The quantity* 



(F.-Np. f 



L -V- (16) 



s 



is a measure of the deviation of the sample from the ex- 

 pectation, where F . is the number of observed frequencies 

 in the tth interval, and Np . is the number of expected fre- 

 quencies in the ith interval as predicted by the theoretical 

 distribution. Karl Pearson proved that the above quantity, 

 in the limit, is the ordinary \ 2 distribution which is now 

 tabulated in most statistics books. 



The x 3 computed with equation 16 is compared with 

 s 

 the 5 per cent point for (^-1) degrees of freedom from a x 

 distribution table. The tabulated value of \ 3 at the 5 per 

 cent probability level with 29 degrees of freedom is 42.6. 



Now, if x 2 > as calculated by equation 16, is greater than 



s 

 42.6, then the hypothesis is rejected by this test; that is, 

 the sample is non-gaussian. 



The application of the x 3 test to the data was as fol- 

 lows (fig. 7). Let f(x) represent a probability density 

 curve obtained from the PDA. Divide the curve into 30 

 intervals from x - -3. to x = +3. 0. Let £. be the mid- 

 point of Ax., one of the intervals, and /( £ • ) the value of 

 the probability density at £ . The area under the curve is 

 then estimated by "LA . ', where A . ' =/(£.) Ax. Let A = 

 zi 



<p (z)dz, where <p (z) is the theoretical probability 



z 



t-1 



-See ref. 5, pp. 197-200. 



22 



