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XLII. On a Brief Proof of the Fundamental Formula for 

 Testing the Goodness of Fit of Frequency Distributions, 

 and on the Probable Error of " P." By Karl Pearson, 

 F.R.S* 



1. TN a contribution to the ' Philosophical Magazine ' in 

 A 1900 (vol. 1. pp. 157-175) I gave the first proof of 

 the fundamental formula for testing goodness of fit of a 

 theoretical frequency curve or surface to observational data. 

 The observational data were supposed to constitute a sample 

 of definite size drawn from an indefinitely large population 

 obeying the theoretical curve or surface of frequency. The 

 problem proposed was : To measure the probability that 

 a sample of the proposed size drawn from the indefinitely 

 large theoretical population would present as great or 

 greater divergence than the observed material does from 

 the sampled population. Was, in fact, the observed popu- 

 lation so improbable a sample of the theoretical distri- 

 bution that it must be rejected as a sample by the prudent 

 statistician ? 



Let the theoretical population consist of M individuals 

 failing into q classes with the frequencies m l5 m 2 , . . . m^ 

 all these classes being indefinitely large, but having finite 

 ratios. Then, if a sample of size JST give the classes 

 7i 1? n 2 , ... n 2 , the mean value of n s in a long series of 

 samples will be « s =Nm,/M, and the distribution of n s will 

 follow the binomial 



(l+K)}"- 



and accordingly, if no class be extremely small as compared 

 with N, the distribution of any frequency like n s will be 

 approximately normal and the combined system of fre- 

 quencies will follow the generalized Gaussian law 



j l Ass xs 2 \ 9< (Ass'xsXs' \ { 



: e I v A **' > A «" r «>/1 , . . . (i.) 



where 



x s =n s — n ( 



*(i-5) 



the squared standard deviation of the binomial, and the 

 correlation of x 8 and x 8 > or 7v, is given by 



n s n s ' 



CTsO- s <r S s< = j^t 



* Communicated by the Author. 

 Phil. Mag. S. 6. Vol. 31. No. 184. April 1915. 2 C 



