# Confidence interval of quantile / percentile of the normal distribution

What is the formula (if it exists) for the sample variance / confidence interval of a quantile / percentile of the normal distribution?

For example, the 5th percentile for a standard normal population distribution is -1.64, but what is the 95% one-sided confidence interval if I have a sample of n=1,000? I.e. on average the 5th percentile of a standard normal sample will be -1.64 and 95% of the time the sample 5th percentile of a sample of n=1,000 will be below -1.54 (approximate from simulations).

There seems to be some kind of formula as it's an option in SAS (CIQUANTNORMAL) and I think it may be related to the Probit but haven't found an explanation.

Suppose that $X\sim N(\mu,\sigma)$, where $\mu$ and $\sigma$ are unknown. Since $\bar{x}$ and $s$ are independent, it is pretty easy to calculate confidence bounds for any linear combination of $\mu$ and $\sigma$.
Now, all you need to remember is that the 5th percentile of $X$ is, as you note, $\mu-1.64\sigma$.