Curious Sample Correlation Property

Question

I'm examining correlations in a data set with a large number of variables but small sample sizes. To get a feel for how these quantities behave, I generated some random data and looked at the distribution of correlations:

n = 4
y = matrix(rnorm(1000 * n), 1000, n)
x = matrix(rnorm(1000 * n), 1000, n)
p = as.numeric(cor(t(x),t(y)))
hist(p)

To my surprise, the distribution is almost perfectly uniform:

Does anyone have an explanation for this phenomenon? It makes some sense in that for n=2 we have either p=1 or p=-1, and as n->infinity the distribution becomes normal, so this distribution falls somewhere inbetween. But why uniform? I'm stumped.

Answer 1

For independent Normal variates, the distribution of the correlation coefficient $r$ is proportional to $(1 - r^2)^{{1\over2} (n-4)}dr$. When $n=4$, that's uniform.

Reference

R. A. Fisher, Frequency-distribution of the values of the correlation coefficient in samples from an indefinitely large population. Biometrika, 10, 507. See Section 3. (Quoted in Kendall's Advanced Theory of Statistics, 5th Ed., section 16.24.)