.624 
 
 121. C 
 
 24 
 
 14. 003266 
 
 1.14622934 -20.234275 
 
 
 6.593567 
 
 2.020487 
 
 104.831 
 
 105. C 
 
 25 
 
 15. 003266 
 
 1.17618581 -20. 76o094 
 
 — 
 
 6.3.54091 
 
 1.931144 
 
 85.338 
 
 85. C 
 
 26 
 
 16.003266 
 
 1.20420862 -21.257778 
 
 - 
 
 5.769539 
 
 1.821012 
 
 66. 224 
 
 66. C 
 
 27 
 
 17.003266 
 
 1.23053235 -21.723469 
 
 — 
 
 5.430214 
 
 1.695646 
 
 49.619 
 
 50. C 
 
 28 
 
 18.003266 
 
 1.25535130 -22.160595 
 
 
 5. 128595 
 
 1.559139 
 
 36 , 236 
 
 36. C 
 
 29 
 
 19.003266 
 
 1.27882824 -22.575031 
 
 — 
 
 4.858716 
 
 1.414582 
 
 25.977 
 
 26. C 
 
 30 
 
 20. 003366 
 
 1.30110090 -22.968208 
 
 — 
 
 4.615820 
 
 1.264d01 
 
 18.678 
 
 18. C 
 
 31 
 
 21.003266 
 
 1.32228683 -23.342201 
 
 — 
 
 4.396053 
 
 1.110075 
 
 12. 885 
 
 13. C 
 
 32 
 
 22. 003266 
 
 1.34248715 -23.698795 
 
 — 
 
 4. 196262 
 
 .953272 
 
 8.979 
 
 9.C 
 
 33 
 
 23.003266 
 
 1.36178950 -24.039538 
 
 — 
 
 4.013842 
 
 . 794949 
 
 6. 236 
 
 6. C 
 
 34 
 
 24.003266 
 
 1.38027034 -24.365778 
 
 — 
 
 3. 846621 
 
 .635930 
 
 4.624 
 
 4.( 
 
 35 
 
 25.003266 
 
 1.39799674 -24.678701 
 
 — 
 
 3.692776 
 
 .476852 
 
 2.998 
 
 3.C 
 
 36 
 
 26. 003266 
 
 1.41402790 -24.961697 
 
 — 
 
 3.550764 
 
 . 335868 
 
 2.167 
 
 2.C 
 
 37 
 
 27.003266 
 
 1.43141630 -25.268653 
 
 — 
 
 3.419270 
 
 . 160406 
 
 1.446 
 
 l.£ 
 
 38 
 
 28. 003266 
 
 1.44720869 -25.547435 
 
 - 
 
 3. 297168 
 
 .003726 
 
 1.00$ 
 
 l.C 
 
 39 
 
 29.003266 
 
 1.46244690 -25.816433 
 
 — 
 
 3. 183485 
 
 9.848411 
 
 .705 
 
 l.C 
 
 40 
 
 30. 003266 
 
 1.47716853 -26.076313 
 
 - 
 
 3.077381 
 
 9.694635* 
 
 .495 
 
 0.£ 
 
 41 
 
 31. 003266 
 
 1.49140745 -26.327671 
 
 - 
 
 2.978121 
 
 9.542537 
 
 .348 
 
 0. 
 
 
 
 
 
 
 
 
 969.6 
 
 
 12.043405 
 
 1.08074929 -19.078362 
 
 — 
 
 7.666558 
 
 2. 103409 
 
 126.885 
 
 
 * -10 
 
'61 
 
 of correlation, or the product -morrient coefficient, or simply the 
 correlation coefficient . It is universally represented by the lettei 
 
 r. 
 
 The correlation coefficient varies from -1 to +1. For the ex- 
 tremes of the range the correlation is perfect . In perfectly cor- 
 related series, given one of any pair of numbers, we can find the 
 other by solving a linear algebraic equation. An example of this 
 perfect correlation is that of the radii and circumferences of cir- 
 cles in Which tne correlation coefficient is +1. If a decrease in 
 one of the pairs of a perfectly correlated series is accompanied by 
 an increase in the other, the coefficient is -1, If r=0 there is no 
 tendency to this linear relationship though there may be a close con- 
 nection of another kind. 
 
 The correlation coefficient then may be described as a mieasure 
 of the approach to linear relationship. 
 
 XII. CALCULATION OF r 
 
 Tne correlation coefficient, r, is defined by the formula 
 
 r = 
 
 where x and y are the deviations from their respective means. O' and 
 CTy are the standard deviations of the two distributions and W is the 
 total number of observations. In the given sets of data of heights 
 and weights, y represents height and x represents weight, an arrange- ■ 
 ment which is purely arbitrary, for y might just as well have stood 
 for weight and x for height as far as results are concerned. How- 
 ever, as it has been arranged, the columins give the frequencies of 
 each weight int erval distributed according to height, and the rows 
 give the frequencies of each unit of height distributed according 
 
 
- - - ' • ■■ ' ' '■ 
 
 
 jl' <j£' © 5 " f f 0 aM^:' 
 
 ilvj; i- 0 =^ iX .'- a* 4 fl*iptii 4 ¥a 
 
 ,i!,-swnj #fii!ig.xirtXi}i 
 
 ® , *.r . .. 'U ,.‘'* .* * 7 - ' ^V \ , .* . . , 
 
 fc'*' ' ■ /■ , ■ ,'"'^ ■* -f ■• > F v‘ * ' '*♦, -^'i: V * ' 
 
 ‘ it i ±. ■ ^ 
 
 f' t 
 
 w ,'i ■'■, V .“w •• ' “r^ ' ■■€ ' 
 
 • , v: V'v. •nplv 
 
 ,r / 
 
 :^i 
 
 14 . , 
 
 • >r; ■'.f'. 
 
 
 . < '-K 
 
 ■X.- ♦ ■ ^ ■:.«* '■ ’ 
 
 '.Sehi;*/. ■> . 
 
 ' -./ f .W ^ / ,- '.. ■ f, ■> f rg 
 
 -^5 • ■« '.',Jb,‘'*»^ 
 
 -, ■ \.,.i .cl* 
 
 ) «L u A — . __ ^ A L. 1 - ^ - .'w * L .^r AJ* .. J... . c . _ "^^RH 
 
 K' ■'- . ’. ■ '- 'i 
 
 *. ^ *’ ’i ji *^vi|(. ■" i^-i 
 
 iC-~^ -ui- tl,i -5 t Tiii'.'i f. rM»4 Air.x *'l« f '''■jk.J^t’>'.' ‘?'\ , 
 
 1 A 
 
 ”'f ^ *■ ” J) ‘ * 1 y ■ ''^ 'aJKw 4 ''i|(. "" -,- 
 
 fJt . '■' . ' ■ ■•" -'liF- 
 
 . tit^ -^o. ^rj;. >0 xO -41 _/:,v . d ,^»ji 4:46. . ,« 
 
to weight. The total frequency distributions for tne rows and col- 
 umns give, as is to be expected, tne original frequency distributions 
 (cf Tables II ana Xlii). If there seems at times to be a discrepancy 
 It is due to the fact that in making the correlation tables, tne 
 measurements wnicn occurred on the division point between intervals 
 were not divided as in the original distributions, but were given 
 alternately to the class above and to the class below, i^/iany times 
 the number of such occurrences was not even and one of the classes 
 received one more observation than was given by the original tables. 
 The discrepancy of several hundredths which is seen when tne original 
 standard deviations are compared with those of the correlation tables 
 IS due to the same cause. 
 
 The method used for finding r is given by Table XIII. In that 
 table, d represents tne deviation from any arbitrary point, taken 
 as in the summation method given above, so that tne least possible 
 arithmetical work is involved. In the example (19 year mien) this 
 arbitrary origin was taken at height 68 ana at weight Ibb as was tne 
 case in obtaining the means and standard deviations of the distri- 
 butions in order to compare one with tne other, so in finding the 
 standard deviations which are necessary for the calculation of the 
 correlation coefficient, the method of obtaining the moments can be 
 used, since as was said before the formu.la for the square cf the 
 
 standard deviation is that of the second moment ) . Since x is 
 
 N 
 
 the deviation from tne mean, the -£fx must necessarily be 0, but us- 
 ing d, the aeviation from an arbitrary point, the ^fd is not equal 
 
 to 0 and in order to get tne standard aeviation about the mean tne 
 
 f d^ ^ f d 
 
 summation — ;:r- must be decreased by the square of o(=‘^=^); that is, 
 
 N 
 
 /T ^fd^ ,^f d^ ® 
 
 y “ — W~ " value a given in the vertical column of the 
 
,e' 't.fr ^ 
 
 
 •f '‘if 4 ^ 
 
 I ■ '> ,i’. :; 
 
 'T , <■ 
 
 7 - ' 7 : .fe 
 
 I 
 
 '-h* i 
 
 .. A - 
 
 • i C 2 t' ll 
 
 C * »A, 
 
 ' •; . .] 
 
 . i. 
 
 .'*‘ -a O' 
 
 
 t; ' Jbi- «, 
 
 6 - ^ 
 
 • *■* V ) 
 
 ’* ■ ' , •' ''■ . i“ ' t^'r • ’W '“i '■ ^"'■r’’f -J . . ;• 
 
 • •• * » »» .. -jv"' . '- X r . ^ ■ 4 { *■ vA-**'' ^ 
 
 •• . • * -7 
 
 4 .. l. » . . /■ 
 
 . 1 - ■-■ ‘.rtw : %>-• 
 
 fty 
 
 ■,13 . ;.; 
 
 *' «• ' v;'«' ■': ■ 
 
 ' l.« 
 
 -V 
 
 ) 
 
 I 
 
 - u*. ,.:;.r ■,*>‘^.'i.’,A£i. 
 
 . .< . ^ ■ ", Tf-^' .', 1 ’»/f 
 
 ■J - "y » ,C - '• > ' i* .' ■, 
 
 « 
 
 ■ ii - •: : : ,c ... 
 
 >' 
 
 ■•/' • 1 .zy: vr.c; ?■.■. 
 
 u .v. 
 
 ( w AvkX 
 
 - ^12 4 £»" 
 
 V 
 
 ■:o - ( 
 
 ’•■V«* • ■ '*• ' (t i 
 
 
 
 1 . ;,;,or ' '■i'J! tr; 
 
 1 
 
 4 ^,, ‘‘ j 4 : '7 ■' !• 
 
 • 'i‘' :i . V .: _■■ ■ 
 
 .t- : • -■ 
 
 
 . . V 
 
 
 
 t 
 
 
 ..iV 
 
 . 3 .,^ 
 
 1 
 
53 
 
 table, is the value of the frequency, as given in each square in 
 eacn row, multiplied by the respective deviation d of the weights 
 (given in tne row just below the table). Tne quantity ad then gives 
 tne product of the x and y deviations from an arbitrary point weight- 
 ed by tne proper frequency, and the row and column ad must check. In 
 order to nave the-f^xy about the mean, must be decreased by the 
 product c^Cy as shown in the table. Then we have 
 
 r = 
 
 <^ad « CvCv 
 ~¥ ^ 
 
 (T cr 
 
 ^x y 
 
 In the examiple, r = ,5 approximately. 
 
 The remaining groups show correlations whose values are near 
 that of the example given above. The 20 year group is the only one 
 with a correlation higher than .b; with its coefficient of .516, it 
 seem^s significant that the most miature group averages tallest and 
 heaviest, and also snows the greatest tendency for heaviness and 
 tallness, lightness and shortness to be associated. In agreemient 
 with this we see that the youngest group shows the least assoc iatior 
 between weight and height, with a coefficient of .425. The corre- 
 lation Of the weights and heights of the 18 year group is expressed 
 by a coefficient of .495, while the freshman group has a slightly 
 lower coefficient of .478. Tne heights and iveights of women showed 
 the least correlation of all the groups, their correlation being ,411, 
 
 Since there is perfect correlation when r = ±1 and since the 
 variables are independent when r =0, we conclude that there is a 
 fair amount of correlation between heights and weights, or that weigl: ; 
 is to somje extent dependent upon height. 
 
 1. See 0. B. Davenport ’’Statistical Methods”, p 20,45; also Elderton 
 
TABLE XIII 
 
 
 
 Calculation of Correlation Coefficient for 
 Height, ana weight of 991 Freshmen Age 19 
 
 \Ve L a k. 't 
 
 
 
 Joo 
 
 '7^ 
 
 //O 
 
 /is 
 
 /Jo 
 
 ns 
 
 '70 
 
 ;4r 
 
 /U 
 
 
 /50 
 
 /•a 
 
 /•h 
 
 nis 
 
 /*> 
 
 IZS 
 
 /20 
 
 /fi" 
 
 tio 
 
 
 7oo 
 
 'KtjU 
 
 c/ 
 
 
 
 a 
 
 ad 
 
 
 
 Sh 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 / 
 
 
 
 
 
 z 
 
 vz 
 
 'zV 
 
 
 '5 
 
 4o 
 
 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 I 
 
 -7 
 
 -7 
 
 H 
 
 -4 
 
 //z 
 
 
 
 u 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 ( 
 
 / 
 
 
 
 / 
 
 / 
 
 S' 
 
 -6 
 
 -30 
 
 ri(S 
 
 -'7 
 
 /oZ 
 
 
 
 43 
 
 
 
 
 f 
 
 
 
 
 
 
 
 
 X 
 
 X 
 
 X 
 
 X 
 
 4 
 
 
 3 
 
 X. 
 
 / 
 
 3 
 
 xi 
 
 
 -/ 3o 
 
 kSb 
 
 
 ZTT 
 
 
 
 4V 
 
 
 
 
 
 
 
 
 
 
 ;z. 
 
 
 1 
 
 z 
 
 s 
 
 X 
 
 4 
 
 4 
 
 2f 
 
 4 
 
 
 
 4o 
 
 -4 
 
 -li>0 
 
 L4o 
 
 -/o4 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 / 
 
 X 
 
 /I 
 
 4 
 
 4 
 
 IS 
 
 ;2/ 
 
 4 
 
 7 
 
 
 
 ^4 
 
 -3 
 
 -zsz 
 
 7S'4 
 
 ~IX3 
 
 34y 
 
 
 
 U 
 
 
 
 
 
 
 
 / 
 
 / 
 
 z 
 
 
 
 t 
 
 IS 
 
 IX 
 
 n 
 
 Xo 
 
 4 
 
 /4 
 
 4 
 
 
 
 /zz 
 
 -z 
 
 
 
 -1X3 
 
 
 
 
 ^7 
 
 
 
 
 
 
 
 /7 
 
 / 
 
 
 4 
 
 
 14 
 
 n 
 
 x4 
 
 3X 
 
 13 
 
 xs 
 
 7 
 
 7 
 
 
 
 '^1 
 
 '/ 
 
 -'Sf 
 
 ‘^1 
 
 -64 
 
 44 
 
 
 •Ki 
 
 &7 
 
 1 
 
 
 
 X 
 
 
 z 
 
 
 
 4 
 
 J 
 
 /) 
 
 !% 
 
 z(> 
 
 xi 
 
 /7 
 
 // 
 
 IX 
 
 7 
 
 3 
 
 / 
 
 
 ISS 
 
 0 
 
 0 
 
 0 
 
 
 0 
 
 
 - 
 
 ^7 
 
 ( 
 
 / 
 
 
 z 
 
 
 
 X 
 
 
 k 
 
 /r 
 
 /r 
 
 !$ 
 
 /3 
 
 X7> 
 
 /4 
 
 4 
 
 y 
 
 3 
 
 
 
 
 !3i 
 
 ! 
 
 nL 
 
 /34 
 
 
 z/x 
 
 
 O' 
 
 3: 
 
 7o 
 
 / 
 
 
 
 1 
 
 / 
 
 z 
 
 / 
 
 /5" 
 
 
 7 
 
 /3 
 
 XI 
 
 IS 
 
 XX 
 
 n 
 
 7 
 
 3 
 
 
 
 
 
 /3f 
 
 X 
 
 zii 
 
 SS!> 
 
 X?4 
 
 s4i 
 
 
 
 7/ 
 
 
 
 / 
 
 
 
 / 
 
 i" 
 
 
 /o 
 
 4 
 
 J 
 
 /O 
 
 IX 
 
 S 
 
 X 
 
 3 
 
 / 
 
 
 
 
 
 7/ 
 
 3 
 
 Z/3 
 
 43f 
 
 x^f 
 
 443 
 
 — 
 
 
 r-2 
 
 
 
 
 
 
 
 
 X 
 
 / 
 
 3 
 
 X 
 
 4 
 
 y 
 
 4 
 
 
 
 
 / 
 
 
 
 
 X? 
 
 V 
 
 /tx 
 
 
 4z 
 
 
 
 73 
 
 
 
 
 
 
 / 
 
 
 1 
 
 
 ; 
 
 / 
 
 z 
 
 
 1 
 
 X 
 
 
 
 
 
 
 
 13 
 
 3" 
 
 (.s 
 
 3ZS 
 
 -/r 
 
 Z3£^ 
 
 
 
 77 
 
 
 
 
 X 
 
 
 
 / 
 
 z 
 
 
 
 
 1 
 
 / 
 
 
 
 
 
 
 
 
 
 7 
 
 4 
 
 4x 
 
 xSX 
 
 
 XTZ 
 
 
 
 73" 
 
 
 
 / 
 
 
 
 
 / 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 7 
 
 XI 
 
 
 x4 
 
 /6g 
 
 
 1. 
 
 tiU 
 
 3 
 
 / 
 
 
 /o 
 
 J 
 
 4 
 
 -2/ 
 
 4 
 
 3^ 
 
 xs 
 
 
 /o/ 
 
 ixil^ 
 
 IZ$ 
 
 "1 
 
 fs 
 
 77 
 
 ^3 
 
 3 / 
 
 /O 
 
 4 
 
 yy/ 
 
 
 -o<f 
 
 sm 
 
 
 3^/0 
 
 
 
 J 
 
 o 
 
 /X 
 
 // 
 
 /o 
 
 7 
 
 
 7 
 
 4 
 
 X 
 
 4 / 
 
 3 
 
 z. 
 
 1 
 
 0 
 
 */ 
 
 -z 
 
 -J 
 
 
 
 -4 
 
 -7 
 
 
 
 
 
 
 
 
 
 fd 
 
 3f 
 
 /;?. 
 
 ;i;?. 
 
 /DO 
 
 '^7 
 
 4-i 
 
 
 
 no 
 
 ^;?o 
 
 '7a 
 
 
 u4 
 
 0 
 
 -"7 
 
 ~'7o 
 
 '■*7/ 
 
 ^'a 
 
 
 -to 
 
 -4z 
 
 
 
 
 
 
 
 
 
 U'- 
 
 5o7 
 
 
 2*^i 
 
 V 
 
 
 
 
 '’V. 
 
 Jxi 
 
 J7e 
 
 •^u 
 
 Vo)l 
 
 >x<l 
 
 e> 
 
 "7 
 
 3<4> 
 
 ?73 
 
 
 77r 
 
 3^0 
 
 
 ///3o 
 
 
 
 
 
 
 
 
 d 
 
 3 
 
 / 
 
 /o 
 
 ,r 
 
 
 
 4/5^ 
 
 V 
 
 47 
 
 
 si 
 
 IX 
 
 •7 
 
 7 
 
 -•/s 
 
 7^3 
 
 ''4> 
 
 -P 
 
 -7«. 
 
 -3g 
 
 
 
 
 
 
 
 
 
 
 
 
 /A 
 
 no 
 
 St> 
 
 7x 
 
 y4 
 
 
 ‘Hi 
 
 3-?0 
 
 •lx 
 
 Hi- 
 
 lii- 
 
 n 
 
 0 
 
 4s 
 
 »74 
 
 v& 
 
 >lx 
 
 3to 
 
 
 ic3 
 
 37/0 
 
 
 
 
 
 
 
 - 0 *’= Syi - 01 9i> 73 
 
 ff/ y ■ ^ 
 
 2 
 
 J-y = 
 
 / .o S^JL4Bi> / O 
 j/. iff 'J^Zo i 
 .<fo S' Sy f c>4 
 
 -JYJ- = ^)C .5-00 3-0^" 
 
 z, 
 
 7 /V 
 
 ,<f 0 srg y r c 4 
 f, 7 yiy f s-z -fo 
 
 ^ Ji. - Apji 
 
 77/ M /V 
 
► f. * ^ # 
 
 
 
 .■ is*- 
 
 ’•' * :>/"' ■ *. ' 
 
 ■ ', '■ '^;.' - 
 ■ •■ 5 ?* ii jj-i ’■' ’ , '' .'•* ' ' 'I .Tf" 
 
 , ^ ., y -»r**T- r.^'iSff. ---u . ^,, 
 
 < -<«L 
 
 
 
 !* ^ ■, -^ ^ A}-- 
 
 - ,w ^-'’.-ii'ri '■?;_‘>yii,r^4yj4i 
 
 •Ipifflpl 
 
 i l^.l t 4 t I 
 
 ^ * • >.-»i . Jn < CTu » p. 
 ^ ' I . 
 
 rf'MBf’**" ' -T- ' — , - 
 
 I , J‘ ci‘ii.'ikji;;ii>i„t ^ 
 
 , . i -•' w f?^-'-l 
 V Uj.-.b':.^:^ 
 
 jA \.,\ir. , ■‘<f‘ lU^ f I 
 
 f* r- f - He . ■■] 
 
 ;V> » •>- - 1 
 
 t= 
 
 
 A. !' i 
 
 ■Ul.' y 
 
 i ,--U4 . ,| .^'.K:?; ‘*’i' T 
 
 ^ » * 't' /A ‘ ■**;*' 'j / t -ft 
 
 
 < « 
 
 
 ?5 
 
 , »* - '10 . ’'»« ■>! fi, - - ■ -■ ’• J .•', , ■•yitV.vr-;, 
 
 li/ 1 ifcJESTrSR?^ ■ *» \*^ 4 t-...'s 
 
 
 p *■ ■ ■ '1 ■ ■'’»■'*' < 8 ^ < 34 « 
 
 . ' ♦'.. ' '‘' . ;d’-> -^I#- ^ -’Xy^ 
 
 •i." --1?' '<*•- * . 
 
 ^'4 . 4 '^ ‘.i idOifi, ■ ;r.A-#' 
 
 '■"A.! 
 
 
 '>dyir^ 
 
 
 ,TT!t: 
 
55 
 
 XIII. SUMARY 
 
 To sum up this investigation I have made a table to show in a 
 concise way how much the means and correlation coefficients of the 
 heights and weights of the various classes of freshmen differ. I 
 have expressed the means with their probable errors, which give the 
 measure of unreliability of values calculated; the probable error is 
 a pair of values lying one above and the other below the value deter' 
 mined, and the true value lies some\7here between these limits. The 
 fig-ares which follow, sum up the various distributions and indicate 
 the closeness with which anthropometric measurements are fit by the 
 normal curve and by several other types. 
 
 
 Group 
 
 Mean 
 
 Height 
 
 Mean 
 
 Weight 
 
 Correlation 
 
 Coefficient 
 
 Freshman Class 
 
 67.699±. 047 
 
 135.94+.34 
 
 .478 
 
 17 
 
 year 
 
 Freshmen 
 
 67.826±.064 
 
 134. 75±. 48 
 
 .425 
 
 18 
 
 year 
 
 Freshmen 
 
 67.73G±.050 
 
 135. 44±. 36 
 
 .495 
 
 19 
 
 year 
 
 Freshmen 
 
 67.859±.051 
 
 137.51±. 36 
 
 .500 
 
 20 
 
 year 
 
 Freshmen 
 
 67.985±. 053 
 
 139,01±.40 
 
 .516 
 
 V/omen 
 
 
 63.313±,048 
 
 117.93+.38 
 
 .411 
 
eta /v t t.T\ t 7L £ 
 

 ( 
 
 i 
 
 V' . * 
 
\ 
 
 1 
 
LQ ki. 
 

I 
 
• Me iak't fn. incA.es 
 
* 
 
HO ns tZ9 Its >y> iss /fp /fr 'Sa iss /io i(,s Ho ns /So /fs i<fo i‘js' zpp 
 
\ 
 
 V. 
 
 1* 
 
 , J 
 
 r *''• 
 
 'I 
 
 
 * , I 
 
 ' , 4 
 
 . /r • 
 
 ,J»’ 
 
 -T 
 
 » ‘V 
 
 -j.. . . . 1. ; 
 
 
 • t p.fP>4 
 
 : •• Xirt 
 
 1 1 
 
 i. 
 
 ♦ 
 
 j 
 
 ■» 
 
Itr Ho nS IZO /Zjr /So ISS' /fo if^ /S^S^ /(.o ttys' /Jo /J^ /io ffo Ijs Zoa 
 
i 
 
 i 
 


V 
 
 io 
 
 4 / 
 
 L2 
 
 43 
 
 M 
 
 M 
 
 i>7 
 
 ,(>f 
 
 %] 
 
 71- 
 
 n 
 
 13' 
 
 n<f-\ 
 
 i£\ 
 
 % 
 

BIBLIOGRAPHY 
 
 Davenport, 0,B. 
 
 ”Stat istical Methods” 
 
 Elderton, W.P. 
 
 "Frequency Curves and Correlation" 
 
 Pearson, Karl 
 
 •^kew Variations in Homogeneous Material" , Phil. 
 Royal Society London, (a) , Vol.183, Part I, 
 
 Yule, G, U. 
 
 "An Introduction to the Theory of Statistics" 
 
 Trans.