boas] THE CEPHALIC INDEX 45 I
and the observed values may be expressed by l % -\- x 2 , ^s + x Zi l±-\-x^ . . . The values x ly x 2% . . . are determined by a great many very small causes which we will call e. Then we may say —
x x = "ti *i + a %% e 2 + a is € s + • • •
x t = a 2\ £ 1 + a 22 € i + ^23 *3 + • • •
where the a are constants. When we wish to investigate the cor- relations of the series of organs with the first organ, we may sub- stitute the value of e x from the first equation and we find :
- 2 = ?21 X \ + /^12 € 2 +As *3 + • • •
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L ■*„ = ?m X \ + Ana f 2 + ^n 3 f 3 + • • •
Here the q and ft are new constants which result from the
substitution. Since the causes € are subject to chance only, their
averages will disappear. If, then, we assume x x to be constant,
we have
(2) Average x a = q ni x t .
Now we will form the products x x x n . These may be arranged in such order that all the x x that have equal values shall be grouped together. Then each of these values for x x will be mul- tiplied by the sum of all the correlated values of x a . Supposing there are/ individuals in which the first organ has the deviation from its average x l9 we may substitute for the sum of all the cor- related values A n p times the average of x n , or according to (2), / q ni x x . Therefore the sum total
«*- at j x n = ^2 x i q ni x x = «*- q ni x x .
Now, the sum of all the x x * is equal to the mean square variation (or standard variation) of / lf which we will call n x : Therefore
2. X x X n = 2. ^ni/'i »
or when n the total number of individuals measured
= nq ni Mi 2 -
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