exhibited in Tables of Statistics. 365 



ah—b + \ 



(ai) = 



{a-\)b ' 



( \ _ {<^b-b + l){ab-b + 2) 

 ^^^> - [a-lf.b.{b-\) ' 



. . _ {ab-b + \){ab-b + 2){ab-b + ?>) 

 ^^^^ ~ {a-\f.b.{b-\){b-2) ' 

 and so on. And 



^ ^^1 - (ab-b){flb-b-\) ' 



( \ {a-linb + l){b + 2){b+S) 



^''i) ^(,b-b){ab-b-l){ab-b-2y 



Now before \yorking out numerical results, there is one obser- 

 vation that is evident on the face of these formulaj. It is evi- 

 dent that when we get far on in the scale, these numbers will 

 increase very rapidly ; so that when we get to a number which 

 is much greater or less than b, we see that we should expect b to 

 occur with much greater frequency than such number; and 

 when we get considerably away from b (the average number), 

 these figures will become so immense that the result will be that, 

 even in a very large table of statistics, we shall not expect such 

 a number to occur at all. We may remark, moreover, that to a 

 very rough approximation the numbers would be symmetrically 

 arranged on each side of b, those above it increasing at first more 

 rapidly, and then less so than those corresponding to the num- 

 bers below b. In the case of n not being very large, we should 

 observe also that the numbers increase more rapidly than when 

 it is so. An application of this last observation may be the fol- 

 lowing. Suppose that in a table of statistics the numbers were 

 found more uniform than should be expected from the whole 

 number of the community. The hypothesis that such phsenomena 

 are confined for the most part to a definite section of the commu- 

 nity, might in some cases go far to explain such a uniformity. 



It is easy to see how the expressions above given will, when 

 worked out, afford a test such as was proposed in the outset. 

 But probably a much more useful application of the problem will 

 be, in the case of the extreme numbers which occur, to be able to 

 tell whether tlieir variation is such as to require explanation from 

 the special circumstances of the particular year. 



When we come to work out results numerically, wc shall be 

 probably startled at the extremely small probability of a number 

 far removed from the average one which our results will give. 

 It will be much more than in proportion to the moral surprise 



