ERROR TO CASES OF NORMAL DISTRIBUTION AND CORRELATION. 151 
than the mean square of xfj, the error in Y as obtained by direct observationand 
therefore we obtain a better result for Y by the calculation than by observation. 
(2.) Suppose that we fix on particular class-indices a and /3, and that we require 
the corresponding value of Y. The error in Y, as determined by calculating the 
averages, average squares, and average product, is — Z9 ; while the error for direct 
observation is (§ 30 (3.)) ^ (l — y) [F ((y-|-xp)/a -j-y] + 
= — Z6 (j) — ^ {1 — y) X ~~ i ~ X- Since the mean products of (f), y, y' with 6 
are zero, the mean square of this last error is greater than the mean square of — ZO. 
This result, of course, is identical with (1.) ; for if the observed class-index of X' is 
a, we may consider that we are observing either the class-index of X' or the value 
of L corresponding to class-index a'. 
(3.) If we determine D by tire method of § 30 (3.), the resulting error is 
6 — Z~^ ~ 2 “ y) X ““ i" (1 — S) y'}. The mean product of 6 and ^ ^ (1 — y)y 
— ^(1 — h)x is zero; hence the probable error due to the method of § 30 (3.) is 
greater than that due to the method of § 30 (1.). 
(4.) Similarly, if we take the weighted mean of a number of different values of D, 
as in § 30 (4,), we shall still get an error of the form d -j- where the mean 
value of 6^ is zero. Hence, if the averages, average squares, and average product 
can be determined, the value of D so obtained cannot be inq^roved by direct obser¬ 
vation of the values of Y corresponding to selected pairs of class-indices.t 
(5.) Generally, let R be any quantity wdiich would be known if the true means, 
mean squares, and mean product of L and M were known. Let Ri be the value 
obtained by taking these as equal to the averages, average squares, and average 
product, for the n individuals; and let R., he the value obtained by any other 
method involving observation of the numbers occurring in any set of classes deter¬ 
mined by a finite number of class-indices of L and M, with or without the use of the 
averages, average squares, and average products. Let Bijn and %\ln be the 
mean squares of the errors in R as determined by the tw^o methods. Then the pro¬ 
positions stated in § 23 (4.) hold good. The theorem may be extended to the case of 
any number of mutually correlated attributes. 
§ 32. Test of Hypothesis as to Normal Correlation. —To test whether the distri¬ 
butions of L and of M, in any particular case, may be regarded as normally correlated, 
we use the method of § 24, with the necessary modifications. 
(1.) With the notation of § 31 (5.), let R denote the proportion of individuals for 
which L exceeds X and M exceeds Y, the values of X and Y being fixed beforehand. 
Then, writing -g-F (1 — y) = A, A (l — 8) = B, we have 
* This shows that V (1 — V) is greater than A} (I -f 2AB (l.-t cos D) cos D -f BRl -|- 
-j- {Ax -h By) Z sin D cos D -|- Z^sin'^ D, where A = ^ F (I — 7), B = ^ A (I — o). 
t Of. Kael Peaeson, in ‘Phil. Trans.,’ A, vol. 187 (1896), p. 265. 
