KiRSTiNB Smith 
13 
where a is the S.D. of x calculated from the sample, and Il^y is the product moment 
for X and y determined by 
X\^„ = ~%{xiy,)-xy (20). 
• - - mi 
As in the previous section 2 denotes a sum of products each of which consists 
of factors from the same class. In the sums H each product contains factors from 
at least two classes, and when two factors belong to the same class it is indicated 
by an ' s ' inserted between them. 
For evaluation of the standard deviation of py the S.D. of n^;,;, o" and a are 
required, as well as the product moments for each pair of these three functions. 
(a) Mean Value and Standard Deviation of the Product Moment 11,,-,,. 
The equation (20) may also be written 
= 2 {x,y,) -^S {x,y,) (21). 
By taking the mean value for a great number of samples we therefore tind 
Yi,,j=^T.,ss (22). 
From (21) we find by squaring and taking mean value 
nV = ('^ - 1)' (S («i3/i))= + - 2 (7. - 1) t{x,y,)ii{x,y,) . . .(23). 
Together with the determination of the mean values occurring here, we shall 
determine the other mean values of products required for the evaluation of cr^ . They 
are such as arise from multiplication of S (^i^i) and Six^y^) with each of the two 
groups S(yi''^), X(y^y.^), S (y^y^) and 1{x{-), Six^Xo) and also those which contain a 
fixctoj- of each of the two latter groups. As in the foregoing section, we need, 
however, not consider products of a 2 and an S, because such products may be 
developed into sums of products all containing a factor uncorrelated with all the 
other factors of the product, from which it follows that the mean value for a 
great number of samples is zero for each of these sums of products. It remains 
to determine the following products : 
(2 («i2/i))' 
= 2 + 22 {x,"y,y.) + 28 (x, y,x^ y.,) 
s s 
(S (x,y,)y 
= S {x^-y,") + 28 (x-'y, 2/o) + 2;? {x, y,x. y.^ + €, 
Ji S S 
2 {x,y^) 2 (yr) 
= 2 (x,y,') + 2 {x,y,'y,) + 8 (x, y,y.?) 
S 
S {a^iyO t {y,y^) 
= ^ {^hyi'V'i) + 32 {x,y,y,y,) + 8 (x, y,y., y.) 
s s 
S {x,y,) S {y,y.) 
= S (x, ?/i?/2-) + 28 (x, y,y, y,) + e, 
s s s 
^{x,y,)%{xl) 
= - (*'i'i/i) + 8 (xi'xo y,) 
S (^'i^/i) S {x^x.,) 
s 
= 8 {xi'Xo yi) + e.i 
s 
2(^,^)2(2/r) 
= t(x{-y,^) + 8{x,''y,') 
2 {x,^)t{y,y„) 
= 2 (x,-y,y.^ + 8 {x{'y, y.^ 
S {x,x.;) S (7j,y,) 
s 
= 8{Xi ?/i«,,y-2) + e4 
y (24). 
