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Mean, Average, and Standard Deviation

Date: 04/18/2001 at 21:05:48
From: Matthew
Subject: Statistics

I was wondering if you could help me explain the following:

If you have three groups of data and you find the mean of each group, 
and then find the average of these three means, you get the same 
number as if you put the data from all three groups into a single 
group and calculate the average.

However, if you do the same thing with the standard deviation, the 
average of the three standard deviations of the three groups is not 
equal to the standard deviation of the three groups put together.

I can see why this happens, but I don't know how to explain it.


Date: 04/19/2001 at 14:11:59
From: Doctor TWE
Subject: Re: Statistics

Hi Matthew - thanks for writing to Dr. Math.

Your assertion that 'the average of the means of the groups is the 
same as calculating the mean of a single group consisting of the data 
from all groups' is not entirely correct; it is only true if either 
(a) all original groups are of the same size, or (b) you use a 
weighted average of the means, using group sizes as the weights.

For example, consider the following three groups:

     A = {10,10}                        n = 2   mean = 10
     B = {20,20,20}                     n = 3   mean = 20
     C = {30,30,30,30,30,30,30,30,30}   n = 9   mean = 30

The average of the means is:

     mean = (10+20+30)/3
          = 20

But if we put them into a single group we'd have:

     S = {10,10,20,20,20,30,30,30,30,30,30,30,30,30}   n = 14

     mean = (10+10+20+20+20+30+30+30+30+30+30+30+30+30)/14
          = 350/14
          = 25

Why are these different? Because the third group - with an average of 
30 - has more members, and therefore counts more when averaging the 
combined group S.

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum   
Associated Topics:
High School Statistics

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