Sampling Methods
Sampling Types
| Property | Statement |
|---|---|
| Simple Random | Every subset of size n equally likely |
| Stratified | Population divided into strata; sample from each |
| Cluster | Randomly select clusters; sample all within |
| Systematic | Every k-th element |
Central Limit Theorem
For large n, \(\bar{X}\sim N\!\left(\mu,\frac{\sigma^2}{n}\right)\) regardless of the population distribution.
Examples
Example 1. Population has μ=50, σ=10. n=100. Find P(X̄>52).
Solution. SE = 10/10 = 1. z = 2. P(Z>2) ≈ 0.023.
Background
A sample is a subset of a population selected to make inferences about the whole. The quality of inference depends critically on how the sample is drawn. A biased sampling method — one that systematically over- or under-represents certain groups — produces misleading conclusions regardless of sample size.
Simple random sampling (SRS) gives every subset of size \(n\) an equal chance of selection. It is the gold standard for unbiasedness but may be impractical for large or geographically dispersed populations.
Stratified sampling divides the population into homogeneous subgroups (strata) and samples from each. It reduces variance compared to SRS when strata differ in their means. Proportional allocation samples each stratum in proportion to its size; optimal allocation (Neyman) allocates more to strata with higher variability.
Cluster sampling randomly selects clusters (e.g., schools, cities) and surveys all members within selected clusters. It is cost-effective but less precise than SRS. Two-stage cluster sampling surveys a random sample within each selected cluster, balancing cost and precision. Systematic sampling (every \(k\)th element) is simple to implement but can be biased if the population has periodic structure.
Further Reading & Context
The study of sampling methods connects to many areas of mathematics and its applications. Understanding the foundational definitions and theorems provides the basis for advanced work in analysis, algebra, and applied mathematics.
Historical development: most mathematical concepts evolved over centuries, with contributions from mathematicians across many cultures. The modern axiomatic treatment provides rigor, while computational tools enable practical application.
In modern mathematics, this topic appears in graduate courses and research across pure and applied mathematics. Connections to computer science, physics, and engineering make it a versatile and important area of study. Mastery of the core results and techniques opens doors to research in number theory, analysis, geometry, and beyond.
Recommended next steps: work through the standard theorems with full proofs, explore the connections to related topics listed above, and practice with a variety of problems ranging from computational exercises to theoretical proofs. The interplay between different areas of mathematics is one of the subject's greatest rewards.