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
)