# Does $r$ overestimate true effects for small sample size datasets?

Suppose $N$ is relatively small. Does the Pearson product-moment correlation coefficient generally overestimate the true effects?

This must be something that is well-known in statistics. I would like to find a reference I can cite. Is there any?

• Are you asking if the Pearson correlation is biased for the population correlation, $\rho$? (which - if it were the case - would be something which would tend to be larger at small sample sizes) – Glen_b Oct 11 '16 at 23:57
• @Glen_b I'm pretty sure that's indeed what the asker means. – Kodiologist Oct 12 '16 at 5:25

In fact the sample correlation is biased, but it's not biased upward; it's biased toward 0 (this has been known for at least a century).

For example, I just did a little simulation -- in a sample of 10000 simulations of samples of size 3 where the pairs were generated from a bivariate normal population with $\rho= 0.1$, the average sample correlation was $0.0685$.

Soper (1913) [1] came up with some approximations to the expected value of the sample correlation when sampling from a bivariate normal (including the approximation $E[r]\approx \rho(1-\frac{1-\rho^2}{2n})\,$) and Fisher [2] worked on the problem and did some of the mathematics in detail (and continuing on in later papers).

[1] Soper, H.E. (1913),
"On the probable error of the correlation coefficient to a second approximation",
Biometrika, vol 9, p91-115
pdf

[2] Fisher, R.A. (1915),
"Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population",
Biometrika, vol 10, no. 4, 507-521
pdf