RELATIVE PRECISION OF STRATIFIED RANDOM SAMPLING WITH SIMPLE RANDOM SAMPLING

 

RELATIVE PRECISION OF STRATIFIED RANDOM SAMPLING

WITH

SIMPLE RANDOM SAMPLING

 

-Garima Singh

 

Here we are mainly focusing on a comparative study of SRSWOR & Stratified Random Sampling under various methods of allocations.

 

Let’s start with a simple introduction.

 

What is sampling?

For the purpose of determining population characteristics, instead of enumerating entire population, we take a finite subset of statistical individuals from the population called a sample and this process through which the required number of units are selected is called sampling.

 

What is Simple Random Sampling (SRS)?

In SRS, we draw a sample in such a way that each unit of the population has an equal & independent chance of being selected in the sample. It is of two types:

1.    Simple Random Sampling With Replacement (SRSWR)

2.    Simple Random Sampling Without Replacement (SRSWOR)

 

What is Stratified Random Sampling?

In Stratified Random Sampling, the whole population is divided into homogeneous groups under certain criterion. These groups are termed as strata.

Then the sample is drawn randomly from each stratum independently. The estimates are then calculated from the data obtained from all the strata. The allocation of the sample sizes for different strata is done in the following ways:

a)    Equal Allocation

nh=n/L

where nh is the sample size of hth stratum

L is the total no. of stratums

n is the sample size

 

b)    Proportional Allocation

nh= (n/N) Nh

where Nh is the stratum size of hth stratum

 

c)    Optimum Allocation

 

nh= n (Nh Sh)/∑ (Nh Sh)

Well now we have some information before proceeding further.

 Now let me ask you this,

How to decide which estimator is more efficient?

Suppose we want to compare which estimator is more efficient, one which is obtained using SRSWOR or one which is obtained using the Stratified Random Sampling to estimate, let’s say population mean. One way to answer this is to compare the variance of the estimates.

 

Lesser the variance, more is the efficiency then more precise is the estimate.

 

Let’s take an example to understand how we are using the above-mentioned idea to decide which estimator gives the more precise estimate for population mean.

 

Example-

A sample survey is to be undertaken to ascertain the mean annual income of farms in a certain area. The farms are stratified according to their principal products. A census conducted gave the following results:

 

Type of farm

Number of farms

Mean annual income

Standard deviation

Sheep

161

10946

2236

Wheat

195

6402

2614

Dairy

274

2228

606

Others

382

1458

230

 

For a sample of 120 farms compute the sample sizes in each stratum under proportional and optimum allocation. Compare the efficiency of these methods with that of the SRSWOR by telling which scheme is more efficient and gives more precise estimate of population mean i.e. the mean annual income.

 

Important Formulae used:

1.    Variance of estimate of population mean using SRSWOR:


where 


 

2.    Variance of estimate of population mean using Stratified Random Sampling under proportional allocation:

 



3.    Variance of estimate of population mean using Stratified Random Sampling under proportional allocation:


          

 

Variance of the estimates using R Codes:






Hence, we can see from the above R codes that,

Vsrswor>Vprop>Vopt

This implies that the estimated mean of Annual Income obtained by using Stratified Random Sampling under optimum allocation is most efficient compared to other two schemes hence it gives the most precise estimate for estimating the mean annual income of farms in a certain area.

So now we understand with this example if we have some information about the population, we can easily verify which estimator we should use to estimate the population parameter.

Now you can take any example, any problem and see for yourself which sampling scheme you should take! 

I hope this blog helped you to understand why stratified random sampling gives more precise estimator compared to SRSWOR. 

 

 

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