# Stratified random sampling

Stratified random sampling |
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**Stratified random sampling** is a sampling technique used to ensure that the sample is representative of the population by dividing the population into distinct subgroups (strata) and randomly selecting a sample from each subgroup. It is a type of probability sampling that is used in research and data analysis. Stratified sampling is based on the idea that the population of interest can be divided into distinct subgroups, each of which has a different probability of being selected. The goal is to create a sample that accurately reflects the composition of the population.

## Example of stratified random sampling

- Stratified random sampling can be used in a survey to ensure that all demographics are adequately represented. For example, if a survey is being conducted on the effectiveness of a particular medical treatment, the sample could be stratified by age, gender, and geographical region to ensure that a representative sample is taken from each subgroup.
- In market research, stratified random sampling can be used to ensure that the sample reflects the population in terms of ethnicity, income level, and other demographics. For example, a survey of consumer attitudes towards a new product could be stratified by age, gender, and race to ensure that the results are representative of the population.
- Stratified random sampling can also be used in educational research. For example, a study of student performance could be stratified by gender, race, and socio-economic background to ensure that the sample reflects the population of the school district.

## Formula of stratified random sampling

Stratified random sampling is a sampling technique used to create a sample that accurately reflects the composition of the population. The formula for stratified random sampling is:

$$\begin{equation} \frac{n_{i}}{N_{i}} = \frac{n}{N} \end{equation}$$

Where:

$$n_{i}$$ = the number of people in the sample from the ith stratum

$$N_{i}$$ = the number of people in the population from the ith stratum

$$n$$ = the total number of people in the sample

$$N$$ = the total number of people in the population

This formula can be used to determine the proportion of people from each subgroup that should be included in the sample. For example, if the population consists of 40% men and 60% women, the sample should also include 40% men and 60% women.

## When to use stratified random sampling

Stratified random sampling is a sampling technique used to ensure that the sample accurately reflects the composition of the population. It is often used when the population contains different subgroups of people, such as different genders, ages, or ethnicities. It can be used in a variety of contexts, including:

**Market research**: Stratified random sampling can be used to ensure that a sample of people accurately reflects the population of a specific target market, such as the population of a certain age group or region.**Clinical trials**: Stratified random sampling is often used in clinical trials to ensure that the sample population is representative of the overall population of patients.**Political polling**: Stratified random sampling is used in political polling to ensure that the sample accurately reflects the population in terms of gender, age, ethnicity, etc.**Quality control**: Stratified random sampling is used in quality control to ensure that a sample of products or services accurately reflects the overall quality of the population.

## Types of stratified random sampling

Stratified random sampling is a type of probability sampling that divides the population into distinct subgroups (strata) and randomly selects a sample from each subgroup. The types of stratified random sampling include:

**Proportional Stratified Sampling**: This type of stratified sampling ensures that the sample size is proportional to the size of each strata in the population.**Disproportional Stratified Sampling**: This type of stratified sampling allows for a sample size that is not proportional to the size of each strata in the population.**Cluster Sampling**: This type of stratified random sampling involves dividing the population into clusters, or groups, and randomly selecting a sample from each cluster.**Systematic Stratified Sampling**: This type of stratified sampling involves selecting a sample by randomly selecting a starting point and then selecting every Nth element from the population.**Stratified Random Sampling with Replacement**: This type of stratified random sampling involves selecting a sample with replacement, meaning that the same individual can be selected multiple times.**Stratified Random Sampling without Replacement**: This type of stratified random sampling involves selecting a sample without replacement, meaning that the same individual cannot be selected more than once.

## Steps of stratified random sampling

**Step 1**: Divide the population into strata. Stratified random sampling requires that the population of interest be divided into distinct subgroups, or strata. These strata must be mutually exclusive and collectively exhaustive, meaning that all elements of the population must belong to one and only one stratum.**Step 2**: Determine the sample size for each stratum. The sample size for each stratum should be proportional to the size of the stratum in the population. This means that the sample size should be inversely proportional to the variability within the stratum.**Step 3**: Randomly select the sample from each stratum. Once the sample size for each stratum has been determined, the researcher can randomly select the sample from each stratum. This can be done using a random number generator or a simple random sampling procedure.**Step 4**: Analyze the data. Once the sample has been selected, the researcher can begin to analyze the data. Depending on the research goals, this may involve descriptive statistics, correlation analysis, or other types of statistical analysis.

## Advantages of stratified random sampling

Stratified random sampling is a type of probability sampling method that ensures a sample accurately reflects the population of interest. It has several advantages, including:

**Greater statistical power**: Stratified random sampling can increase the statistical power of the sample by ensuring that each subgroup is adequately represented. This can help researchers to make more accurate inferences about the population of interest.**More efficient sampling**: Stratified random sampling can be more efficient than other sampling methods, as it reduces the amount of data that needs to be collected. This can save time and money.**Increased precision**: Stratified random sampling reduces the variance of the sample, resulting in increased precision of the results. This can help researchers to more accurately estimate population parameters.**Reduces sampling bias**: Stratified random sampling reduces sampling bias by ensuring that all subgroups are adequately represented. This can help to ensure the sample is representative of the population of interest.

## Limitations of stratified random sampling

Stratified random sampling has several limitations. These include:

- It can be difficult to identify the appropriate strata, which can result in an unrepresentative sample if the strata are not chosen carefully.
- It can be difficult to ensure that all members of each stratum are included in the sample, which can lead to a biased sample.
- There is the potential for overlap between strata, which can lead to an inaccurate representation of the population.
- Stratified sampling can be more time-consuming and costly than other sampling methods.
- It can be difficult to determine the appropriate sample size for each stratum, which can lead to an unrepresentative sample.

Stratified random sampling is not the only sampling technique used to ensure that the sample is representative of the population. There are several other approaches related to stratified random sampling, including:

**Cluster Sampling**: This method involves dividing the population into clusters and selecting a sample from each cluster. This reduces the cost of selecting a sample since only a few clusters need to be selected.**Systematic Sampling**: This approach involves selecting every nth item from the population. This method is useful for selecting a sample when the population is ordered in some way, such as by age or income level.**Multi-stage Sampling**: This method involves selecting a sample from a population in multiple stages. This method can be used to ensure that the sample is representative of the population by selecting a sample from each stage.

In conclusion, there are several approaches related to stratified random sampling that can be used to ensure that the sample is representative of the population. These include cluster sampling, systematic sampling, and multi-stage sampling. Each of these techniques has its own benefits and can be used in different contexts.

## Suggested literature

- Aoyama, H. (1954).
*A study of stratified random sampling*. Ann. Inst. Stat. Math, 6(1), 1-36. - Kadilar, C., & Cingi, H. (2005).
*A new ratio estimator in stratified random sampling*. Communications in Statistics—Theory and Methods, 34(3), 597-602. - Kadilar, C., & Cingi, H. (2003).
*Ratio estimators in stratified random sampling*. Biometrical journal, 45(2), 218-225.