Cluster sampling is a probabilistic sampling approach wherein the population is divided into clusters, and a random subset of clusters is chosen for examination. Various cluster sampling methods exist, mainly differentiated by the formation of clusters and the manner in which sampling occurs within them.

Cluster sampling is a suitable method in several scenarios, and its appropriateness depends on the characteristics of the population, the research objectives, and practical considerations.

Table Of Content

1. Comparison between Simple Random Sampling and Cluster Sampling
2. Steps Involved
3. Various Tyes Of Cluster Sampling Methods
4. Applications
5. Advantages
6. Some Drawbacks
7. Example
8. Summary

Comparison Between Cluster Sampling and Random Sampling

1. Sampling Unit:
   Cluster Sampling: Involves dividing the population into clusters or groups, and then randomly selecting entire clusters to be part of the sample. The units within the selected clusters are included in the sample.
   Simple Random Sampling: Involves randomly selecting individual units from the entire population without grouping them into clusters.

2. Representation:
   Cluster Sampling: Each selected cluster is expected to be a representative microcosm of the entire population, with internal diversity.
   Simple Random Sampling: Every individual unit in the population has an equal chance of being selected, and the sample is expected to be representative of the entire population.

3. Efficiency and Cost:
   Cluster Sampling: Can be more cost-effective than other methods, especially when the clusters are geographically or physically concentrated, as it reduces travel and logistical expenses.
   Simple Random Sampling: May be more resource-intensive, especially in large and dispersed populations, as each unit needs to be individually identified and included in the sample.

4. Precision and Variability:
   Cluster Sampling: Typically introduces more variability within clusters, leading to larger within-cluster homogeneity and potentially larger standard errors.
   Simple Random Sampling: Tends to have lower variability within the sample since each unit is independently selected.

5. Implementation:
   Cluster Sampling: Requires identifying and sampling entire clusters, making it suitable for situations where the population naturally forms groups.
   Simple Random Sampling: Involves directly selecting individual units without regard to any inherent grouping or structure in the population.

6. Complexity:
   Cluster Sampling: Can be more complex in terms of design and analysis, especially when dealing with unequal cluster sizes or varying intra-cluster correlation.
   Simple Random Sampling: Generally simpler to design and analyze.

Steps Involved:

• Define the population: Clearly define the population of interest. This is the entire group from which the sample will be drawn.
• Identify Clusters: Divide the population into clusters. Clusters are subsets or groups that ideally represent the diversity of the entire population.
• Randomly Select Clusters:Use a random sampling method to select clusters from the identified clusters. This ensures that each cluster has an equal chance of being included in the sample.
• Include All Elements in Selected Clusters: Once clusters are selected, include all elements within those clusters in the sample. This may involve every individual or unit within the chosen clusters.
• Assign Data Collection Units: Within each selected cluster, identify and assign the specific units or elements that will be part of the data collection process. This could involve individuals, households, or other relevant units.
• Assign Data Collection Units: Within each selected cluster, identify and assign the specific units or elements that will be part of the data collection process. This could involve individuals, households, or other relevant units.
• Collect Data: Perform data collection based on the assigned units within the selected clusters. This could involve surveys, interviews, observations, or other data collection methods.
• Analyze Data: Analyze the collected data to draw conclusions about the entire population. Use appropriate statistical techniques to account for the cluster sampling design.
• Interpret Results: Interpret the results in the context of the entire population. Recognize any limitations or biases introduced by the cluster sampling method.

Various Types Of Cluster Sampling Methods

Single Stage Cluster Sampling

In this sampling is done only once. Every element within the selected clusters is included in the final sample. For eg. If we want to survey the rich people in 5 towns then we will randomly select any 2-3 of them and survey for rich people within those towns.

Two stage Cluster Sampling

Involves two stages of sampling. Clusters are randomly selected in the first stage, and then a random sample of elements is taken from within each selected cluster in the second stage. In our survey for rich people we again sample within each town.
The main advantages of two-stage cluster sampling include increased efficiency and cost-effectiveness compared to single-stage cluster sampling, especially when the clusters are large and contain a diverse set of elements. This approach allows for a more detailed analysis of the selected clusters without the need to include every individual from each cluster.

Multi Stage Cluster Sampling

Similar to two-stage sampling but involves more than two stages. Clusters may be further divided into sub-clusters, and samples are taken at multiple levels.
Multi-stage cluster sampling is especially useful when the population has a complex structure, and sampling at each level provides a more efficient and cost-effective way to obtain a representative sample. This method allows for greater flexibility in addressing the hierarchical organization of the population.

Area Cluster Sampling

Clusters are defined based on geographical areas or administrative units. This is useful when the population is spread over a large geographic region.

Random Cluster Sampling

Clusters are randomly chosen from the population, and all individuals within the selected clusters are included in the sample.

Systematic Cluster Sampling

Clusters are selected at fixed intervals from a list or sequence. This method may be suitable when there is a logical order or structure to the population.

Stratified Cluster Sampling

Clusters are formed within strata (subgroups) of the population, and samples are taken from these strata to ensure representation from different subgroups.

Applications

Cluster sampling is a practical and efficient method used in various fields for conducting surveys and studies when it is not feasible or cost-effective to survey the entire population. Some common applications of cluster sampling include -

Public health surveys
When conducting health surveys, especially in large geographical areas, cluster sampling can be used to select specific regions or neighborhoods as clusters. This method is effective for studying disease prevalence, health behaviors, or access to healthcare.

Educational research
In education studies, researchers might use cluster sampling to select schools as clusters and then randomly sample students within those schools. This approach is useful for studying educational outcomes, teaching methods, and student performance.

Market research
Businesses often use cluster sampling to study consumer behavior and preferences. Geographic regions or retail outlets may be selected as clusters, and then customers within those clusters are surveyed to gather information about product preferences or market trends.

Agricultural research
In agricultural studies, cluster sampling can be employed to study crop yields or farming practices. Geographic regions or farming communities may be selected as clusters, and then individual farms within those clusters are surveyed.

Social science research
Cluster sampling is commonly used in social science research to study populations such as communities, neighborhoods, or social groups. Researchers might use clusters to sample specific social units and then study individual behaviors or attitudes within those clusters.

Advantages of cluster sampling

Cost-Effective:
Cluster sampling is often more cost-effective than other sampling methods, especially when the population is widely dispersed. It reduces the expenses associated with travel and data collection by focusing efforts on selected clusters.

Logistically Feasible:
When dealing with large populations or widespread geographic areas, it may be logistically challenging to survey every individual. Cluster sampling simplifies the process by grouping individuals into clusters, making data collection more manageable.

Statistical Efficiency:
Clustering can lead to increased statistical efficiency, especially if there is homogeneity within clusters. This means that variability within clusters is lower than variability in the overall population, resulting in more precise estimates.

Representativeness:
If the clusters are selected properly and are representative of the overall population, the results obtained from the sampled clusters can be generalized to the entire population.

Reduced Sampling Frame Requirements:
Instead of creating a comprehensive sampling frame for the entire population, cluster sampling only requires a list of clusters. This can be particularly advantageous when creating a sampling frame for the entire population is impractical or impossible.

There are some drawbacks for the method

Increased Variability Within Clusters:
One of the main drawbacks of cluster sampling is the potential for increased variability within clusters. If clusters are not homogeneous, this can lead to less precision in the estimates compared to simple random sampling.

Risk of Underrepresenting Small Subgroups:
In situations where clusters are chosen based on some grouping, there is a risk of underrepresenting small subgroups if they are not well-distributed across clusters. This can limit the generalizability of the findings to specific population segments.

Loss of Precision:
Cluster sampling may result in less precision compared to simple random sampling, especially if the intra-cluster variation is high. This can affect the reliability of the study results.

Potential for Cluster Bias:
If clusters are not selected randomly or if there is bias in the selection of clusters, the study results may be biased. It is crucial to ensure that the chosen clusters are representative of the entire population.

Complex Sample Design:
The implementation of a cluster sampling design, especially with multiple stages, can be more complex than simple random sampling. This complexity may lead to increased chances of errors in the sampling process.

Despite these disadvantages, cluster sampling remains a valuable method in various research contexts. Researchers need to carefully consider the characteristics of the population, the nature of the clusters, and the study objectives when deciding on the most appropriate sampling method.

Lets see an example
We want to survey how many students are there schools with in a state. So we can use single stage sampling for a start. We can sample a subset of the schools then survey them we can get a better estimate of the number we are interested in.

import random
import pandas as pd
import numpy as np

schools = pd.DataFrame({
    "school_ids":range(1, 100),
    "student_count":np.random.randint(20, 5000, 100)
})

school_ids	student_count
0	           1172
1	           3595
2	           1577
3	           3354
4	           4866
...	           ...
95	           3021
96	           582
97	           4933
98	           3592
99	           2322

How to select sample size?

• One general guideline suggests employing a minimum of 5 to 10 clusters, as this range is commonly recognized as the minimum necessary to achieve a sample that accurately represents the population.
• An alternative method involves using the square root of the population size divided by the desired number of sample units per cluster. This approach helps guarantee that the clusters maintain a size significant enough to be representative of the population while avoiding excessive sampling error.
• Additionally, it is advisable for the sample size to be a multiple of the number of clusters selected to ensure that the sample effectively mirrors the characteristics of the entire population.
• Another crucial consideration is the similarity among units within each cluster. Units within the same cluster should exhibit greater similarity to one another compared to units in different clusters.

clusters = random.sample(schools['school_ids'].to_list(), 10)

to_survey = schools[schools['school_ids']isin(clusters)]

Then we can just survey a subset i.e. 10 in this case as the output of the code.

school_ids	student_count
0	2316
2	2751
4	389
5	654
6	2452
8	2784
18	1195
21	3285
25	4456
28	249

This serves as a straightforward illustration showcasing the application of cluster sampling. In more complex scenarios, additional factors can be taken into account for the implementation of multistage cluster sampling.

Summary

Cluster sampling is a statistical technique where the population is divided into clusters, and a random sample of clusters is selected for analysis.It is particularly useful when the population is heterogeneous and difficult to divide into distinct strata.

The initial step involves randomly selecting clusters from the population, ensuring each cluster has an equal chance of being chosen. This randomness helps in achieving a more unbiased and representative sample. After cluster selection, all elements within the chosen clusters are included in the sample, simplifying data collection and analysis. Statistical techniques like stratification and weighting may be employed to enhance precision. If clusters are too homogeneous, there is a risk of underrepresenting the diversity of the population, leading to less reliable results. Careful consideration of cluster characteristics is essential to mitigate this risk. The external validity of findings in cluster sampling depends on the representativeness of the selected clusters. Researchers must carefully consider the potential impact of cluster characteristics on the generalizability of results.

Cluster sampling is a powerful and efficient technique for obtaining representative samples from large and diverse populations. Careful consideration of cluster characteristics, random cluster selection, and appropriate statistical analysis are critical elements in ensuring the reliability and validity of study findings.