Suppose I have drawn n samples from a population of known mean and variance ( for example, a normal distribution with mean zero and variance 1.0 ).

I then calculate the mean and standard deviation of the sample.

How do I calculate the pdf of these sample values, given that I know the population values?

  • 1
    $\begingroup$ The procedure is illustrated at stats.stackexchange.com/questions/68984, as well as in many more of the thousands of posts found by searching sampling distribution. Most of them focus on the sample mean; a few on the sample correlation; and probably none of them describe how to compute the sampling distribution of the SD because (with few exceptions) it is an intractable calculation. $\endgroup$
    – whuber
    Feb 23, 2014 at 16:25
  • 1
    $\begingroup$ The variance is generally substantially more straightforward than the standard deviation. $\endgroup$
    – Glen_b
    Feb 23, 2014 at 16:38

1 Answer 1


The distributions of the sample mean and variance of a normal distribution are well-known (normal for the mean, Chi square for the variance). As whuber says, you can't find the pdfs of the sample mean $\overline{x}$ and, especially, the variance $s^2$ except in special situations. Given only the population mean $\mu$ and variance $\sigma^2$ and nothing else, all you can find exactly are sample mean and variance of $\overline{x}$ and the mean of $s^2$ (but not $s$):

Let the sample size be $n$. Then every introductory text on statistical theory demonstrates that:

$$ E(\overline{x})= \mu $$

$$ Var(\overline{x}) = \frac{\sigma^2}{n} $$ and $$ E(s^2) = \sigma^2 $$

If you know, in addition to $\mu$ and $\sigma^2$, the population fourth central moment $ \mu_4 = E[(X =\mu)^4]$, you can also compute the exact variance of $s^2$

$$Var(s^2) = \frac{(n-1)^2}{n^3}\left(\mu_4 - \frac{n-3}{n-1}\sigma^2\right) $$


CR Rao (1965) Linear Statistical inference and its applications, Wiley, New York, p.368.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.