Introduction
Since studying an entire population is usually impractical, statistics relies on drawing conclusions from a representative sample — this chapter covers how samples should be drawn and how sample-based statistics behave, providing the theoretical bridge to formal inference in Chapter 7.
1. Population, sample, parameter, and statistic
The population is the complete set of individuals or items of interest; a sample is a subset of the population actually observed. A parameter is a numerical characteristic of the population (usually unknown, and the actual target of investigation, e.g., the true population mean, μ); a statistic is the corresponding numerical characteristic calculated from the sample (e.g., the sample mean, x̄), used to estimate the unknown parameter.
2. Probability sampling methods
Simple random sampling gives every possible sample of a given size an equal chance of selection. Stratified sampling divides the population into relevant subgroups (strata) and samples separately (often proportionally) from each, ensuring representation of every important subgroup and typically improving precision when the strata differ meaningfully from each other. Systematic sampling selects every kth individual from an ordered list after a random starting point. Cluster sampling divides the population into naturally occurring groups (clusters), randomly selects entire clusters, and samples all (or a further sample of) individuals within the chosen clusters — often more practical and cost-effective for geographically dispersed populations, though generally less statistically efficient than stratified sampling for a given sample size.
3. Sampling and non-sampling errors
Sampling error is the unavoidable difference between a sample statistic and the true population parameter, arising simply because a sample is not the entire population — it can be reduced (though never eliminated) by increasing sample size. Non-sampling error arises from flaws in the data collection process itself — such as measurement error, non-response bias (certain types of people systematically failing to respond), or undercoverage (some part of the population having no chance of being selected at all) — and, critically, is not reduced by simply increasing sample size, since a larger flawed sample simply repeats the same flaw more often.
4. Sampling distributions
A sampling distribution is the probability distribution of a statistic (such as the sample mean) computed over all possible samples of a given size drawn from the population — a theoretical construct essential for inference, since it describes how much a sample statistic would be expected to vary from sample to sample purely by chance, providing the basis for assessing how reliable any single observed sample statistic actually is.
5. Standard error and the Central Limit Theorem
The standard error is the standard deviation of a sampling distribution (most commonly, of the sample mean, where it equals σ/√n) — it decreases as sample size increases, showing that larger samples give more precise (less variable) estimates of the population parameter. The Central Limit Theorem states that, for a sufficiently large sample size, the sampling distribution of the sample mean approaches a normal distribution, regardless of the shape of the original population’s distribution — a remarkably powerful result that justifies using normal-distribution-based methods (confidence intervals, hypothesis tests, Chapter 7) even when the underlying population itself is not normally distributed.
6. Sample size and survey design
Determining an appropriate sample size balances the desired precision (a smaller standard error, achieved with a larger sample) against practical constraints of cost, time, and feasibility — formal sample-size calculations typically specify a desired margin of error and confidence level in advance and solve for the required n. Good survey design requires deliberately anticipating and minimising non-sampling error sources (careful, unambiguous question wording, appropriate sampling frame, and strategies to maximise response rate) since, as established in point 3, these errors cannot be corrected after the fact simply by collecting more data.
Key Takeaways
- A statistic (from a sample) estimates a parameter (of the population); the four common probability sampling methods (simple random, stratified, systematic, cluster) differ in structure and typical efficiency.
- Sampling error is unavoidable but shrinks with larger samples; non-sampling error stems from flawed data collection and is not fixed by increasing sample size.
- A sampling distribution describes how a statistic varies across all possible samples, underlying the standard error and all of formal statistical inference.
- The Central Limit Theorem’s power lies in guaranteeing approximate normality of the sample mean’s distribution for large samples, regardless of the original population’s shape.
- Sample-size determination and careful survey design both aim to control precision and error, but only sample size addresses sampling error — survey design must separately address non-sampling error.