Difference Problem 
391 
(2) Let the figure represent any frequency distribution given hy y = N<^{x), where we may 
suppose the Hmits, if finite, to be extended, if necessary, from +00 to - qo by zero ordinates. 
We make no hypothesis as to the nature of the distribution, or the position of the origin ; as a 
(VI' IVI 
corollary we will deal with the case of normal distribution. Let N be the number of individuals 
or the area of the curve*, A the area to the left of any ordinate PM=y, at a character-value 
OM=x. Thus the area to the right is N — A. Then, if a = AIN, we shall have : 
r +x 
dx (i), 
o 
an integral which may be supposed known when the distribution of the general population is 
known. 
We first note that the chance of any random individual having a chai'acter less than 
X =AIN=a, and having a character greater than x ={N — A)IN= \ — a. Now let OM=Xp 
correspond to the p^^ individual's character reckoned downwards and Oi/' = .rp+j, to the next or 
{p + XY^ individual's character. Then we require first to find the mean value of M'M=Xp — x^^-^, 
there being p—\ individuals to right of PAI and n —p — 1 individuals to left of P'M' in the 
sample of n individuals we are selecting out of the population. The chance of an individual 
falling at if is given hj y^bXpfN, and of one at M' "^J >/„ ^i^x^^JN ; the chance of an individual 
to left of P'M' =Ap^JN and to right of PM ={N-Ap)IN. The total chance therefore of an 
individual at M, another at M' and n-p — 1 to left of P'M' and jd— 1 to right of PM 
N 
N 
N 
71— p — 1 
iV- 
N 
But clearly we could permute the two individuals as well as those to right and left of PM and P'M' 
and must introduce the factor / 1 jjo - l).t To get the average we must multiply the 
chance thus obtained by the corresponding Xp-x^^-^^ and first integrate from Xp + ^= —ca to Xp 
and then for Xp from -00 to +00. For, the and (jd + I)*'' individuals may be anywhere in 
the range provided (i) there are no individuals between them, (ii) the (p + l)*'' is anywhere below 
the p^^, (iii) jD— 1 individuals fall above the latter, and (iv) n — p- 1 individuals below the former. 
Hence if we write x' for Xp+j, x for Xp, a for Ap + JJV, a for Ap/JV, for yp + JJV, t/q for 
we have for the average interval between the^"" and (^ + 1)* individuals : 
Xp = 
-p — I \p 
/■+» f+x 
dx' 3/0 /yo' n'" -^il-aY-\x-x') (ii), 
where by (i) 
da _ , 
d^^y^' 
da 
dx 
.(iii). 
Since 
y dx = N, it follows that 
x) dx = l. 
t We have to find the permutations of n things which may be distributed into four groups which 
contain respectively p-1, n-p-1, 1, and 1 individuals. This is the same as the number of ways in 
which out of n factors (x + y + z + u) we can pick out {p - 1) .t's, {n-p -1) y's, one z and one u, i.e. the 
coefficient of x^-^y'^-P-'' zu in {x + y + z + uy'. But this coefficient is |n ^ (|p - 1 |)i - 1 jl |1 ). I owe 
this method of looking at the factor to Dr L. N. G. Filon. 
