Karl Pearson 
293 
Accordingly to a first approximation 
1 + 5. ySy 
where -q, k and y are now mean values and Sry, 8k and 8y are variations from those 
means. Squaring, summing for all values, and dividing by the number of cases: 
2 2 (1-^')* 22, / 2 2y'<(l + y')AT s ^ r■■^ 
^ = (1 + ^2)2 '^'^'c' + (1-^72)2 - ,72)3 ^^^"^ ^^"^^^ 
Hence to find a, we require to evaluate a^, and Mean (8yS«:). We shall 
now proceed to the consideration of these quantities. 
1 _i_^2 
We have v-, N = — = e dx. 
Hence BvJN = — Sy e~ 
V '2tt 
= - Sy2, 
where z is the ordinate of the frequency curve of the marginal total at the 
boundary of the alternative categories. 
Thus ^\J^'" = f^y'^'- 
But if the sample has been taken from an indefinitely large population 
= (1 - uJN). 
Accordingly cr^2 _ 
Again k8k = S (ksSkts), 
where = ^ yj^ 
K^a,2 = S {kJ^(t\^) + 2S {k,k,' Mean (8K,hK, )} , (v). 
We require therefore to find ct^^^ and Mean (Sk^Sk/). Now 
2k,8k, = ^ + 2 y,8y, (vi). 
But . n,, = -ri^ I e dx. 
V2tt Jy, 
8ni, n^.Sn 
2 
= - 8ys2s (vii). 
And accordingly: 
- Mean (Sn^.Sn,) + a\ = ^/a^,^. 
