Cluster Sampling, Idea of two stage sampling and Multi stage sampling
2.2
Cluster Sampling
It is one of the basic assumptions in
any sampling procedure that the population can be divided into a finite number
of distinct and identifiable units, called sampling units. The smallest units
into which the population can be divided are called elements of the population.
The groups of such elements are called clusters.
In many practical situations and many
types of populations, a list of elements is not available and so the use of an
element as a sampling unit is not feasible. The method of cluster sampling or
area sampling can be used in such situations.
In
cluster sampling,
-Divide
the whole population into clusters according to some well-defined rule.
-Treat
the clusters as sampling units.
-Choose
a sample of clusters according to some procedure.
-Carry out a complete enumeration of the selected
clusters, i.e. collect information on all the sampling
units available in selected clusters.
Area
sampling
In case, the entire area containing the
populations is subdivided into smaller area segments and each element in the
population is associated with one and only one such area segment, the procedure
is called as area sampling.
Examples:
1.
In a city, the list of all the individual persons staying in the houses may be
difficult to obtain or even maybe not available but a list of all the houses in
the city may be available. So every individual person will be treated as
sampling unit and every house will be a cluster.
2.
The list of all the agricultural farms in a village or a district may not be
easily available but the list of village or districts are generally available.
In this case, every farm is sampling unit and every village or district is the
cluster.
It is easier, faster, cheaper and
convenient to collect information on clusters rather than on sampling units
In
above examples, draw a sample of clusters from houses/villages and then collect
the observations on all the sampling units available in the selected clusters.
Real life situations where Cluster Sampling is used
1.
Geographical
areas: When the population is spread over a large geographical region and it is
difficult to reach every individual, like conducting household surveys across
cities.
2.
Educational
studies: Sampling schools (clusters) rather than individual students for
educational assessments.
3.
Healthcare: For
health surveys, clusters can be formed from hospitals or regions, and then
surveys are conducted within selected clusters.
4.
Market research:
When targeting a specific demographic spread over various localities, a company
might sample entire communities.
ii)
Even if the list of elements is available, the location or identification of
the units may be difficult
iii)
A necessary condition for the validity of this procedure is that every unit of
the population under study must correspond to one and only one unit of the
cluster so that the total number of sampling units in the frame may cover all
the units of the population under study without any omission or duplication.
When this condition is not satisfied, bias is introduced.
The clusters are constructed such that
the sampling units are heterogeneous within the clusters and homogeneous among
the clusters.
This is opposite to the construction of
the strata in the stratified sampling. There are two options to construct the
clusters equal size and unequal size. We discuss the estimation of population
means and its variance in equal size case.
Stratified Sampling:
In this method, the population is
divided into strata (groups) based on shared characteristics, and these
strata are internally homogeneous but externally heterogeneous.
This means that the members within a stratum are similar, but different strata
differ from each other. The idea is to improve precision by ensuring each
stratum represents its subgroup of the population.
In cluster
sampling, the population is divided into clusters, which are internally
heterogeneous but externally homogeneous. That is, the elements
within a cluster are diverse (representing the population), but clusters as a
whole are similar to each other. The aim is to sample a few clusters that act
as mini-populations representing the entire group.
Stratified
sampling focuses on reducing variance and ensuring representativeness of each
subgroup. Cluster sampling focuses on reducing costs and making data collection
more practical.
N: Total number
of clusters in the population.
n: Number of
clusters selected for the sample.
M: Number of
elements within each cluster (assuming equal cluster sizes).
Yij: jth element in the ith
cluster in population (j=1, 2, .. , M ; i=1,2,…,N)
yij: jth element in the ith
cluster in sample (j=1, 2, .. , M ; i=1,2,…,n)
Population size = NM, Sample size = nM
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