Introduction to Probability and Statistics for Engineers and Scientists

13.2Control Charts for Average Values: TheX-Control Chart 551

we see from Equations 13.2.2 and 13.2.3 that

E[S 1 ]=

√

2 (n/2)σ

√

n− 1 (n− 21 )

Hence, if we set

c(n)=

√

2 (n/2)

√

n− 1 (n− 21 )

then it follows from Equation 13.2.1 thatS/c(n) is an unbiased estimator ofσ.
Table 13.1 presents the values ofc(n) forn=2 throughn=10.

TECHNICAL REMARK

In determining the values in Table 13.1, the computation of (n/2) and (n−^12 ) was
based on the recursive formula

(a)=(a−1) (a−1)

TABLE 13.1 Values of c(n)

c(2) = .7978849

c(3) = .8862266

c(4) = .9213181

c(5) = .9399851

c(6) = .9515332

c(7) = .9593684

c(8) = .9650309

c(9) = .9693103

c(10) = .9726596

which was established in Section 5.7. This recursion yields that, for integern,

(n)=(n−1)(n−2)··· 3 · 2 · 1 · (1)

=(n−1)! since (1)=

∫∞

0

e−xdx= 1

The recursion also yields that

(

n+ 1

2

)

=

(

n− 1

2

)(

n−

3

2

)

···

3

2

·

1

2

·

(

1

2

)