Applied Statistics and Probability for Engineers

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

We proved that the sample variance is an unbiased estimator of the population variance, that

is, E(S^2 ) ^2 , 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 Xfollows a normal

distribution.

Let X 1 , X 2 ,p, Xnbe a random sample of size nfrom a normal population with mean 

and variance ^2. Now it can be shown that the distribution of the random variable

is chi-square with n1 degrees of freedom, denoted (the chi-squared distribution

was introduced in our discussion of the gamma distribution in Chapter 4, and the above re-

sult will be presented formally in Chapter 8). Therefore the distribution of S^2 is

times a random variable. So when sampling from a normal distribution, the expected

value of S^2 is

because the mean of a chi-squared random variable with n1 degrees of freedom is n1.

Now it follows that the distribution of

is a chi distribution with n1 degrees of freedom, denoted. The expected value of Scan

be written as

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

where the gamma function Then

Although Sis a biased estimator of , the bias gets small fairly quickly as the sample size

nincreases. For example, note that cn0.94 for a sample of n5, cn0.9727 for a sample

of n10, and cn0.9896 or very nearly unity for a sample of n25.

cn 

E 1 S 2 

B

2

n 1

 1 n 22

 31 n (^12)  24


 1 r 2 



0

yr^1 ey^ dy.

E (^1) n 12  22
 1 n 22
 31 n (^12)  24
E 1 S 2 E a

1 n 1
(^) n 1 b

1 n 1
E (^1) n 12
(^) n 1
11 n 12 S

E 1 S^22 E a
^2
n 1
(^2) n^  1 b
^2
n 1
E (^1 2) n^  12 
^2
n 1
1 n 12 ^2
(^2) n 1
^2  1 n 12
(^2) n 1
1 n 12 S^2
^2
7-1
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