Stratified Sampling

 1.3 Stratified Sampling

 Introduction: 

An important objective in any estimation problem is to obtain an estimator of a population parameter which can take care of the salient features of the population. If the population is homogeneous with respect to the characteristic under study, then the method of simple random sampling will yield a homogeneous sample, and in turn, the sample mean will serve as a good estimator of the population mean. Thus, if the population is homogeneous with respect to the characteristic under study, then the sample drawn through simple random sampling is expected to provide a representative sample. Moreover, the variance of the sample mean not only depends on the sample size and sampling fraction but also on the population variance. In order to increase the precision of an estimator, we need to use a sampling scheme which can reduce the heterogeneity in the population. If the population is heterogeneous with respect to the characteristic under study, then one such sampling procedure is a stratified sampling.

 The basic idea behind the stratified sampling is to

i. divide the whole heterogeneous population into smaller groups or subpopulations, such that the sampling units are homogeneous with respect to the characteristic under study within the subpopulation and

ii. heterogeneous with respect to the characteristic under study between/among the subpopulations. Such subpopulations are termed as strata.  

iii. Treat each subpopulation as a separate population and draw a sample by SRS from each stratum. (Note: ‘Stratum’ is singular and ‘strata’ is plural).

Example: In order to find the average height of the students in a school of class 1 to class 12, the height varies a lot as the students in class 1 are of age around 6 years, and students in class 10 are of age around 16 years. So one can divide all the students into different subpopulations or strata such as

Students of class 1, 2 and 3: Stratum 1

Students of class 4, 5 and 6: Stratum 2

Students of class 7, 8 and 9: Stratum 3

Students of class 10, 11 and 12: Stratum 4

Now draw the samples by SRS from each of the strata 1, 2, 3 and 4. All the drawn samples combined together will constitute the final stratified sample for further analysis.

Some real life situations where Stratified Random Sampling is used:

ⅰ) To study the income tax returns considered strata as a income group.

ii) To study the socioeconomic factor considers either rural or urban population as strata.

ⅲ) To study the life time of trees using stratified random sampling, consider strata as trees of a same species.


Estimation of parameters in stratified sampling

 Divide the population of N units in k strata. Let the ith stratum has, Ni,  i = 1, 2,…, k number of units. Note that there are k independent samples drawn through SRS of sizes n1, n2,..., nk , from each of the strata. So, one can have k estimators of a parameter based on the sizes n1, n2,..., nk respectively. Our interest is not to have k different estimators of the parameters, but the ultimate goal is to have a single estimator. In this case, an important issue is how to combine the different sample information together into one estimator, which is good enough to provide information about the parameter.

We now consider the estimation of population mean and population variance from a stratified sample. 

     

    




                                                                    
                            
















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