XI.—CORRELATION : MISCELLANEOUS THEOREMS. 215
or from (9)
ov Mie,
§2 = TES = 20, + 0,2) (10)
9. Correlation between Indices—(Ref.11.) The following problem
 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 indices
 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