12 



Carl C. E}io_bcrz 



first case discussed in this paper, Professor Pearson finds the 

 average percentage error to be 5.75, which he considers by no 

 means bad. If, however, we apply I\Ir. Elderton's test, we find 

 the probability, that in a random sampling a deviation system 

 as great or greater shall exist, to be about .000000, that is, not 

 once in a million samplings would he get a greater deviation. 

 This probability P is given by the equation 





for n even, and by 



+ • 



P^ 



r 



fx , X 



!_ -_._!_. 



I I ' 1-3 I-3-5 



1. 3. 5.. (w— 3) 



+- 



,n-Z 



L ' 2 ' 2.4 ' 2.4.6 ' ' 2.4.6..(;i— 3) 



for « odd, where n is the number of groups, and 



X'=-J — =sum 



j squares of dift'erences of theo- \ "^ 

 I retical and observed frequency i 

 theoretical frequencv 



The trouble with this formula for x' is that it assumes that, 

 if, say, 1,000 observations distributed in n groups give a mean 

 error of 4 per cent, 16,000 observations distributed in the same 

 way shall give a mean error of i per cent. Now some objects 

 are more variable than others, so that it is easily conceived that 

 1,000 observations on one plant or animal give as accurate a 

 result as 16,000 observations on another more variable plant or , 

 animal, but whatever the value of P in the first case, it is very 

 large as compared with the P of the second case. One example 

 is sufficient to show this. 



98 



