It is known that for the normal distribution, the mean and the standard deviation are independent; it is also the case for samples as sample size tends to infinity.
However, for small samples, the sample mean and the sample standard deviation are not independent. Indeed, the sample mean is used to compute the sample standard deviation.
As an example, simulating 100,000 samples of size 1000, and computing for each means and standard deviations, I get a correlation of 0.0005 between the 2 statistics. For 1,000,000 samples of size 3, I get a negative correlation of -0.0003.
Is there a way to find the expected correlation as a function of sample size? Its distribution? Or is it zero on average?