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