Page 6
Semester 3: Sampling Techniques
Basic concepts of sample surveys, Sampling unit, Sampling frame, Census, Probability Sampling
Sampling Techniques
Basic Concepts of Sample Surveys
Sample surveys are methods used to collect data from a subset of a population to make inferences about the entire population. They allow researchers to gather information efficiently and cost-effectively. Key elements include defining the research objective, selecting the sample size, and determining data collection procedures.
Sampling Unit
A sampling unit refers to the individual elements or groups from which data is collected in a study. These can be people, organizations, or objects. The definition of the sampling unit is crucial as it affects the overall design and outcome of the survey.
Sampling Frame
A sampling frame is a list or representation of all the sampling units in a population. It serves as the basis for selecting a sample. A good sampling frame is essential for obtaining a representative sample and reducing bias in the survey results.
Census
A census is a complete enumeration of a population, where data is collected from every member. While it provides comprehensive data, conducting a census can be time-consuming and expensive. It is often impractical for large populations, making sample surveys more viable alternatives.
Probability Sampling
Probability sampling is a technique in which each member of the population has a known, non-zero chance of being selected in the sample. Types of probability sampling methods include simple random sampling, systematic sampling, stratified sampling, and cluster sampling. This method helps minimize bias and allows for statistical inferences.
Simple random sampling with and without replacement, Properties of estimates, Finite population correction
Sampling Techniques
Simple Random Sampling
Simple random sampling is a basic and widely used sampling technique in which each member of the population has an equal chance of being selected. This method ensures that every potential sample has the same probability of occurring. It can be conducted with or without replacement.
Simple Random Sampling Without Replacement
In simple random sampling without replacement, once a member is selected for the sample, it is not returned to the population for subsequent draws. This means that each selection decreases the total number of available members, which can affect the probabilities of selection for subsequent members.
Simple Random Sampling With Replacement
In simple random sampling with replacement, each selected member is returned to the population after selection. This method maintains the same sample size throughout the process, as every member can be selected multiple times. This also means that the probabilities of selection remain constant throughout the sampling.
Properties of Estimates
Estimates derived from simple random sampling possess certain desirable properties. These estimates are unbiased, meaning that the expected value equals the true population parameter. Furthermore, the variance of these estimates can be minimized by increasing the sample size. The precision of the estimates is closely tied to the sample size.
Finite Population Correction
The finite population correction (FPC) is an adjustment made to variance estimates when sampling from a finite population without replacement. The FPC accounts for the reduced variability in the sample due to the limited size of the population. It is applied in calculations to improve the accuracy of estimates when the sample size is a significant fraction of the total population.
Stratified random sampling, principles of stratification, Estimation of population mean and variance, Allocation techniques
Stratified Random Sampling
Definition of Stratified Random Sampling
Stratified random sampling is a technique where the population is divided into distinct subgroups or strata that share similar characteristics. A random sample is then selected from each stratum to ensure that the sample accurately reflects the diversity of the population.
Principles of Stratification
The key principles of stratification include homogeneity within strata and heterogeneity between strata. Each stratum should be internally homogeneous while being distinctly different from other strata. This ensures that the samples drawn effectively represent the various segments of the population.
Estimation of Population Mean
To estimate the population mean from a stratified sample, the mean of each stratum is calculated, and then these means are weighted according to the size of the strata in relation to the entire population. The overall mean is computed as a weighted average.
Estimation of Population Variance
Variance in stratified sampling can also be estimated by calculating the variance within each stratum and then combining these variances using the proportions of each stratum in the total population. This approach can lead to lower overall variance compared to simple random sampling.
Allocation Techniques
Common allocation techniques include proportionate allocation, where sample sizes are proportional to the sizes of the strata, and optimal allocation, which takes into account the variance within each stratum, allocating more samples to strata with higher variance.
Systematic sampling, relation to cluster sampling, Estimation of population mean and sampling variance
Sampling Techniques
Systematic Sampling
Systematic sampling is a method where elements are selected from an ordered population at regular intervals. It starts with a randomly selected element and then selects every kth element in the population. This technique is easy to implement and ensures that the sample is spread evenly across the population.
Relation to Cluster Sampling
Systematic sampling and cluster sampling are both methods of sampling used in statistics. While systematic sampling involves selecting members at fixed intervals, cluster sampling involves dividing the population into clusters and then randomly selecting whole clusters to sample. Systematic sampling can be more efficient than cluster sampling, especially when there is no natural grouping in the population.
Estimation of Population Mean
In systematic sampling, the estimation of the population mean is achieved by calculating the mean of the sampled data. The formula applied is the sum of all sampled values divided by the number of samples taken. This approach allows for a more consistent estimate of the population mean.
Sampling Variance
Sampling variance in systematic sampling measures how much the sample means would vary if different samples were taken. It is calculated by taking the sum of the squared differences between each sample mean and the overall population mean, divided by the number of samples minus one. Understanding sampling variance is crucial for assessing the reliability of the estimates.
Varying Probability sampling, PPS Sampling, Estimator for population total and variance
Varying Probability Sampling, PPS Sampling, Estimator for Population Total and Variance
Varying Probability Sampling
Varying Probability Sampling refers to sampling techniques where different units in a population have different probabilities of being selected. This approach allows for a more flexible design and can improve the efficiency and representativeness of the sample. The selection probability can be based on factors such as size, cost, or other characteristics relevant to the research.
PPS Sampling
Probability Proportional to Size (PPS) Sampling is a specific method of varying probability sampling. In this technique, the selection probability of each unit is proportional to a measure of its size. This is particularly useful in situations where larger units are expected to provide more information. PPS sampling helps to ensure that larger entities are adequately represented in the sample while still allowing for smaller entities to be included.
Estimator for Population Total
An estimator for the population total is used to infer the total value of a characteristic in the entire population based on a sample. In varying probability sampling, estimators are adjusted to account for the different probabilities of selection. The Horvitz-Thompson estimator is commonly used in this context, where each sampled unit's contribution is weighted by the inverse of its probability of selection.
Estimator for Variance
Estimators for variance in probability sampling are crucial for understanding the reliability of the estimates obtained from the sample. In varying probability sampling, the variance can also be estimated taking into account the differing probabilities of inclusion. The variance estimator can be calculated using the sample data while including adjustment factors based on the probabilities to reflect the sampling design accurately.
