xxxii Tables for Statisticians and Biometricians [III 



Suppose a = 1T52, p= 7'5, then 



11 -52 -6-5 5-02 



= 1-969,006. 



Ve 7 ^ 2-549,510 

 Hence, by linear interpolation from the table, 



^ -42630. 



Accordingly from (iv) 



log / (a, p) = 7-5 log 11-52 + log -4263 - 1 T52 log e - % log 6-5 - log T (7'5). 



The value of logr(7'5), found by aid of Table XXXI of Part I, is 3'272,1329, 



and so we have 



log I (a, p) = 2-908,9468, 



I (a, p)=*-0811. 



Its true value is '0834, and as found from the normal curve with the same area, 

 mean and standard deviation as y = x*' 5 e~ x , it is '0787. We have therefore con- 

 siderably improved our approximation by taking for the desired area a normal 

 curve fitting better at the tail. 



Illustration (ii). Find the mean errors of (a) all errors greater than the probable 

 error, and (6) all errors less than the probable error in the case of a normal 

 distribution of standard deviation a. 



The distance from the mean of the probable error ( (1 a) = -25, or a = -50) is 

 known from Table III of Part I to be '674,4898. Interpolating into the present 

 table between 67 and 68 we find @l x = '78672. Hence, if x z be the mean error of 

 all errors greater than the probable error, 



= l-271,1003o-. 



Let i?i be the mean error of all errors less than the probable error, and #i +2 the 

 mean error of the whole distribution beyond its mean, then 



a?i +a = 0-/1-25331 = '797,8872o- 

 by the present table. 



But 



or ,*i 



= (1-595,7744 - 1-271,1003) a 

 = '324,6741o-, 



or, the mean error of all errors greater than the probable error is slightly less than 

 four times* greater than the mean error of all errors less than the probable error. 

 The exact numbers are 



i= l-27l,1063o-, xt = -324,6628o-, 



so that the above values, as we might expect, fail in the fifth decimal place, the 

 table itself being only to that number of places. 



* Actually 3-915,0037, the more exact value being 3 '915, 1587: see p. xxviii above. 



