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

###### Asked by 1andrew.charles1957 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?

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###### March 30th, 2022 at 5:35 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.

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###### December 13th, 2018 at 2:49pm

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.

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###### October 30th, 2019 at 12:32am

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)