Fundamentals of Probability and Statistics for Engineers

9.1.2 Sample Variance

The statistic

is called the sample variance of population X. The mean of S^2 can be found by
expanding the squares in the sum and taking termwise expectations. We first
write Equation (9.7) as

Taking termwise expectations and noting mutual independence, we have

where m and^2 are defined in Equations (9.4). We remark at this point that the
reason for using 1/(n 1) rather than 1/n in Equation (9.7) is to make the mean
of S^2 equal to^2. As we shall see in the next section, this is a desirable property
for S^2 if it is to be used to estimate^2 , the true variance of X.
The variance of S^2 is found from

var

Upon expanding the right-hand side and carrying out expectations term by
term, we find that

where 4 is the fourth central moment of X; that is,

Equation (9.10) shows again that the variance of S^2 is an inverse function of n.

262 Fundamentals of Probability and Statistics for Engineers

S^2 ˆ

1

n 1

Xn

iˆ 1

…XiX†^2 … 9 : 7 †

S^2 ˆ

1

n 1

Xn

iˆ 1

‰…Xim†…Xm†Š^2

ˆ

1

n 1

Xn

iˆ 1

…Xim†

1

n

Xn

jˆ 1

…Xjm†

"# 2

ˆ

1

n

Xn

iˆ 1

…Xim†^2 

1

n…n 1 †

Xn

i;jˆ 1

i6ˆj

…Xim†…Xjm†:

EfS^2 gˆ^2 ; … 9 : 8 †









fS^2 gˆEf…S^2 ^2 †^2 g: … 9 : 9 †

varfS^2 gˆ

1

n

 4 

n 3

n 1

^4



; … 9 : 10 †



 4 ˆEf…Xm†^4 g: … 9 : 11 †