Page 279 - Fundamentals of Probability and Statistics for Engineers

262 Fundamentals of Probability and Statistics for Engineers
9.1.2 SAMPLE VARIANCE

The statistic
n
1 X 2
2
S ˆ X i X† 9:7†
n 1
iˆ1
2
is called the sample variance of population X. The mean of S can be found by
expanding the squares in the sum and taking termwise expectations. We first
write Equation (9.7) as
n
1 X 2
2
S ˆ ‰ X i m† X m†Š
n 1
iˆ1
" # 2
n n
1 X 1 X
ˆ X i m† X j m†
n 1 n
iˆ1 jˆ1
n
n
1 X 2 1 X
ˆ X i m† X i m† X j m†:
n n n 1†
iˆ1 i; jˆ1
i6ˆj
Taking termwise expectations and noting mutual independence, we have
2
2
EfS gˆ ; 9:8†
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
2
of S equal to 2 . As we shall see in the next section, this is a desirable property
2
for S if it is to be used to estimate 2 , the true variance of X.
2
The variance of S is found from
2 2
2
2
varfS gˆ Ef S † g: 9:9†
Upon expanding the right-hand side and carrying out expectations term by
term, we find that

1 n 3 4
2
varfS gˆ 4 ; 9:10†
n n 1
where 4 is the fourth central moment of X; that is,
4
4 ˆ Ef X m† g: 9:11†
2
Equation (9.10) shows again that the variance of S is an inverse function of n.

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