CHAPTER XVI, 
NORMAL CORRELATION. 
1-3. Deduction of the general expression for the normal correlation surface 
from the case of independence—4. Constancy of the standard- 
deviations of parallel arrays and linearity of the regression—5. The 
contour lines: a series of concentric and similar ellipses—6. The 
normal surface for two correlated variables regarded as a normal 
surface for uncorrelated variables rotated with respect to the axes of 
measurement : arrays taken at any angle across the surface are normal 
distributions with constant standard-deviation : distribution of and 
correlation between linear functions of two normally correlated 
variables are normal : principal axes—7. Standard-deviations round 
the principal axes—8-11. Investigation of Table II1., Chap. IX., to 
test normality : linearity of regression, constancy of standard-deviation 
of arrays, normality of distribution obtained by diagonal addition, 
contour lines—12-13. Isotropy of the normal distribution for two 
variables—14. Outline of the principal properties of the normal dis- 
tribution for n variables. 
1. THE expression that we have obtained for the “normal” dis- 
tribution of a single variable may readily be made to yield a 
corresponding expression for the distribution of frequency of pairs 
of values of two variables. This normal distribution for two 
variables, or “normal correlation surface,” is of great historical 
importance, as the earlier work on correlation is, almost with- 
out exception, based on the assumption of such a distribution ; 
though when it was recognised that the properties of the correla- 
tion-coeflicient could be deduced, as in Chap. IX., without reference 
to the form of the distribution of frequency, a knowledge of 
this special type of frequency-surface ceased to be so essential. 
But the generalised normal law is of importance in the theory of 
sampling : it serves to describe very approximately certain actual 
distributions (e.g. of measurements on man) ; and if it can be 
assumed to hold good, some of the expressions in the theory of 
correlation, notably the standard-deviations of arrays (and, if 
more than two variables are involved, the partial correlation- 
coefficients), can be assigned more simple and definite meanings 
than in the general case. The student should, therefore, be 
familiar with the more fundamental properties of the distribution. 
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