XXV] Introduction cxxxiii 



Thus the chance that a value of x' should occur as Urge as or larger than this is 

 '185,2418, taking positive and negative excesses in x' together. The odds are only 

 about 4'5 to 1 against such occurrence. 



Now let us consider the two characters m and which have been combined in 

 " Student's " test separately. 



The means in the samples are distributed normally with standard deviation of 

 n. = in our case 3'1623, or the ratio of the observed deviation in the sample 

 mean to the standard deviation of sample means is 2*2136. 



From Table II of Part I of these Tables for Statisticians : 

 MO - '986,4474, S a wo = -77, 

 MJ = -986,7906, 8 2 u! = -75, 

 B = -36, </> = -64, 10$ = '0384. 

 Accordingly 14 = -986,5709,5 + '000,0008,8 



= -986,5718. 



Thus the chance of a mean as great as or greater than this occurring 

 = '013,4282, or taking both positive and negative excesses = '026,8564. Thus the 

 odds against such a mean occurring in a single sample are of the order 36 to 1, 

 while those as judged by " Student's " test are about 4*5 to 1. 



Now turn to the standard deviation, which is 14'64 against the 10 of the 

 parent population. 



If we judged roughly, assuming the distribution of standard deviations to be 



2 



approximately normal with a standard deviation - = 2*2361 about a mean of 



v2w 



2 = 10, the deviation 14'64 10 = 4'64 would be 2-075 times the standard devia- 

 tions, or deviations as great as or greater than this would only occur about 38 times 

 in 1000 trials, or the odds are of the order 25 to 1 against such an occurrence. 



For a more accurate appreciation of the odds, we must note that the curve of 

 distribution of s in samples from a normal population is 



where '/(2/'V / 2n) in our present notation. But this curve has not yet had its 

 probability integral tabled for various values of n and x'. 



If, however, we write z = x' 2 , the probability integral becomes 



F 

 I 



z * cr'dz 



z a e- 

 Jo 



dz 



= Probability Integral of a Type III curve as tabled in the Tables of the Incomplete 

 T -Function*. 



* Published by H.M. Stationery Office, 19*2. 



