Sample Variance 2 - Free Download | Page 3
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[] []
[]
2
2
22
var
1
μ
+==
∑
i
x
n
xEYE
[] []
22
2
22
1
μσ
+==
∑
n
xEYE
[] []
22
2
22
)(
1
μσ
+== n
n
xEYE
[] []
2222
1
μσ
+==
n
xEYE
We can substitute this stuff for the second term on the RHS of equation 1
. Also, note that
the first term on the RHS of eqation 1
is the second moment of X, so that can also be re-
written. Doing both substitutions gives us:
[][ ]
⎥
⎦
⎤
⎢
⎣
⎡
+−+=
−
μσμσ
2222
11
n
SE
n
n
[]
222
11
σσ
n
SE
n
n
−=
−
[
]
22
σ
=SE
Whew! That was hard, but solvable. This is why S
2
with the n-1 denominator is an
unbiased estimator.
