What is the mean and variance of the median of a set of i.i.d normal random variables?

Question

Let $X_1$, ..., $X_n$ be identically independently distributed random variables with $N(\mu, \sigma^2)$

It is easy to show that sample mean $\bar{X} = \frac{1}{n}\sum^n_{i = 0}{X_i}$ is a random variable with $N(\mu, \frac{\sigma^2}{n})$.

However, I am having a hard time finding what the distribution of sample median $median(X)$ is, especially in terms of its mean and variance.

I ask because I am trying to summarize some features in predefined groups in order to reduce the number of tests that I have to do between two conditions.

If there is no simple answer to this, as I suspect, I would be interested in knowing the variance of $median(X)$, especially how it differs from $\bar{X}$.

Answer 1

The median is the central order statistic when the number of observations is odd. If $n$ is even then the median is either an order statistic, or the mean of 2 order statistics (or something else) depending on which definition of median you use.

So the exact distribution of the median can be worked out based on the distribution of order statistics. For odd $n$ where all the $x$'s are iid from a pdf $f$ with cumulative distribution $F$ the distribution of the median is:

$\binom{n-1}{(n-1)/2} F(x)^{\frac{n-1}2} f(x) (1-F(x))^{\frac{n-1}2}$

You can google "distribution of order statistics" to get more details and derivation.

For the normal we don't have a closed form solution for $F(x)$, but there are computational tools that can help estimate the above (see the distr package for R for one possibility).

If your main goal is just an estimate of the variance of the median, then a simpler approach is just to simulate a bunch of datasets and compute the variance of their medians (and the variance of their means for comparison).

The Wikipedia article on "Median" also has information that may be of interest.