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7-2.2 Proof That S is a Biased Estimator of (CD Only)

We proved that the sample variance is an unbiased estimator of the population variance, that
2
2
is, E(S ) , and that this result does not depend on the form of the distribution. However,
the sample standard deviation is not an unbiased estimator of the population standard devia-
tion. This is easy to demonstrate for the case where the random variable X follows a normal
distribution.
Let X , X , p , X be a random sample of size n from a normal population with mean
1
2
n
2
and variance . Now it can be shown that the distribution of the random variable
1n 12 S 2
2
2
is chi-square with n 1 degrees of freedom, denoted n 1 (the chi-squared distribution
was introduced in our discussion of the gamma distribution in Chapter 4, and the above re-
2
2
sult will be presented formally in Chapter 8). Therefore the distribution of S is 1n 12
2
times a n 1 random variable. So when sampling from a normal distribution, the expected
2
value of S is
2 2 2
2 2
2
E1S 2 E a n 1 b E 1 n 1 2 1n 12
2
n 1 n 1 n 1
because the mean of a chi-squared random variable with n 1 degrees of freedom is n 1.
Now it follows that the distribution of
11n 12S

is a chi distribution with n 1 degrees of freedom, denoted n 1 . The expected value of S can
be written as

E1S2 E a n 1 b E1 n 1 2
1n 1 1n 1

The mean of the chi distribution with n 1 degrees of freedom is

1n 22
E 1 n 1 2 22
31n 12 24
r 1 y
where the gamma function 1r2 y e dy. Then
0
2 1n 22
E 1S2
Bn 1 31n 12 24

c n
Although S is a biased estimator of , the bias gets small fairly quickly as the sample size
0.94 for a sample of n 5, c 0.9727 for a sample
n increases. For example, note that c n n
of n 10, and c 0.9896 or very nearly unity for a sample of n 25.
n
7-1