June 12, 1914] 



SCIENCE 



877 



law of variation. Natural phenomena, as well 

 as chance, tend to fluctuate in a manner in- 

 dicated by this curve. Chest measurements 

 on 5,738 soldiers^* show the close agreement 

 with theory. The stature of 1,052 English 

 women^" was found by Karl Pearson to 

 closely obey the Gaussian law. Some of the 

 lower mental traits can be measured in the 

 psychological laboratory. Thorndike^" found 

 twelve-year-old pupils to be distributed ac- 

 cording to the Gaussian curve as regards their 

 accuracy and rapidity of perception. Memory 

 tests yielded similar results. When it comes 

 to tests of higher intellectual powers, records 

 are discordant. Different examiners have 

 varied to such a marked degree in marking 

 the same individuals that conclusions can not 

 be safely drawn from their estimates. On 

 account of the presence of constant errors, 

 the lopsidedness of individual markings can 

 not be altogether eliminated by taking the 

 averages of many grades from different ex- 

 aminers. ■ A curve constructed from 1,487 

 grades in mathematics given by 19 different 

 teachers in three high schools in Colorado ex- 

 hibits two peaks with a valley between. The 

 first peak is at 70 per cent., the passing mark ; 

 the other peak is just above 85 per cent. Evi- 

 dently the peak at 70 per cent, is due to a con- 

 stant error arising from the practise of rais- 

 ing marks of some pupils to the passing grade. 

 Such constant errors arise also where a mark 

 of 85 per cent, on the daily work exempts 

 students from final examinations. It is found 

 that in such cases medium grade students are 

 advanced to the exempt limit. Seldom are 

 marks given between 55 and 59, where 60 is 

 the passing grade. If a doubtful student is 

 finally passed, some teachers give him a mark 

 considerably above passing, the idea being^^ 

 that, if passed at all, he ought to be passed 



18 L. A. Quetelet, ' ' Lettres sur la tbgorie des 

 probabilites, " p. 400. See also A. L. Bowley, 

 "Elements of Statistics," London, 1902, p. 278; 

 Dearborn, op. cit., p. 8. 



19 Cattell, op. cit., p. 371 ; Dearborn, op. cit., 

 p. 9. 



20 Thorndike, "Educational Psychology," p. 15. 



21 Finkelstein, op. cit., p. 42. 



" handsomely." The tendency to mark high 

 is inherent in human nature. Dr. Euffner 

 says :22 



A temporizing professor who loves popularity 

 and desires, like the old man in the fable, to please 

 everybody, is sure to be guilty of this fault, and, 

 like many a politician, to sacrifice permanent good 

 for temporary favor. 



For these reasons, available statistics as to 

 the distribution of mental abilities are incon- 

 clusive. Some empirical curves indicate con- 

 siderable skewness, others follow the Gaussian 

 curve. President Poster found that 8,969 

 grades in 21 elementary courses for two years 

 at Harvard obeyed the normal curve of fre- 

 quency. Dearborn makes similar reports for 

 472 high school pupils, also for freshman 

 grades of these same pupils at the University 

 of Wisconsin. It is doubtless the principle 

 of continuity that has led not only English 

 statisticians like Galton and Pearson, but 

 also American investigators, Foster, Meyer, 

 Smith, Dearborn, Finkelstein and others, to 

 aver that the Gaussian curve or normal curve 

 is the proper curve for the distribution of 

 marks in school. In what follows we assume 

 that the Gaussian curve can be so used. 



The question then arises, what marks should 

 be assigned to a random group or " fair sam- 

 ple " of, say, twenty students, whose order of 

 rank is known by the tests suggested in Part 

 I. This question involves some intricate sta- 

 tistical theory, which has been worked out by 

 Karl Pearson. Pearson^^ states the problem 

 thus: 



A random sample of n individuals is taken from 

 a population of N members which when N is very 

 large may be taken to obey any law of frequency 

 expressed by the curve y ^=N<t>{x), ydx being the 

 total frequency of individuals with characters or 

 organs lying between x and x -f- dx. It is required 

 to find an expression for the average difference in 

 character between the pth and the (p -^ l)th in- 

 dividual when the sample is arranged in order of 

 magnitude of the character. 



In answering this question, Pearson de- 



22 Quoted by Finkelstein, op. cit., p. 47. 



23 Karl Pearson, "Note on Francis Galton 's 

 Problem," Biometrica, Vol. 1, pp. 390-399. 



