Introduction to Probability and Statistics for Engineers and Scientists

6.2The Sample Mean 203

Since the value of the sample meanXis determined by the values of the random variables
in the sample, it follows thatXis also a random variable. Its expected value and variance
are obtained as follows:

E[X]=E

[

X 1 +···+Xn

n

]

=

1

n

(E[X 1 ]+···+E[Xn])

=μ

and

Var(X)=Var

(

X 1 +···+Xn

n

)

=

1

n^2

[Var(X 1 )+···+Var(Xn)] by independence

=

nσ^2

n^2

=

σ^2

n

whereμandσ^2 are the population mean and variance, respectively. Hence, the expected
value of the sample mean is the population meanμwhereas its variance is 1/ntimes
the population variance. As a result, we can conclude thatXis also centered about the
population meanμ, but its spread becomes more and more reduced as the sample size
increases. Figure 6.1 plots the probability density function of the sample mean from
a standard normal population for a variety of sample sizes.

n = 10

n = 4

n = 2

n = 1

− 4 − 224

FIGURE 6.1 Densities of sample means from a standard normal population.