# Why the sample covariance estimator is unbiased, but the sample pearson correlation coeficient is not?

Why the sample covariance estimator is unbiased, but the sample Pearson correlation coefficient is not?

I'm a bit confused because the sample Pearson coefficent was built using the sample covariance estimator and the sample variance estimator, that are both unbiased.

Amazingly, transforming an unbiased estimator often results in a biased estimator. This is how the sample standard deviation is a biased estimator, despite the sample variance being unbiased. This fact comes from something called Jensen’s inequality. For a concave function $$f$$, such as a square root:

$$f(\mathbb{E}[X])\ge \mathbb{E}[f(X)]$$

Equality holds if and only if $$f$$ is a straight line (or if $$X$$ is constant).

So why is $$\hat{\rho}=\dfrac{\widehat{cov}(X,Y)}{s_X s_Y}$$ biased? The standard deviation estimators are biased!

• The Wikipedia article on Jensen’s inequality has some pictures that might give some intuition.
– Dave
Commented Apr 28, 2021 at 10:16