## What is best?

I have been learning a lot browsing through John D Cook’s blogs. I found *Are men better than women at chess *particularly illuminating, as well as the associated hyperlinks, esp. the 1961 paper he refers to (available in full text).

I reproduce the paper’s equations:

Consider, a random sample of size drawn from a standard normal distribution () and let be the probability density function and and the corresponding cumulative distribution function. In order to determine the order statistics, consider the sample ordered:

The probability distribution of the -order statistic can be derived (as shown in the paper) as follows:

- the number of ways of choosing k statistic: (there are (k-1)! ways of arranging the numbers below the k-statistic and (n-k)! ways of arranging the numbers above the k-statistic
- the density function follows from: k-1 observations are less than y
**and**n-k observations are greater than y**and**density of y - this leads to the probability density distribution:

- the corresponding cumulative distribution function:

The -statistic is the maximum of the sample. As observed in the paper, and is evident from above this yields:

Using the notation for the paper, let be , e.g median .

I explored these considerations by simulations and analytically. The 100 samples of sizes 100, 1000 and 10000 yielded the following:

The analytic relationship with the simulation data overlaid is shown in the following graphics (the second logarithmic horizontal axis):

In relation to the observations of John D Cook, the *gap *between median of maximums of samples of size n and size 0.1 n are shown in the next graphic: