Description and properties of stratified random sampling

 Description of Stratified Random Sampling

Stratified random sampling (SRS) is a probability sampling method in which the population is divided into groups or strata, and then a random sample is selected from each stratum. Stratification is often used when the population is heterogeneous, meaning that there is a lot of variation in the variable of interest within the population. By stratifying the population, we can ensure that the sample is representative of all of the different groups in the population.

Properties of Stratified Random Sampling

SRS has a number of desirable properties, including:

• Unbiasedness: SRS samples are unbiased, meaning that they have the same expected value as the population they are drawn from.
• Efficiency: SRS is a relatively efficient sampling method, meaning that it requires a smaller sample size than other sampling methods to achieve the same level of precision.
• Representativeness: SRS samples are more representative of the population than samples drawn using other methods, such as simple random sampling. This is because SRS ensures that all of the different groups in the population are represented in the sample.

Mathematical Proofs of the Properties of SRS

Unbiasedness:

To prove that SRS samples are unbiased, we can use the following mathematical argument:

Let X be the population mean and S be the sample mean. Then, the expected value of the sample mean is given by:

E(S) = n/N * Σ(x_i)

where n is the sample size, N is the population size, and x_i is the value of the variable of interest for the ith individual in the population.

Under SRS, each member of the population has an equal chance of being selected for the sample. Therefore, the expected value of each x_i in the sample is equal to the population mean X. This means that the expected value of the sample mean S is also equal to the population mean X.

Efficiency:

To prove that SRS is a relatively efficient sampling method, we can compare the variance of the sample mean under SRS to the variance of the sample mean under other sampling methods.

The variance of the sample mean under SRS is given by:

Var(S) = σ^2 / n

where σ^2 is the population variance.

The variance of the sample mean under other sampling methods, such as simple random sampling, is typically greater than the variance of the sample mean under SRS. This means that SRS is a more efficient sampling method than simple random sampling.

Representativeness:

To prove that SRS samples are more representative of the population than samples drawn using other methods, we can use the following mathematical argument:

Let P be the proportion of the population that belongs to a particular stratum. Then, the expected proportion of the sample that belongs to the stratum is also equal to P. This is because SRS ensures that each stratum is represented in the sample in proportion to its size in the population.

Examples of Stratified Random Sampling

Here are some examples of SRS:

• A researcher wants to estimate the average height of all students at a university. The researcher could stratify the population by gender and then select a random sample of students from each stratum.
• A marketing company wants to know what percentage of consumers are interested in a new product. The company could stratify the population by age and then select a random sample of consumers from each stratum.
• A government agency wants to estimate the unemployment rate in the country. The agency could stratify the population by region and then select a random sample of people from each stratum.

Conclusion

Stratified random sampling is a powerful sampling method that can be used to obtain representative samples from heterogeneous populations. SRS is unbiased, efficient, and relatively simple to implement. It is often used in research studies and surveys to ensure that the results are generalizable to the population as a whole.