Download SAMPLING VARIANCE OF THE SAMPLE MEAN

PLS 802
Spring 2017
Professor Jacoby
SAMPLING VARIANCE OF THE SAMPLE MEAN
I. Objective:
2 . The square
We will derive the formula for the sampling variance of the sample mean, σX̄
root of this formula is the standard error, or the standard deviation of the sampling
distribution for X̄. Assume that the data consist of n observations, sampled randomly
from the population.
II. The Sampling Variance of the Sample Mean:
Our strategy is to start with the definition of the sample mean, recognize that it is a
linear combination of the data values, and then employ the definition of the variance
of a linear combination in order to reach the desired result. Note that the derivation
employs the variance operator and the covariance operator. That is, var(X) refers to
2 , and cov(XY ) refers to the covariance of the variables
the variance of the variable X, σX
X and Y (note that all summations are taken over the n observations in the sample):
P
X̄
var X̄
=
=
X1
X2
Xn
+
+ ... +
+
n
n
n
=
1
1
1
X1 +
X2 + . . . +
Xn
n
n
n
=
var
=
=
=
=
=
2
var(X̄) = σX̄
X
n
=
1
1
1
X1 +
X2 + . . . +
Xn
n
n
n
2
2
2
1
1
1
var(X1 ) +
var(X2 ) + . . . +
var(Xn ) +
n
n
n
1 1
1 1
2
cov(X1 X2 ) + . . . + 2
cov(Xn−1 Xn )
n n
n n
2
1
n
2
σX
+
2
1
n
2 1
2
2
2
σX
+ σX
+ . . . + σX
n
2 h X
1
n
1
n2
2
σX
+ ... +
2
n σX
2
σX
=
2
1
n
2
σX
i
1 2
σ
n X
2
σX
n
Thus, the sampling variance of the sample mean is equal to the variance of X divided
by the sample size.