XI.—CORRELATION : MISCELLANEOUS THEOREMS. 215 
or from (9) 
ov Mie, 
§2 = TES = 20, + 0,2) (10) 
9. Correlation between Indices—(Ref.11.) The following prob- 
lem affords a further illustration of the use of the same method. 
Required to find approximately the correlation between two ratios 
I =2X,|X,, Zy=X,/X,, X; X, and X, being uncorrelated. 
Let the means of the two ratios or indices be 7; 7, and the 
standard-deviations s; s,; these are given approximately by (9) 
and (10) of the last section. The required correlation p will be 
given by 
(X X. 
¥psysy=3(y' - n)( - r,) 
XZ 
=x( =) “NTT 
MM, ( i ( Zy X Za ¥: Ar 
Neglecting terms of higher order than the second as before and 
remembering that all correlations are zero, we have 
MM, 
psi8y=—. yr (1 +307) - 1], 
Le 
=3: 
where, in the last step, a term of the order v,* has again been 
neglected. Substituting from (10) for s; and s,, we have finally— 
v2 
= me {11 
P= Jol + v5%)(25? + 25%) gab 
This value of p is obviously positive, being equal to 05 if 
vy =v, =7;; and hence even if X; and X, are independent, the in- 
dices formed by taking their ratios to a common denominator X will 
be correlated. The value of p is termed by Professor Pearson the 
“spurious correlation.” Thus if measurements be taken, say, on 
three bones of the human skeleton, and the measurements grouped 
in threes absolutely at random, there will, nevertheless, be a 
positive correlation, probably approaching 05, between the indices 
formed by the ratios of two of the measurements to the third. To 
give another illustration, if two individuals both observe the same 
series of magnitudes quite independently, there may be little, if