Introduction

Let $\bar{X}$ and $S$ be the sample mean and sample standard deviation, respectively, of a random sample of size $n$. What is

$$ \hbox{cov}(\bar{X}, S)? $$

My post ( https://larryzhangnz.netlify.app/post/cov-of-xbar-and-s/ ) gives an answer. In this small note, I present a quick solution, where the result is less accurate, but this is an interesting exercise.

Solution

According to Zhang (2007),

$$ \hbox{cov}(\bar{X}, S^2)=\frac{\mu_3}{n}, $$

where $\mu_3$ is the third central expectation of the population. As sample size $n$ is large, the sample standard deviation $S$ is almost surely equal to the population standard deviation $\sigma$. Therefore,

$$ \begin{array}{cl} & \hbox{cov}(\bar{X}, S)\\ =& \hbox{cov}(\bar{X}, S^2/S)\\ \approx & \hbox{cov}(\bar{X}, S^2/\sigma)\\ =&\frac{\displaystyle \mu_3}{\displaystyle n\sigma}. \end{array} $$

Reference

Zhang, L. (2007). Sample Mean and Sample Variance: Their Covariance and Their (In)Dependence. The American Statistician, 61, 159-160.