# 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)  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  worked on the problem and did some of the mathematics in detail (and continuing on in later papers).

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

 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