ERROR TO CASES OF NORMAL DISTRIBUTION AND CORRELATION. 125 
(ii.) Denote the error in Lj by w. Then the calculated value of is 
S (z + e) {x — ojY =: S (z + e) ^ w) ; 
and therefore the error in is 
Sx'^e — ptzx'^' “' _ 1 CO = X (cc^ — _ 1 re) e. 
Hence this error is distributed normally with mean square 
\tz{x‘^-p\-xxf - {Xz(.x^’ -pK-xx)}~]ln—{\^ — ,X^,_i + ;3-X^_i X, — \l)/n. 
In particular, the mean square of the error in X, is (X^ — X:^/n. 
(ill.) The mean product of the errors in Lj and in is 
[tzx (x^’ — p)\--\x) — Xzx. Xz (.U’ — p>K--^ x)]ln = (\,+i — p)\-x X^ln. 
In particular, the mean product of the errors in the mean and in the mean square 
of deviation is X 3 /W. 
(iv.) The mean product of the errors in X^, and in \ is 
{Xz (x^’ — p\,_i x) {x'^ — 5 X ^-1 x) — Xz (x^ — p\,-i x). Xz (x/^ — q\-i x)}ln 
= iK+i —pK-i K+i — \-i + p^K-i — KK)/^^- 
§ 19. Error in Class-Index .—Let the values Lj + Xi, Li -f Xo, Li + X 3 . . . ., in 
§ 18, be supposed to be in order of magnitude, Lj + Xj being least; and let X be any 
possible value of L, not coinciding with any one of these actual values."^' Let 
the two classes for which L is respectively less and greater than X be denoted by 
C' and C, and let the numbers in these classes be in the ratio of 1 + ot : 1 — a ; then a 
will be called the class-index of X for classification according to values of L. Its 
value ranges from — 1 to + 1 . 
If a representative selection of n individuals were made, the numbers coming 
from the two classes would be ?q = l?i(l -j- a) and n., = -g-n. (l — a); so that 
a. = {ny — n,)/(5q + n.i). Suppose however that the selection is a random one, the 
errors being as in § 18. Then, if we take X as lying between X,. and X,.+i, the 
observed value of a is (zj + € 1 ) + ( 2:2 + ^ 2 ) + .... + (z,. -j- u) — {'^r+\ + ^r+i) — ...., 
and therefore the “ error ” in a is ej + €2 + . . . . fi- e,. — e,.+ i — .... Hence :— 
(1.) By considering the division of the community into the two classes C' and C, 
we see from § 15 (i.) and (ii.) that the error in a is distributed normally v/ith 
mean square (1 — a^)/n about a mean value zero. 
* This limitation does not introduce any difficulty in the case of continuous variation, since the 
frequency of any single value is then indeSnitely small. (Cases in which the curve of frequency has an 
infinite ordinate are excluded from consideration.) 
