UNEQUAL CLUSTER SAMPLING

 

GADHA M KURUP

2148130

DEPT. OF STATISTICS

CHRIST DEEMED TO BE UNIVERSITY, BANGALORE


INTRODUCTION

In statistics, Cluster sampling is a sampling method in which the entire population of the study is divided into externally homogenous but internally heterogeneous groups called clusters. Essentially each cluster is a mini-representation of the entire population.

Clusters are generally made up of neighboring elements and thus the elements within a cluster tend to have similar characteristics. The general rule is that the number of elements in a cluster should be small and the number of clusters should be large.




There exists two types of cluster sampling :

EQUAL CLUSTER SAMPLING: Under this , the number of elements in each cluster will be different.

UNEQUAL CLUSTER SAMPLING: Clusters are of different size.


Here, we will be dealing with Unequal Cluster Sampling


MERITS AND DEMERITS OF CLUSTER SAMPLING

MERITS:

  • Economical
  • Quick or less time consuming
  • Requires fewer resources
  • More feasible
  • Larger sample size can be used due to increased level of accessibility of perspective sample group members
  • Easy to be used from practicality viewpoint


DEMERITS :

  • Biased sample
  • Sampling error
  • Cluster sampling may fail to reflect the diversity in the sampling frame


FORMULAS :

Suppose there are N clusters. Let the ith cluster consist of Mi elements (i=1,2,3…..N)

And   


 .

The population mean per element Ybar is defined by   

 


    i=1,2,3,…..N where yibar is the mean per element of the ith cluster.

 

Let a random sample wor of n clusters be drawn and all elements of the clusters surveyed,. The estimator of ybar can be given by





     i=1,2,…..n

 

Similarly the unbiased estimator of Var(ybar) can be given by



APPLICATIONS :

  • Cluster sampling is area sampling or geographical cluster sampling. Each cluster is a zero graphical area because a geographically dispersed population can be achieved by grouping several respondents with in a local area into a cluster. 
  • It is used to estimate high mortalities increase such as wars, famines and natural disasters.
  • Image analysis.
  • Cluster sampling is also typically used in market research. For instance, it Is used when a researcher can’t get information about the population as a whole, but then they can obtain information about the clusters. For instance, suppose a researcher may be interested in data about City taxes of a particular State (suppose Karnataka) The researcher would obtain data from selected cities and compile them to get a picture about the state Condition. Here, the individual cities would be considered as clusters. 
  • Usually , it is difficult to find equal size clusters from the population which depict homogeneous characteristics between clusters. Hence unequal cluster sampling is used.

QUESTION :

A survey on pepper was conducted to estimate the number of pepper standards and production of pepper in Kerala. For this 3 clusters from 95 were selected by srswor. The information on the number of pepper standards  recorded is given below:


 Cluster No.

 Cluster size

 No. of pepper standards

 1

11 

 41,16,19,15,144,454,212,57,28,76,199

 2

12 

 39,70,38,37,161,38,27,219,46,128,30,20

 3

 7

 115,59,120,36,411,197,17






  1. Find the estimator of population mean per element
  2. Find Sb(square) which is an unbiased estimator of variance and hence the standard error.
  3. Examine the relative efficiency of unequal cluster sampling w.r.t. simple random sampling


ANALYSIS USING R CODES :





















INFERENCE :

The clusters are of three different sizes – 11,12 and 7.

The mean of each cluster is calculated and the average cluster size is found to be 10.

The average pepper standards is calculated and is found to be 102.3.

The standard error for the region is found to be 81.22741.

The relative efficiency of unequal cluster sampling w.r.t. simple random sampling is calculated and is found to 0.87 which implies it is 87% more efficient because the sampling variance between the clusters is small.

 

CONCLUSION :

Cluster sampling is a great way for researchers to study an entire population –without having to survey the entire population. It’s cost-effective, efficient, offers easier analysis, and is generally very reliable. In most cases it is impossible to find equal size clusters from the population which depict homogeneous characteristics between clusters and remain heterogenous within a cluster so that it can be a true representative of the population. Hence unequal cluster sampling is used. It is the most economical and practical solution for statisticians doing research.


















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