550 Chapter 13:Quality Control
To estimate σ, let Si denote the sample standard deviation of theith subgroup,
i=1,...,k. That is,
S 1 =
√√
√√∑n
i= 1
(Xi−X 1 )^2
n− 1
S 2 =
√√
√
√
∑n
i= 1
(Xn+i−X 2 )^2
n− 1
..
.
Sk=
√√
√√∑n
i= 1
(X(k−1)n+i−Xk)^2
n− 1
Let
S=(S 1 +···+Sk)/k
The statisticSwill not be an unbiased estimator ofσ— that is,E[S]=σ. To transform
it into an unbiased estimator, we must first computeE[S], which is accomplished as
follows:
E[S]=
E[S 1 ]+···+E[Sk]
k
(13.2.1)
=E[S 1 ]
where the last equality follows sinceS 1 ,...,Skare independent and identically distributed
(and thus have the same mean). To computeE[S 1 ], we make use of the following
fundamental result about normal samples — namely, that
(n−1)S 12
σ^2
=
∑n
i= 1
(Xi−X 1 )^2
σ^2
∼χn^2 − 1 (13.2.2)
Now it is not difficult to show (see Problem 3) that
E[
√
Y]=
√
2 (n/2)
(n− 21 )
whenY∼χn^2 − 1 (13.2.3)
Since
E[
√
(n−1)S^2 /σ^2 ]=
√
n− 1 E[S 1 ]/σ