Equal size cluster sampling is a method used in survey sampling where the population is divided into clusters of equal size, and a fixed number of clusters are selected for the sample. Within each selected cluster, all units are included in the sample, resulting in a subset of the population that is representative of the entire population. Estimating the population mean from equal size cluster samples involves the use of specific estimators tailored to this sampling design. In this article, we will explore the concept of equal size cluster sampling, discuss the estimators of population mean commonly used in this context, examine their calculation methods, and highlight their applications, advantages, and limitations.
Concept of Equal Size Cluster Sampling:
Equal size cluster sampling involves dividing the population into clusters of equal size and selecting a fixed number of clusters for inclusion in the sample. Each selected cluster is then sampled in its entirety, with all units within the cluster being included in the sample. This sampling design is particularly useful when the population is large and geographically dispersed, making it impractical or costly to sample individual units directly. By grouping units into clusters, equal size cluster sampling reduces the logistical burden of sampling and allows for more efficient data collection.
Estimators of Population Mean in Equal Size Cluster Sampling:
Several estimators can be used to estimate the population mean from equal size cluster samples. Two commonly used estimators are the cluster mean estimator and the ratio estimator.
1. Cluster Mean Estimator: The cluster mean estimator calculates the sample mean within each selected cluster and then takes the average of these cluster means to estimate the population mean. Mathematically, the cluster mean estimator (\( \hat{\mu}_{\text{CM}} \)) is expressed as:
\[ \hat{\mu}_{\text{CM}} = \frac{1}{m} \sum_{i=1}^{m} \bar{y}_i \]
Where:
– \( \hat{\mu}_{\text{CM}} \) = Cluster mean estimator of population mean
– \( \bar{y}_i \) = Mean of the \( i^{th} \) cluster
– \( m \) = Number of clusters in the sample
2. Ratio Estimator: The ratio estimator calculates the ratio of the total observed variable (\( Y \)) to the total number of units (\( N \)) within each selected cluster and then takes the average of these ratios to estimate the population mean. Mathematically, the ratio estimator (\( \hat{\mu}_{\text{R}} \)) is expressed as:
\[ \hat{\mu}_{\text{R}} = \frac{1}{m} \sum_{i=1}^{m} \left( \frac{Y_i}{N_i} \right) \cdot N \]
Where:
– \( \hat{\mu}_{\text{R}} \) = Ratio estimator of population mean
– \( Y_i \) = Total observed variable in the \( i^{th} \) cluster
– \( N_i \) = Total number of units in the \( i^{th} \) cluster
– \( N \) = Total population size
– \( m \) = Number of clusters in the sample
Calculation Methods:
1. Cluster Mean Estimator Calculation: To calculate the cluster mean estimator, first compute the sample mean within each selected cluster by taking the average of the observed values (\( \bar{y}_i \)). Then, take the average of these cluster means to obtain the overall estimate of the population mean (\( \hat{\mu}_{\text{CM}} \)).
2. Ratio Estimator Calculation: To calculate the ratio estimator, first compute the ratio of the total observed variable (\( Y_i \)) to the total number of units (\( N_i \)) within each selected cluster. Then, take the average of these ratios and multiply by the total population size (\( N \)) to obtain the estimate of the population mean (\( \hat{\mu}_{\text{R}} \)).
Applications, Advantages, and Limitations:
1. Applications: Equal size cluster sampling is commonly used in various fields, including public health surveys, agricultural surveys, and economic surveys. It allows researchers to efficiently sample large and geographically dispersed populations while maintaining a representative sample.
2. Advantages: Equal size cluster sampling reduces the logistical burden of sampling by grouping units into clusters. It also allows for efficient data collection and can be cost-effective compared to sampling individual units directly.
3. Limitations: One limitation of equal size cluster sampling is the potential for increased sampling variability within clusters compared to simple random sampling. This can lead to larger standard errors and reduced precision in population estimates.
Conclusion:
Equal size cluster sampling is a valuable technique for sampling large and geographically dispersed populations. By dividing the population into clusters of equal size and selecting a fixed number of clusters for inclusion in the sample, researchers can efficiently collect data while maintaining a representative sample. Estimating the population mean from equal size cluster samples involves the use of specific estimators, such as the cluster mean estimator and the ratio estimator, which account for the cluster sampling design. While equal size cluster sampling offers advantages in terms of efficiency and cost-effectiveness, it is important to consider its limitations, such as increased sampling variability within clusters. Overall, equal size cluster sampling is a powerful tool that enables researchers to obtain reliable population estimates in a wide range of survey settings.