374 



Prof. Karl Pearson on Testing the 



Hence generally, 



^(W)-^* 2 ))' 



The values of P for the value of % 2 and for q and q~2 

 can be taken directly from the Tables for Goodness of Fit *. 

 The value of a 2 is given by 



V = {%' 



1) 



'/ 1 9(9-2 ) 

 H + N 



}'■ 



where q= number of cells as before and H is the harmonic 

 mean of their contents f. It is thus fairly easy to 

 determine the standard deviation of P, and so ascertain 

 whether the degree of " goodness of fit " is sufficiently 

 stable to provide in any case an adequate measure of fit. 



Illustration I. — We may illustrate the method on the data 

 provided in the introduction to my ( Tables for Statisticians,' 

 p. xxxii, for the goodness oE fit of the cephalic index of 900 

 Bavarian crania to a Gaussian distribution. The following 

 table gives the observed and Gaussian frequencies : — 



Cephalic Index. 

 Under 75'5 



Observed. 



Gaussian. 



9-5 

 12-5 



17 



37 



55 



71-5 



82 

 116 



98 

 107 



82 



74 



34-5 



19 



10 



8 



9 



124 



12-7 



221 



35-3 



51-9 



70-1 



87-0 



99-4 



104-2 



100-5 



89-1 



72-6 



54-3 



37-4 



23-7 



13-8 



7-4 



6-3 



75'5-76 - 5 



76'5-77'5 



77-5-78-5 



78-5-79-5 



79-5-80-5 . 



80-5-81-5 



81-5-82-5 .. . 



82-5-83-5 



83-5-84-5 



84-5-85-5 



85-5-86-5 



86*5-87-5 



87-5-88-5 



88-5-89-5 



89-5-90-5 



90-5-91-5 



Over 91-5 





These 18 groups give % 2 = 10-27, leading to P = '8909. 



The ?t's are, of course, given by the Gaussian column, and 

 from this H can be calculated by aid of tables of reciprocals. 

 Thus 



s (ff) = s (i)=- 786 ' 272 ' 53 - 



* ' Tables for Statisticians/ p. 26. 



t Young and Pearson, he. cit., where we must note that x 2 =N0 2 . 



