for a System of Correlated Variables. 3 



5. Generally, if two variables are correlated, their mean 

 product differs from zero, even though the means of each 

 separately, or of one of them, are zero. For the mean ^yo- 

 duct is xy=§\ (f)(xy)xydxdy, and if x and?/ are correlated, 

 so that (j>(xy) cannot be expressed as f\(jf)f%(jf) 3 this ex- 

 pression for xy is not in general zero. If <p(^y) =fi(x)f 2 (y) ,. 

 then ley = \\&{xy)xy dx dy = yf\(x)x dx . \fi(ij)y dy, and since 

 x= \f x (x)xdx, and y = \fi(y)ydy^ the mean product is the 

 product of the means, and is zero if either x or y is zero. As 

 an example of this theorem let x and y be two vibrators, 

 having the same period, but different phases, so that we 

 may have x — A sin nt, y — B sin (nt + a) . Then # = 0, ?/ = 0, 

 but #y=JABcos«. Also in this case x 2 = ^A. 2 , y 2 =±B 2 . 

 And xy may change, while x 2 and y 2 remain unchanged. 

 Similarly, any two variables x and y may, with given 

 numerical values, be more, or may be less, likely to have the 

 same sign, than, with the same numerical values, to have 

 opposite signs. If the chance of their having the same sign 

 be (fj^xij), and the chance of their having opposite signs be 

 (£ 2 (<r?/), then 



*y = S OM 2 ^) - fcixy))*!/ (Lc d u- 



But the mean squares x 2 , y 2 are independent of the difference 

 <t>\tx!/)~4>2( d "!/)> and ma y therefore be constant while xy 

 changes. 



Of very small Correlations. 



6. It may be that /(a/) of art. 1, although it is a function 

 of y as well as of x', yet is very little affected by change in y: 



that is ' may be very small or negligible. Similarly 

 may be very small or negligible. In the same case 



(tX 



<f>(xij) may differ inappreciably from a product of the form 

 f\(x)f 2 (y), so that for some purposes we may without appre- 

 ciable error treat x and y as not correlated. For instance, in 

 the kinetic theory of gases, if x 1 ... x n are the vector velocities, 

 ??*! . . . m the masses, of n molecules, n being a very great 

 number, X\ ... x n are, strictly speaking, correlated by virtue 

 of the relation 2w« 2 =2T where T is the kinetic energy, 

 supposed constant. For if x n be given we alter the limits of 

 integration for x l ... x n _ v so that the chance of x x having a 



B2 



