380 Miscellanea 
III. 
Adjustment of Moments. 
The quadrature formulae given above can be used very conveniently for adjusting the 
statistical moments. 
We may first take the calculation of moments when the ordinates at equidistant points are 
rn-h 
known. Let )/o, yi, .y2 •••.yn-i be given and we require J ^ y^dx. Now 
rn-h ri ri rn-i 
I , yxdx= \^y:,dx+ \ y^dx+...+ \ ^^^^ y^dx. 
If we now ai^ply Formula I. we see that we can use it for all the integrals on the right- 
hand side of this equation except the hrst two and the last two and the value of these can 
be taken from Formula IV. Summing the values we have 
, y.dx = [6463^0 + 4371^1 + 6669^^2 + 5537^3 
+ 5760 (;/4 +,?/5 + • • • +y» - a) + ■'>5373/„ - 4 
+ 6669?/„_3 + 4371v/„_.3 + 64G.3?/„_i} X., 
which means that we can multiplj' the first and last ordinates by (or 1"122205), the second 
43*71 6669 
and last but one by (or -758854), the third and last but two by (or 1-157813), the 
5537 
foui'th and last but three by ^J^^ (or 'SGI 285), leave all the other ordinates unaltered and work 
out the moments in the usual way from the modified series of ordinates which will now give the 
proper values for equating to the moments from the formula for the ciu-ve. 
If there be high contact at each end of the curve — and ordinates are known — there is 
no reason why Formula I. should not be used even for the end groups, and this tells us that the 
rough statistical moments require no modification in such a case. 
The above rules have been applied in actual cases, thus Formula VI. was found when tested, 
/"I dx 
by using 12 ordinates to approximate to I :j , to give an error of — -000000205 which, 
though it is greater than the best formulae given by Professor Pearson in Biometrika^ Vol. i. 
pp. 278, 279, is sufficiently accurate for almost all conceivable purposes. The rule in cases 
of high contact was tested by adding 12 ordinates of the normal curve calculated to 5 decimal 
places, which gave 1-24998 instead of 1-25000. A type III. curve which had high contact 
gave 24473 with 9 ordinates instead of 24475, so that in each case the result was practically 
exact. A further test can however be applied, for if we assume that areas and not ordinates are 
known we can apply the formulae and reach Sheppard's adjustments. For if areas are known 
we are given as the rough moment the expression 
j^_j^dxXt + y^dx{X+lY + j'^ yAv{X+2y+ + j y^dx{J+7i-iy, 
rn-i 
and we require I ^ (X+x)' y^^dx. The series of integrals can be written by the help of 
{. . . + [51 78 A' + 308 {{k - 1 )< + (A + 1 )'[ _ 17 {(k - 2)' + (A + 2)'}] y, + ...}, 
where A" is neglected in order to simplify, and working out this general coefhcient we have 
{5760 A' + 240;; (< - 1 ) A' - 2 + 3< (i! - 1 ) (C - 2) (< - 3) A' - < etc.}. 
