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Topic: Expectation of the variance
Replies: 2   Last Post: Mar 23, 2011 6:28 PM

 Messages: [ Previous | Next ]
 Steven D'Aprano Posts: 21 Registered: 3/22/11
Expectation of the variance
Posted: Mar 23, 2011 5:16 PM

I'm trying to demonstrate numerically (rather than algebraically) that
the expectation of the sample variance is the population variance, but
it's not working for me.

The variance of a population is:

?^2 = 1/n * ?(x-?)^2 over all x in the population

where ^2 means superscript 2 (i.e. squared). In case you can't read the
symbols, here it is again in ASCII-only text:

theta^2 = 1/n * SUM( (x-mu)^2 )

If you don't have the entire population as your data, you can estimate
the population variance by calculating a sample variance:

s'^2 = 1/n * ?(x-?)^2 over all x in the sample

where s' is being used instead of s subscript n.

This is unbiased, provided you know the population mean mu ?. Normally
you don't though, and you're reduced to estimating it from your sample:

s'^2 = 1/n * ?(x-m)^2

where m is being used as the symbol for sample mean x bar = ?x/n

Unfortunately this sample variance is biased, so the "unbiased sample

s^2 = 1/(n-1) * ?(x-m)^2

What makes this unbiased is that the expected value of the sample
variances equals the true population variance. E.g. see

http://en.wikipedia.org/wiki/Bessel's_correction

The algebra convinces me -- I'm sure it's correct. But I'd like an easy
example I can show people, but it's not working for me!

Let's start with a population of: [1, 2, 3, 4]. The true mean is 2.5 and
the true (population) variance is 1.25.

All possible samples for each sample size > 1, and their exact sample
variances, are:

n = 2
1,2 : 1/2
1,3 : 2
1,4 : 9/2
2,3 : 1/2
2,4 : 2
3,4 : 1/2
Expectation for n=2: 5/3

n=3
1,2,3 : 1
1,3,4 : 7/3
2,3,4 : 1
Expectation for n=3: 13/9

n=4
1,2,3,4 : 5/3
Expectation for n=4: 5/3

As you can see, none of the expectations for a particular sample size are
equal to the population variance. If I instead add up all ten possible
sample variances, and divide by ten, I get 1.6 which is still not equal
to 1.25.

What am I misunderstanding?

--
Steven

Date Subject Author
3/23/11 Steven D'Aprano
3/23/11 bert
3/23/11 RGVickson@shaw.ca