For a bivariate normal distribution, the distribution of correlation coefficients is given by
where is the population correlation coefficient, is a hypergeometric function, and is the gamma function (Kenney and Keeping 1951, pp. 217-221). The moments are
where . If the variates are uncorrelated, then and
so
But from the Legendre duplication formula,
(12)
so
The uncorrelated case can be derived more simply by letting be the true slope, so that . Then
(17)
is distributed as Student's t with degrees of freedom. Let the population regression coefficient be 0, then , so
(18)
and the distribution is
(19)
Plugging in for and using
gives
so
(27)
as before. See Bevington (1969, pp. 122-123) or Pugh and Winslow (1966, ยง12-8). If we are interested instead in the probability that a correlation coefficient would be obtained , where is the observed coefficient, then
Let . For even , the exponent is an integer so, by the binomial theorem,
(31)
and
For odd , the integral is
Let so , then
But is odd, so is even. Therefore
(38)
Combining with the result from the cosine integral gives
(39)
Use
(40)
and define , then
(41)
(In Bevington 1969, this is given incorrectly.) Combining the correct solutions
(42)
If , a skew distribution is obtained, but the variable defined by
(43)
is approximately normal with
(Kenney and Keeping 1962, p. 266).
Let be the slope of a best-fit line, then the multiple correlation coefficient is
(46)
where is the sample variance.
On the surface of a sphere,
(47)
where is a differential solid angle. This definition guarantees that . If and are expanded in real spherical harmonics,
Then
(50)
The confidence levels are then given by
where
(56)
(Eckhardt 1984).
See alsoCorrelation Coefficient,
Fisher's z-'-Transformation,
Spearman Rank Correlation Coefficient,
Spherical Harmonic Explore with Wolfram|Alpha ReferencesBevington, P. R. Data Reduction and Error Analysis for the Physical Sciences. New York: McGraw-Hill, 1969.Eckhardt, D. H. "Correlations Between Global Features of Terrestrial Fields." Math. Geology 16, 155-171, 1984.Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951.Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, 1962.Pugh, E. M. and Winslow, G. H. The Analysis of Physical Measurements. Reading, MA: Addison-Wesley, 1966. Referenced on Wolfram|AlphaCorrelation Coefficient--Bivariate Normal Distribution Cite this as:Weisstein, Eric W. "Correlation Coefficient--Bivariate Normal Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CorrelationCoefficientBivariateNormalDistribution.html
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