Distribution of correlation coefficient (r)

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Assume you got correlation r=0.6 between two variables (N=10), is 0.6 significantly bigger than 0? Or if you observed r=0.98, is it significantly different from r=0.9? For these questions, we need to know the distribution of r of our sample, given the population correlation coefficient ro (i.e. the “true” correlation coefficient) and sample size n.

This distribution is complicated:

For more info, you can find from the link at the bottom. But let’s plot some distributions for different ro and n:

As you can see from the blue curves, if our sample size is 10 (small), a big correlation like 0.4 could be purely by chance and thus is not significant; but if the sample size is 100, a 0.4 correlation is significant and trustworthy.

You can simply calculate the area under the distribution curve to find the p-values.

Link:
http://mathworld.wolfram.com/CorrelationCoefficientBivariateNormalDistribution.html

MatLab code for the distribution:

function y = corrdist(r, ro, n)

y = (n-2) * gamma(n-1) * (1-ro^2)^((n-1)/2) * (1-r.^2).^((n-4)/2);
y = y./ (sqrt(2*pi) * gamma(n-1/2) * (1-ro*r).^(n-3/2));
y = y.* (1+ 1/4*(ro*r+1)/(2*n-1) + 9/16*(ro*r+1).^2 / (2*n-1)/(2*n+1));



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4 Replies to “Distribution of correlation coefficient (r)”

  1. My kid lot his PhD in math (entropy theory) a year ago at Albany NY, and is still unemployed. Where should he look? It would be nice if he could move to California.

    Richard (the engineer)

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