Why can't you average an average ?

  • Thread starter Thread starter Blewyn
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Blewyn

What's the formal or definitive explanation for why you can't average
dataset of averages ?

I understand about the skew effect if no account is taken of th
original sample sizes for the averages to be averaged, but what if the
are all the same size ? Would it then be valid to average them ?

Cheers

Blewy
 
You can.

Example: The average of 2 and 4 is 3. Another average of
4, 5, 7 and 8 is 6.

Now the average of averages is *not* (3+6)/2 = 4.5 but
(3*2 + 6*4)/6 = 5 because the first average has to be
weighted by 2 (2 numbers went into it) and the second one
by 4.

HTH,
sulprobil
 
You already understand the critical point, the size of the samples that make
up each of the averages.

Apply your high school algebra to see the proof. Write out formulas for the
average of cells A1:A10, cells A11:A20. Don't write the AVERAGE function,
write it the way you calculate it, i.e. SUM(A1:A10)/10, SUM(A11:A20)/10

Then write the formula for the average of those 2 averages, which is

[SUM(A1:A10)/10 + SUM(A11:A20)/10 ] / 2


See if you can rearrange the above to get SUM(A1:A20)/20

Then try it when the two averages are for the ranges A1:A5 and A6:A20.
 
This approach is called a weighted average. The weights (2/6 and 4/6)
are nonnegative and add to one, thus it is an average. The simple
arithmetic average is a special case of a weighted average, where the
weights all equal 1/n.

Jerry
 
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