# What is the mean and std dev of the minimum of 6 random variables having a normal distribution?

###### Asked by peaceharris on January 16, 2014

Let’s say the average height of people is 150cm, with a std dev of 20cm, and their heights are normally distributed. Now if I group these people randomly in groups of 6 people, and I just take the height of the shortest person in each group. What is the mean and std dev of these shortest people?

## Answers

## The One Who Knows (Swamy)

## August 13, 2007 at 1:33 am

Without doing any maths, I would guess that the mean would be 147 cm and the standard deviation would remain same. But I could be wrong since I left statistics long ago.

## Champoleon

## August 13, 2007 at 1:44 am

This is “Order Statistics”

It is straigthforward to write the expression of the mean and variance but the computation is to be given to a math engine (MATLAB is my prefered)

## Wala Lang

## August 13, 2007 at 4:46 am

The cumulative distn of the minimum is 1-[1-F(x)]^6 and its pdf is 6{[1-F(x)]^5}f(x), where F(x) is the normal cdf and f(x) is normal pdf.

You can use either the cdf or pdf to derive the mean and variance.