# How to calculate standard error of sample quantile from normal distribution with known mean and standard deviation?

I know that the standard error of the mean for an iid sample is calculated as $$\frac{\sigma}{\sqrt{n}}$$

However, assuming a normal distribution with known mean and standard deviation, how do you calculate the standard error of an arbitrary quantile?

For example, assume

• normal distribution
• population mean = 0
• population standard deviation = 1
• n=100
• quantile = .95

What would be the standard error of this quantile?

I ran this little simulation to explore the properties, but I'm still interested in the closed form solution:

set.seed(1234)
generate_x <- function(n) x <- rnorm(n)
k <- 10000
n <- 100

results <- lapply(seq(k), function(X) generate_x(100))

Z <- seq(.01, .99, .01)
qresults <- sapply(results, function(X) quantile(X, Z))

sd_quresults <- apply(qresults, 1, sd)
var_quresults <- apply(qresults, 1, var)

plot(Z, sd_quresults, type='l') • This plot looks remarkably like the output of curve(sqrt(x*(1-x)/100) / abs(dnorm(qnorm(x))), 0, 1) :-).
– whuber
Aug 5, 2014 at 1:16
• @whuber Is there a need to put abs if you have dnorm? Shouldn't dnorm be non-negative? Aug 5, 2014 at 3:01
• @Glen_b Thank you; that's correct. The abs is a vestige of an earlier (messier) incarnation of this formula (when it had been applied to qnorm(x), which could be negative). In my haste I did not remove it.
– whuber
Aug 5, 2014 at 12:04

In the case of sample quantiles, the standard error depends on which definition of sample quantiles you actually use. I believe R, for example, includes 9 different definitions of quantiles in its quantile function.
I encountered the problem of computing the standard error for sample quantiles of a normal distribution while attending a course of Quantitative Finance during my MSc. The main topic related to this problem regarded the analysis of Value at Risk. This is the closed form solution: $$s.e.\left( \widehat{z}_{\alpha}\right)=\sigma \cdot \sqrt{\dfrac{1}{n}} \cdot \sqrt{1+2z_{\alpha}^2\left( \dfrac{n-1}{2}-\dfrac{\Gamma^2(n/2)}{\Gamma^2((n-1)/2)} \right)}$$ unfortunatly until now I can't provide you any reference for the proof.