TRANSACTIONS OF THE SECTIONS. 3 



and I will only further add, that I feel much gratified to find so large an attendance 

 of eminent men of science here, ready to correct oversights and supply deficiencies. 

 Thev, I am well aware, are far more competent to preside here than I can be ; but, 

 with their assistance, the duty will be light ; and as the Council, no doubt on good 

 grounds, have made the present arrangement, I will, without hesitation or misgiving, 

 at once proceed with the business. 



On the Probability of Uniformity in Statistical Tables. By R. Campbell. 



The object of this paper is to find a test for ascertaining whether an observed 

 degree of uniformity or the reverse in statistical returns is to be considered 

 remarkable. 



Suppose the population to consist of n persons, which we will suppose nearly con- 

 stant. Let a be the number of years during which observations are taken, and sup- 

 pose the whole number of phenomena of a certain class occurring to, or presented 

 by, the individuals in the population to be ab. We will suppose the phenomena of a 

 kind which are not likely to occur to the same person more than once in the same 

 year. [Of this nature are most important facts, of which such Tables are formed.] 

 Now suppose we know nothing of the laws by which these facts occur, except 

 that above given, namely, that the total number in a years is ab. Let us see what 

 kind of uniformity (starting from that fact alone) we should expect the Tables to 

 present. 



Let the people alive in a particular year be A u A 2 . . . A». The probability then 



of the phenomenon being presented in that year by Aj is - . The probability that 



it will be presented by A, and not by A, will be _( 1 — ). Hence we can find 



n\ aw— 1/ 



the probability of its being presented by Aj only ; the probability of its being pre- 

 sented by one person only ; the probability of its being presented by two persons only, 

 and so on. These expressions will be very complicated. That for the probability 

 of the phenomenon being presented by b persons only will be 



(n— 1) (n— 2) . . . (w— 6+1) ab — 1 ab— 2 ab— 6+1 A_ ab — b \ 



2. 3. 4... b ' 'an—l'an — 2 ' ' " an — 6+1 \ cm— 6/ 



ab — b \ /, ab— b \ 



(i- ab ~ b \ .A- 



^ an-Q+D) ^ 



■(«-!), 



Now though these expressions are complicated, we get some very simple results. 

 The above expression would be deducible from the expression for the probability of 

 the phenomenon being presented by b — 1 persons only, by multiplying by a factor 



b(a— 2)+n+l 

 which reduces itself to 1 + (w _ 6)(a _ 6)6 . 



It would be deduced from the expression for the probability of 6+1 persons pre- 

 senting it by multiplying by a factor, which reduces to 



, (a— l)w— (a— 1)5+1 

 («— 6)(o-l)6 



Now these are always greater than 1. This shows that the average number is the 

 most probable one to occur in a particular year. 



The ratios of the probabilities of 6 occurring, to that of (6 — 1), (6—2), &c. are 

 6 to 6-1, 



n-6+1 06-6+I 



■ — j, 7 — =^fi (a) 



an — ab — n — b 



6 to 6-2, 



(w-6+l)(w-6+2) n.5-6+ 1 ab-b+2 



6.(6-1) 'an-ab-^b'an-ab-iT^b+l 1 ' ' ' ' (b) 



• ••• • • • • • • • • 



and 50 on. 



1* 



