Sample Variance 2 - Free Download
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Proof that Sample Variance is Unbiased
Plus Lots of Other Cool Stuff
Scott D. Anderson
http://www.spelman.edu/~anderson/teaching/437/unbiased/unbiased.html
Fall 1999
Expected Value of S
2
The following is a proof that the formula for the sample variance, S
2
, is unbiased. Recall
that it seemed like we should divide by n, but instead we divide by n-1. Here's why.
First, recall the formula for the sample variance:
1
)(
)var(
2
1
2
−
−
==
∑
=
n
xx
Sx
n
i
i
Now, we want to compute the expected value of this:
[]
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−
−
=
∑
=
1
)(
2
1
2
n
xx
ESE
n
i
i
[]
⎥
⎦
⎤
⎢
⎣
⎡
−
−
=
∑
=
2
1
2
)(
1
1
n
i
i
xxE
n
SE
Now, let's multiply both sides of the equation by n-1, just so we don't have to keep
carrying that around, and square out the right side, just like we did with that shortcut
formula for SSX, above.
[]
⎥
⎦
⎤
⎢
⎣
⎡
+−=−
∑
=
n
i
ii
xxxxESEn
1
222
2)1(
