Why is the average of the highest value from 100 draws from a normal distribution different from the 98% percentile of the normal distribution? It seems that by definition that they should be the same. But…

Code in R:

`NSIM <- 10000 x <- rep(NA,NSIM) for (i in 1:NSIM) { x[i] <- max(rnorm(100)) } qnorm(.98) qnorm(.99) mean(x) median(x) hist(x)`

I imagine that I’m misunderstanding something about what the maximum of a 100 draws from the normal distribution should be. As is demonstrated by an unexpectedly asymetrical distribution of maximum values.

**Answer**

The maximum does not have a normal distribution. Its cdf is $\Phi(x)^{100}$ where $\Phi(x)$ is the standard normal cdf. In general the moments of this distribution are tricky to obtain analytically. There is an ancient paper on this by Tippett (*Biometrika*, 1925).

**Attribution***Source : Link , Question Author : russellpierce , Answer Author : Macro*