Introduction to Probability and Statistics for Engineers and Scientists

6.4The Sample Variance 213

1.2

1.0

0.8

0.6

0.4

0.2

0.0

0.0 0.5 1.0 1.5 2.0 2.5

n = 1

n = 5

n = 10

FIGURE 6.4 Densities of the average of n exponential random variables having mean 1.

smaller sample sizes. Indeed, a sample of size 5 will often suffice for the approximation
to be valid. Figure 6.4 presents the distribution of the sample means from an exponential
population distribution for samples of sizesn=1, 5, 10.

6.4The Sample Variance

LetX 1 ,...,Xnbe a random sample from a distribution with meanμand varianceσ^2. Let
Xbe the sample mean, and recall the following definition from Section 2.3.2.

Definition

The statisticS^2 , defined by

S^2 =

∑n

i= 1

(Xi−X)^2

n− 1

is called thesample variance.S=

√
S^2 is called thesample standard deviation.
To computeE[S^2 ], we use an identity that was proven in Section 2.3.2: For any
numbersx 1 ,...,xn

∑n

i= 1

(xi−x)^2 =

∑n

i= 1

xi^2 −nx^2