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78 August 31, 2006 2:55 Process Variations and Their Estimates
-- that is, s does not provide an accurate estimate of σ. In fact, s will generally provide
an underestimate of σ, that is s is smaller than the true value of σ. This is due to the
fact that s is based on the sample mean, while σ is based on the population mean. By
2
virtue of the property of the sum of squared deviations, (x i − ¯x) is minimal when ¯x
2
2
is the sample mean. Consequently, (x i − ¯x) < (x i − μ) when ¯x = μ.
Taking the square root of both sides of equation (6.1) yields a new random variable,
(n − 1) 1/2 s
Y = . (6.2)
σ
It turns out that Y follows a distribution known as the chi distribution with n − 1
5
degrees of freedom. For the chi distribution, it can be shown that
√ (n/2)
E(Y) = 2 .
((n − 1)/2)
Substituting (6.2) into (6.3), and rearranging, we obtain

2 (n/2)
E(s) = σ
n − 1 ((n − 1)/2)
= c 4 σ,

where (•) is the gamma function and the unbiased estimator of σ is
s
ˆ σ =
c 4
in which c 4 is given by

2 (n/2)
c 4 = .
n − 1 ((n − 1)/2)
Table 6.1 gives the values of c 4 for 2 ≤ n ≤ 25. For simplicity, for n > 25, the value of
c 4 can be approximated as
4(n − 1)
c 4 .
4n − 3

Table 6.1 Values of c 4 for sample size from 2 to 25.
n c 4 n c 4 n c 4

2 0.7979 10 0.9727 18 0.9854
3 0.8862 11 0.9754 19 0.9862
4 0.9213 12 0.9776 20 0.9869
5 0.9400 13 0.9794 21 0.9876
6 0.9515 14 0.9810 22 0.9882
7 0.9594 15 0.9823 23 0.9887
8 0.9650 16 0.9835 24 0.9892
9 0.9693 17 0.9845 25 0.9896