# Curious Sample Correlation Property

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.

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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.