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Population and Sample

Population

A population is the complete set of individuals, objects, or measurements that possess some common characteristic that the researcher is interested in studying.

  • Population size: \(N\)
  • Parameters: Fixed numerical characteristics of the population (e.g., \(\mu\), \(\sigma\))
  • Examples: All students at a university, all voters in a country, all products from a factory

Sample

A sample is a subset of the population selected for study. It should be representative of the population to allow valid inferences.

  • Sample size: \(n\)
  • Statistics: Numerical characteristics calculated from the sample (e.g., \(\bar{x}\), \(s\))
  • Purpose: To estimate population parameters and make inferences

Sampling

Why Sample?

  1. Practicality: Studying entire populations is often impossible or impractical
  2. Cost-effectiveness: Samples require fewer resources than censuses
  3. Time efficiency: Faster data collection and analysis
  4. Accuracy: Well-designed samples can provide highly accurate estimates

Sampling Methods

Probability Sampling (Random)

  • Simple Random Sampling: Every member has equal chance of selection
  • Stratified Sampling: Population divided into strata, then random sampling within each
  • Cluster Sampling: Population divided into clusters, random clusters selected
  • Systematic Sampling: Selecting every kth member from a list

Non-Probability Sampling

  • Convenience Sampling: Using readily available participants
  • Purposive Sampling: Selecting specific individuals who meet criteria
  • Snowball Sampling: Participants refer other participants

Sampling Error

Sampling error is the difference between a sample statistic and the corresponding population parameter. It occurs because we're studying a subset rather than the entire population.

  • Reduced by: Larger sample sizes, better sampling methods
  • Quantified by: Standard error, confidence intervals

Representativeness

A sample is representative if its characteristics closely match those of the population. Key factors:

  • Sampling method: Probability methods generally produce more representative samples
  • Sample size: Larger samples tend to be more representative
  • Response rate: High participation reduces potential bias

Relationship to Parameter Estimation

Sampling provides the foundation for [[Parameter Estimation]]. Through careful sampling, we obtain data that allows us to:

  1. Calculate sample statistics (\(\bar{x}\), \(s\), etc.)
  2. Use these statistics to estimate population parameters (\(\mu\), \(\sigma\), etc.)
  3. Quantify the uncertainty in our estimates
  4. Make valid inferences about the population