340 Mr. F. Y. Edgeworth on the 



the odds against the Probability-curve being exact are 

 enormous*. But, if we must adopt some simple representative 

 of the experienced data, the case is as if we had to choose 

 between different explanations of a fact which has occurred ; 

 each of which hypotheses is a priori improbable, but one of 

 which must be true. Upon this view, the odds in favour of 

 the Probability-curve are stupendous. 



It is not to be thought that the method of examining 

 wholes rather than small parts is confined to the comparison 

 of areas. We might regard the sum of the number of obser- 

 vations as a particular case of the sums of powers of errors. 

 The sums of the apparent errors all taken positively, the sums 

 of their squares and other powers, all supply empirical data 

 proper to this method. We might put the facility-curve 

 which is on trial in the form a<f>(ISx) ; determine the con- 

 stants a and /3 by means of any two of the above-named 

 data ; then for the curve so determined, calculate some other 

 sum of powers ; and then compare that computed value with 

 the corresponding datum of experience. 



For instance, let us apply the First Exponential to the 

 group of 25878 height-measurements mentioned above. Let 

 us take for our basis of computation the observed sum-of- 

 squares-of-errors, 166072, and the observed zero-power of 

 errors, viz. the number of observations. The resulting 

 curve is of the form -i _x 



y= 25878^* °, 



where c is to be determined from the equation 

 166072 = Se 2 = 25878i 



Whence 



2 I X 



= 25878 c 2 1 #V- a tae = 25878x2c*. 



c 2 = 3'21, c = V 



For the curve so determined we have now to compute the 

 sum, or Mean, of errors. The Mean Error =c = l # 8. This 

 result is to be compared with experience, which gives for the 

 Mean Error 2 — a discrepancy whose significance may thus 

 be tested. By a formula of Laplace already referred to, the 

 Modulus-squared for testing the error of the mean (first 



* A complex curve formed by the superposition of several Probability- 

 curves having different centres (corresponding to different provinces) 

 would doubtless be a nearer approximation. See on this point my letter 

 to l Nature/ Sept. 22, 1887. 



