oo 2 THEORY OF STATISTICS. 
It should be noticed that if we define the principal axes of any 
distribution for two variables as being a pair of axes at right 
angles for which the variables &, &, are uncorrelated, equation 
(9) gives the angle that they make with the axes of measurement 
whether the distribution be normal or no. 
7. The two standard-deviations, say 2; and 2, about the 
principal axes are of some interest, for evidently from § 2 the 
major and minor axes of the contour-ellipses are proportional 
to these two standard-deviations. They may be most readily 
determined as follows. Squaring the two transformation equations 
(8), summing and adding, we have 
212 =0+ 03 JX (10) 
Referring the surface to the axes of measurement, we have for 
the central ordinate by equation (7) 
, J 
y 12 = 2051 = 72) 
Referring it to the principal axes, by equation (3) 
Ta 
120 ron 
But these two values of the central ordinate must be equal, 
therefore 
D= ay05(1 CE 5)’ (11) 
(10) and (11) are a pair of simultaneous equations from which 
2, and Z, may be very simply obtained in any arithmetical case. 
Care must, however, be taken to give the correct signs to the 
square root in solving. 2; +2, is necessarily positive, and 2, — 2, 
also if 7 is positive, the major axes of the ellipses lying along &; : 
but if » be negative, 3; — 2, is also negative. It should be noted 
that, while we have deduced (11) from a simple consideration 
depending on the normality of the distribution, it is really of 
general application (like equation 10), and may be obtained at 
somewhat greater length from the equations for transforming 
co-ordinates. 
8. As stated in Chap. XV. § 13, the frequency-distribution 
for any variable may be expected to be approximately normal 
if that variable may be regarded as the sum (or, within limits, 
some slightly more complex function) of a large number of other 
variables, provided that these elementary component variables 
are independent, or nearly so. Similarly, the correlation between 
two variables may be expected to be approximately normal if 
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