SECTION IV. TYPE OF CURVE AND il GOODNESS 



OF FIT". 



We shall now test the u goodness of fit" with our " normal " 

 curve K. Pearson ' has shown how this may be done. He shows 

 that 2 if 



m J' 



** = S 



where S denotes a summation, m' and m are observed and 

 theoretical values in each sub-group, then the chances of a 

 system of errors with as great or greater frequency than that 

 denoted bv %*• is given 



by 



P = 



ix-. 



cix I . &%.-> . dx < 



Q/Xifi 



J J J -e dx x . dx. z . dx s .... dx n 



/ -\x* 



J e . x n ~ 1 dx 



X 



/OO 

 e . x n ~ ] . 



dx 



which reduces to for n' odd 



- ±rr.1 



X 



n' -3 



\ I + — + + + T 7-7- ; 



2 2.4 2 . 4 . 6 . . . (n — 3) 



and n' even 





X 



n'-Z 



" '1 .3.5 . .. (n'-3) 



i 



Tables 3 have been calculated to facilitate calculation of P 

 when x % is known. 



Pearson then shows 4 that if x l for the sample is so small as to 

 warrant us in speaking of the frequency distribution as a random 



1 K. Pearson: "On the Criterion that a Given System of Deviations from 

 the Probable in the Case of a Correlated System of Variables is such that it can be 

 reasonably supposed to have arisen from Random Sampling." Phil. Mag. July 

 1900, p. 157. 



2 x' 2 is thus quite easy to calculate ; it is given by 



, _ / square of difference of theoretical and observed values\ 



v 2— V) UrT1 ( : — )• 



V theoretical value of frequency / 



- W. Palin Elderton : " Tables for Testing Goodness of Fit." Biom.\o\ 

 1 (1902), pp. 155 — 163. Reprinted as Table XII on p. 26 of Tables for Statisti- 

 cians, etc. 



* Pearson, paragraph 5 and following of reference 1. 



