civ Tables for Statisticians and Biometricians [XVII 



of 2 = a and o- 2 = er/v 2. Table XVII indicates the degree of exactness involved in 

 such usage in the case of small samples. It is clear that in small samples, we may 

 err considerably in judging the accuracy of a standard deviation by the 'probable 

 error' 67449a-/ / v / 2n, although by the time we reach samples of 50, there will not be 

 much error in assuming for most practical purposes a normal distribution for 2. 



Illustration. In a sample of 16, what is the probability that we shall have a 

 standard deviation twenty per cent, greater than that of the parent population? 

 The mean from Table XVII is '9523cr, and we require the probability of a value 

 as great as, or greater than l'2cr. By Table XVII the standard deviation is '1752o-. 

 Hence, if we were to work this by the normal curve theory we should take 



(l-2o- - 9523er)/-1752o- = 1-4138, 

 which corresponds to a probability integral (Part I, Table II) of 



1(1+ ) = '9213, or 1 (1 -)= "0787, 



thus the odds against such a sample standard deviation would be about 12 to 1. 

 Actually, however, the curve of standard deviations for samples of 16 is not normal 

 but has a skewness of '0961 (Table XVII), and it is reasonable to doubt whether 

 the above approximation is legitimate. We proceed therefore to find the true 

 probability integral. This is given by equation (i) above for any value Xo- as 



00 *7? 



PU4 

 y Tf 

 y M 



) A<r 



Put v = n2?o z and we have 



) 1 / S \"- 2 1 / s V / T \ 

 _L / ^ \ _ l I . I - / ^^ \ 

 7 6 * \ffl\f 2n/ d 

 9n-2n / M -!\ \o-/v / 2n/ \o-/\/2?i/ 

 . v 2 / 



In 

 r^!!^^ 

 2 



where F^ ( - ) denotes as usual the incomplete F- function, and P A(T is the pro- 

 \ * / 



bability integral of the Type III curve. The easiest method to evaluate this is to 

 look it up in the Tables of the Incomplete Y* -Function *. 



For our special case, n = 16, \ 1'2, we have 



p _ 1^.52(7-5) , r/ fi .c\ 



*- F(7 . 5) -I(u,(>5), 



where u= H'52/\/7 > 5 = 4'206,509, and on interpolation we have 



P 1 . 2<r =l- -91671 = -08329, 

 or the odds against such an occurrence are about 917 to 83, say 11 to 1. 



For a small sample of 16, the ordinary theory is likely to be about 10 / in 

 error in computing the odds. For many purposes, however, this would not make a 

 serious difference in our conclusions. 



* Published by H.M. Stationery Office, Kingsway, London. 



