ERROR TO CASES OF NORMAL DISTRIBUTION AND CORRELATION. 147 
the normal curve of semi-parameter unity and central ordinate 1/277- (area = 1/^277) ; 
T\-here x = (X — Li)/a, y = (Y — Mi)/ 6 . 
Applications of the- Theory of Error. 
§ 30. Prohctble Error in Value of Divergence, as Obtained by Different Methods .— 
Let the distributions of L and of M be normally correlated, the means, mean squares 
of deviation, and mean product of deviation, being- Lj, Mj, a", E, and ab cos D. 
If a random selection of n individuals is made, the divergence can be found by any 
one of several different methods. We require to .find the probable error in D, due 
to the use of each method. 
( 1 .) Suppose that we take the averages, average squares, and average product, as 
equal to the means, mean squares, and mean product for the complete community. 
The general expression for the resulting probable error in cos D = Si_ i / \/ hyp-i 
been found in § 20 . To find the values of 83 ^ 1 , 82 , 2 ; and 81 ^ 3 , in the case of normal 
correlation, we write L — Li = {a/b) cos D.(M — Mj) -f- L'; then M — Mj and L' 
are independent, and their mean squares are respectively D and cr sin^ D. The 
mean fourth power of M ~ Mi is 36^; and thus we find 83,1 = ?>cdb cos D, 
82^2 = (1 + 2 cos'^ D), Si ^3 — 3a6® cos D. The table in § 20 becomes 
^2 
Si.i 
r-i 
X, 
2a‘ 
2a?h cos D 
2a-h^ COS' D 
S:,i 
a^Jr (1 + cos^ D) 
2ab’' cos D 
/‘2 
2b* 
and hence we find that the probable error in D, due to adopting this method, 
is Q sin D/\/ n. 
( 2 .) Let D be determined by the method of § 27. Let the medians as given by 
the data be respectively L'l and M'l, and let the result of the double median 
classification be 
Below L'l. 
Above L'l. 
Below M'l .... 
P' 
R' 
Above M'l .... 
R' 
P' 
so that n = 2 (P'+ L'). Let P = n (tt — D)/27r, P = 7il) j2Tr ; and let the classifi- 
u 2 
