For large $N$ the sample median is approximately normally distributed with mean $μ$ and variance $π/2N$. The efficiency for large $N$ is thus $2/π≈0.64$

  • Can somebody explain this for me?
  • Where does that variance come from?
  • and why then ≈0.64?
  • 2
    $\begingroup$ The sample mean has variance $\frac{\sigma^2}N$ (optimal for a sample from a normal distribution) and $\dfrac{\sigma^2/N}{\pi\sigma^2/(2N)} \approx 0.64$ $\endgroup$
    – Henry
    Nov 21 '21 at 15:51
  • 1
    $\begingroup$ The distribution of the median is like a beta distributed variable transformed by the quantile function. For increasing $n$ the beta distribution approaches a normal distribution with decreasing standard deviation. Then you can apply the Delta method to describe the distribution of the median. $\endgroup$ Nov 21 '21 at 16:18
  • 2
    $\begingroup$ Here you see that (as Laplace derived) the variance will be $\frac{1}{4nf(m)^2}$. The distribution density pops up because it relates to the derivative of the quantile function. If you fill in the density of a standard normal distribution then you get your result. (So your example counts as the sample median for a sample taken from a normal distributed population). $\endgroup$ Nov 21 '21 at 16:24
  • 3
    $\begingroup$ stats.stackexchange.com/questions/45124 $\endgroup$
    – whuber
    Nov 21 '21 at 16:46

The most accessible theoretical demonstrations may be linked in the second Comment of @SextusEmpiricus and in @whuber's link.

Hoping that $n = 100$ is large enough to see a suggestive approximation of the ratio $2/\pi$ (for normal data), perhaps the following simple simulation in R of $10^5$ samples of size $n=100$ might give a view of this fact.

n = 100                   # obs per sample (col)
m = 10^5                  # samples (row)
x = rnorm(n*m, 50, 7)
MAT = matrix(x, nrow=m)
a = rowMeans(MAT)         # 10^5 sample means
h = apply(MAT, 1, median) # 10^5 sample medians
[1] 0.4879295
[1] 0.49

[1] 0.7555167
[1] 0.6458223  # aprx 2/pi [0.6406 for n=200]
[1] 0.6366198

enter image description here

The distribution of sample means is exactly normal; the distribution of sample medians is very nearly normal (ever closer as $n \rightarrow \infty).$ R code for the figure is shown below.

 hist(a, prob=T, br=30, xlim=c(45,55), col="skyblue2", 
      main="Dist'n of Means")
  curve(dnorm(x, 50, 7/10), add=T, col="orange", lwd=2)
 hist(h, prob=T, br=30, xlim=c(45,55), col="skyblue2", 
      main="Dist'n of Medians")
  curve(dnorm(x, mean(h), sd(h)), add=T, col="orange", lwd=2)

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