6 On the Probability of Error [January, 



was (say) as low as 200*0156 ; one was as high as (say) 

 200*0159. Further observations might have yielded results 

 still more extreme. But assuming they would not, still the 

 •number of possible values between 200*0156 and 200*0159 

 is infinite. The arithmetic mean a is but one of these, and 

 although more likely than any other that could be named, 

 it is not more likely than one or other of all the possible 

 values. The odds are more than 1000 to 1 that a is not the 

 truth ; but they are also more than 1000 to 1 that a is very 

 near the truth. The question — How near ? — cannot be an- 

 swered. Alter the question to — What is the probability 

 that the truth is comprised within the limits a±k} — and 

 the answer may easily be given, however small k may be. 

 Thus, if & = o'oooi. In other words, if the question be — 

 What is the degree of likelihood that the truth lies between 

 200*0157 and 200*0159 ? — the answer is given by the 

 formulae — 



7T=H„ 



where k, 0*0001, 



„ *£/ = 715000000, and as log. -^ = 8*854, log. \'w =4*427, 



,, \'w = 26800. 

 Then t=kv w = 2 m 68, and from Table I.— 

 7r=H 2 . 68 = o*99985. 



The result, 0*99985, is so near to unity (the measure of cer- 

 tainty) that for every practical purpose it maybe considered 

 certain that the truth is really comprehended within the 

 limits named. 



Take six other observations, the arithmetic mean of 

 which is — 



= 203*870 ; 



and the sums of the squares of the six differences between a 

 and each separate observation is — 



^ 2 = o*7i5i5. 

 Then the weight of a is — 



3 6_i8_ 18 

 2?e* 2e* 0*71515 D 



The smallness of this figure, contrasted with the 715000000 

 of the preceding example, indicates a comparatively low 

 degree of probability that the true number is comprised 

 within the limits ± k, when k is very small. For instance, 

 suppose the question to be — What is the likelihood that the 

 true number is comprised between 203*77 and 203*97 ? As 



