I III] Introduction xvii 



(i.e. 92 - 88 x (+ -4361)) above the average undergraduate. The range of Second 

 Class men is from + 40'50 to + 95'75 mentaces above the average undergraduate, 

 and the range of First Class men all those with more than 95'7o mentaces above 

 the average. The "genius" corresponds to an excess of no less than 287'02 

 mentaces. If we suppose that one individual in 1000 is completely feeble- 

 minded or practically wanting in all intelligence, we should credit roughly the 

 average man with 300 mentaces, and we should then have our range of intel- 

 ligence on a Gaussian scale : 



Poll : below 296 mentaces ; 



Third Class : above 296 and below 340 mentaces ; 



Second Class: above 340 and below 396 mentaces; 



First Class : above 396 mentaces ; 



" Genius " : above 587 mentaces. 



In rough numbers : Poll, below 300 ; Third Class, 300 to 350 ; Second Class, 

 350 to 400 ; First Class, over 400 ; " Genius," over 600. 



Of course there is much that is hypothetical here, but the numbers give us 

 some appreciation of the distribution of ability, and they serve to illustrate the 

 construction of a Gaussian or normal scale. When more than three or four 

 significant figures are needed Tables II and III must be used. 



TABLES II AND III (pp. 210) 



Talks of the Probability Integral: Area and Ordinate of the Normal Curve in 

 terms of the Abscissa ; and Abscissa and Ordinate in Terms of Difference of Areas. 

 (Calculated by Dr W. F. Sheppard, and published in Biometrika, Vol. n. pp. 

 174_190.) 



" Sheppard's Tables" were the first to express the Gaussian* or normal 

 probability integral in terms of the standard deviation ; they are so familiar to 

 statisticians that it would almost seem a work of supererogation to explain their 

 ' use," which is further too manifold for full description. We can only give a few 

 sample illustrations. 



It is most important ivlten using these tables to pay attention to the signs of 

 the differences recorded at the tops of the columns. 



Illustration (i). The mean length of cubit in 1063 adult English males is 

 recorded as 18"'31 '019 and of their 1063 adult sons as 18"'52 "021. Determine 

 the odds against these two measurements being really identical, i.e. random 

 samples from the same population. We assume that the deviation of means and 

 their differences follow the normal law. The difference is 0"'21 and the probable 

 error of this difference = V(-019^ + ('021) a = 0"'0283. Since the probable error 



* The term is usual, but inaccurate. Laplace had reached the probability integral and suggested 

 its tabulation several years before Gauss. 



B. e 



