Cluster Sampling
A cluster sample is a probability sample in which each sampling unit is
a collection of elements. Two common reasons for using cluster sampling
are i) a frame of elements is either impossible or very costly, and ii) the
cost of sampling increases with the distance between the elements. When
using cluster sampling, the …rst decision is what to use as a cluster, several
examples of these considerations are discussed in the text. Once the clusters
are chosen, a frame of clusters is obtained and then a simple random sample
of clusters is taken.
Notation for cluster sampling:
N= the number of clusters in the population,
n= the number of clusters sampled,
mi= the number of elements in cluster i; i = 1;2;3; :::; N
m=1
n
n
X
i=1
mi=the average cluster size for the sample,
M=
N
X
i=1
mi=the number of elements in the population,
M=M=N =the average cluster size for the population,
yi=the total of all observations in the ith cluster.
Estimation of a population mean :
Our estimator of the population mean is just the total of all elements in
the sample divided by the number of elements in the sample:
y=
n
X
i=1
yi
n
X
i=1
mi
with b
V(y) = Nn
N 1
M2s2
r
n, where s2
r=
n
X
i=1
(yiymi)2
n1.
Note that the estimator yis a ratio estimator. The estimated variance
above is biased, so it is advisable to have n20 unless the miare equal.
Example: the number of hours of television watched per day
Suppose we visit a small community of N= 150 households, and we
randomly sample n= 10 households. For each sampled household we …nd
out how many people live at the household, and how many hours of TV are
watched per day by each of them:
1
