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
x
x
S
x
n
i
i
Now, we want to compute the expected value of this:
[]
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−
−
=
∑
=
1
)
(
2
1
2
n
x
x
E
S
E
n
i
i
[]
⎥
⎦
⎤
⎢
⎣
⎡
−
−
=
∑
=
2
1
2
)
(
1
1
n
i
i
x
x
E
n
S
E
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
i
i
x
x
x
x
E
S
E
n
1
2
2
2
2
)
1
(
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−
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 Spring '10
 Dr.Dmitry
 Standard Deviation, Variance, probability density function

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