6.4 Sampling and Estimation

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Understand the difference between a sample and a population, and explain why randomness is necessary when choosing samples.
  2. Explain in simple terms why a given sampling method may be unsatisfactory, including a basic understanding of how random numbers are used to produce random samples.
  3. Recognise that a sample mean is a random variable, and use the results: E(X̄) = μ and Var(X̄) = σ²/n.
  4. Use the fact that X̄ has a normal distribution when X itself has a normal distribution.
  5. Use the Central Limit Theorem (CLT) where appropriate — for large samples, X̄ is approximately normally distributed.
  6. Calculate unbiased estimates of the population mean and variance from a sample, using raw or summarised data.
  7. Determine and interpret a confidence interval for a population mean when the population is normal with known variance, or when the sample is large.
  8. Determine an approximate confidence interval for a population proportion from a large sample.

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1. Population vs. Sample

What is a Population?

A population is the complete set of all items you are interested in studying. For example, if you want to know the average height of all 16-year-olds in a country, the population is every single 16-year-old in that country.

Populations are usually described using parameters — these are numerical measurements that describe the whole population, such as:

  • μ (mu) — the population mean (average)
  • σ² (sigma squared) — the population variance (how spread out the data is)

We use Greek letters for population parameters by convention.

What is a Sample?

A sample is a smaller group chosen from the population. Instead of measuring every single person or item in a population (which is often impossible or too expensive), you study a sample and use it to draw conclusions about the whole population.

Values calculated from a sample are called statistics and are written using ordinary (Roman) letters:

  • x̄ (x-bar) — the sample mean
  • s² — the sample variance

When a sample statistic is used to estimate a population parameter, it is called an estimate. For example, if your sample mean is x̄ = 23.4, you use 23.4 as your estimate of the true population mean μ.

Why Do We Take Samples?

There are two main reasons for taking samples:

  • To estimate the values of population parameters (such as the mean or variance).
  • To test a hypothesis about the population.

What is a Sampling Frame?

A sampling frame is a list or representation of all the items available to be sampled. For example, an electoral register is a sampling frame for voters. In some situations, no sampling frame exists — for example, you cannot list all fish in the ocean.

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