£ = 5 = 3 = .... = ............................. (8), 2/1 2/2 2/3 *2
the sign of the last term depending on the sign of r. Hence the statement that two variables are “ perfectly correlated implies that relation (8) holds good, or that all pairs of deviations bear the same ratio to one another. It follows that in correlation, where the means of arrays are not collinear, or the deviation of the mean of the array is not a linear function of the deviation of the type, r can never be unity, though we know from experience that it can approach pretty closely to that value. If the regression be very far from linear, some caution must evidently be used in employing r to compare two different distributions.
In the case of normal correlation, aly /l—r- is the standard deviation of any array of the x variables, corresponding to a single type of i/’s. C2t / I —r2 is similarly the standard deviation of any array of the y variables, corresponding to a single type of In the general case, the first expression may be interpreted as the mean standard deviation of the ^-arrays from the line of regression, and the second expression as the mean standard deviation of the y-arrays from the line of regression. Otherwise wre may regard ax —r2
as the standard error made in estimating a? from the relation x — and ____ <t2v/1— r
as the standard error made in estimating y from the relation y = these interpretations being independent of the form of the correlation.
(2.) Case of Three Variables. Let the three correlated variables be Xi, X2, X3, and let *1, denote deviations of these variables from their respective means. Let us write, for brevity,
S (a.12) = XV, S (*,*) = XV SO/) = N<r/ S(xxx>) — Xr^ffo S(£c2£c3) = SO**!) — Nr8i«f3<ri
