Introduction to Probability and Statistics for Engineers and Scientists

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