Why is the average of the highest value from 100 draws from a normal distribution different from the 98th percentile of the normal distribution?

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

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