variance of sample median of normal distribution

Question

Does anyone tell me how to find the variance of sample median when random samples are normally distributed?? I know that I need to use the order statistics, but actually I don't understand how to apply it.

Answer 1

I will make some illustrations with R. First, using results discussed at Central limit theorem for sample medians the asymptotic distribution of sample median of $N$ iid observations from a normal distribution is itself a normal distribution centered at the theoretical median $\mu$ and with variance $\frac{2\pi \sigma^2}{4 N}$ where $\sigma^2$ is the variance of the parent distribution. The sample median is the central order statistic, and in the case $N=2s+1$ is odd, it is the order statistic of order $s+1$, which have a distribution we can calculate exactly rather easily (theory given in the cited post and elsewhere at this site). I will use some R function I detail at the end of this post to compare the exact and approximate distributions of the sample median. We use the standard normal for parent distribution.

First with sample size $N=11$:

The approximation is already quite good! Let us look at the case $N=101$:

Here the approximation is virtually exact. Some R functions for distributions of order statistics:

porder  <-  function(x, N=1, r=1, basename="norm", ..., log.p=FALSE) {
     pfun  <-  get(paste("p", basename, sep=""))
     stopifnot(r  <=  N)
     prob  <-  pfun(x, ...)
     retval  <-  pbeta(prob, r, N-r+1, log.p=log.p)
     retval
}

dorder <-  function(x, N=1, r=1, basename="norm", ..., log=FALSE)  {
    pfun  <-  get(paste("p", basename, sep=""))
    dfun  <-  get(paste("d", basename, sep=""))
    stopifnot(r <= N)    
    logdens  <-  -lbeta(r, N-r+1) + (r-1)*pfun(x, ..., log.p=TRUE) + 
        (N-r)*pfun(x, ..., lower.tail=FALSE, log.p=TRUE) + 
        dfun(x, ..., log=TRUE)   
    retval  <-  if (log) logdens else exp(logdens)
    retval
}

qorder  <-  function(p, N=1, r=1, basename="norm",  ..., log.p=FALSE) {
    pfun  <-  get(paste("p", basename, sep=""))
    qfun  <-  get(paste("q", basename, sep=""))
    stopifnot( r <= N)
    retval  <-  qfun(qbeta( p, r, N-r+1, log.p=log.p), ..., 
                            log.p=FALSE)
    retval
    }

rorder  <-  function(n, N=1, r=1, basename="norm", ...)  {
    qfun  <-  get(paste("q", basename, sep=""))
    qfun( rbeta(n, r, N-r+1), ... )
}