Sampling Methods in Statistics: Random, Stratified, Cluster

Why We Sample

Suppose you want to know the average monthly allowance of the 30,000 students in a Nigerian university. Asking every student (a census) would be slow and expensive. Instead, you ask a few hundred and use their answers to estimate the average for everyone. The few hundred are a sample, and the way you choose them decides whether your estimate can be trusted. This tutorial explains the main sampling methods in statistics: simple random, systematic, stratified, cluster, multistage, and the non-probability methods such as convenience, quota and snowball sampling. It covers how to carry out each one, their advantages and drawbacks, sources of bias, and how to choose a method for a project or exam question.

Key Terms

Population: the whole group you want to learn about, such as all 30,000 students.
Sampling frame: the actual list you choose from, such as the registry's list of matriculation numbers. A poor frame (one that leaves people out) causes bias before you start.
Sample: the members actually selected.
Parameter: a true value for the population, such as the population mean μ. Usually unknown.
Statistic: the corresponding value calculated from the sample, such as the sample mean x̄, used to estimate the parameter.
Bias: a systematic tendency for the sample to differ from the population in one direction.

Probability Sampling Methods

In probability sampling every member of the population has a known, non-zero chance of selection. Only these methods allow valid margins of error and confidence intervals.

1. Simple Random Sampling

Every possible sample of the chosen size has the same chance of being selected, so every individual has the same chance too.

How to do it: number every member of the sampling frame from 1 to N, then choose n different numbers using a random number generator, a random number table, or by drawing numbered slips from a box ("the lottery method").

Example. To choose 5 students from a class of 60, number them 1 to 60. Reading two-digit numbers from a random number table, ignore numbers above 60, 00 and repeats, and take the first five valid ones.

Advantages: free from selection bias; the basis of most statistical theory.
Disadvantages: needs a complete list; the sample may still, by chance, be unrepresentative (for example, mostly from one department); can be spread out and costly to reach.

2. Systematic Sampling

Choose every k-th member from the list, where k = N/n is the sampling interval, starting from a random point among the first k.

Example. To choose 50 of 1,000 customers from a list, k = 1000/50 = 20. Pick a random start between 1 and 20, say 13, then take numbers 13, 33, 53, 73, … up to 993.

Advantages: quick and easy; spreads the sample evenly through the list.
Disadvantages: bias if the list has a repeating pattern that matches k. For example, if every 20th house on a street is a corner house, or a duty roster repeats every 7 days and k = 7, every selected item may share the same feature.

3. Stratified Sampling

Divide the population into non-overlapping groups called strata (such as faculties, levels, or gender), then take a random sample from each stratum. Usually each stratum's share of the sample matches its share of the population (proportional allocation).

Example. A school has 400 JSS students, 350 SS1–SS2 students and 250 SS3 students (1,000 in total). For a sample of 80:

JSS: 400/1000 × 80 = 32 students.
SS1–SS2: 350/1000 × 80 = 28 students.
SS3: 250/1000 × 80 = 20 students.

Then choose 32, 28 and 20 students at random within each group.

Advantages: guarantees every group is represented; gives more precise estimates when the strata differ from each other; allows separate results for each group.
Disadvantages: needs information about each member's stratum; more planning.

4. Cluster Sampling

Divide the population into groups called clusters (often by location, such as schools, villages or hostels), choose some clusters at random, and survey everyone in the chosen clusters.

Example. To survey secondary school students in a state with 300 schools, randomly pick 10 schools and survey every student in them.

Advantages: cheap and practical when the population is spread out; no full list of individuals needed, only a list of clusters.
Disadvantages: less precise than simple random sampling of the same size, because people in the same cluster tend to be similar (students in one school share teachers and facilities).

Stratified versus cluster: in stratified sampling you sample from every group; in cluster sampling you sample whole groups. Strata should be different from each other but similar inside; clusters should ideally each be a mini version of the population.

5. Multistage Sampling

Large national surveys combine methods in stages. For example: randomly select states, then local government areas within those states, then enumeration areas, then households within each area. Nigeria's Demographic and Health Survey uses a multistage, stratified cluster design of this kind. It makes huge surveys affordable while keeping them random.

Non-Probability Sampling Methods

Here selection depends on convenience or judgement, so the chance of each member being chosen is unknown. These methods are quick and useful for pilot studies or hard-to-reach groups, but results cannot be safely generalised and margins of error are not valid.

Convenience sampling: asking whoever is easiest to reach, such as students passing the library. Strongly biased towards certain kinds of people.
Quota sampling: the interviewer must fill quotas (say 30 male and 30 female respondents) but chooses the people. It looks like stratified sampling, but the final choice is not random.
Purposive (judgement) sampling: the researcher picks people with particular knowledge, such as heads of department, for expert interviews.
Snowball sampling: each participant recruits others. Useful for hidden populations, such as informal miners or people with a rare condition, but the sample follows social networks.
Voluntary response: people choose themselves, as in an online poll. Those with strong opinions are over-represented.

Sources of Bias in Sampling

Selection bias: the method or frame favours some groups. A phone survey misses people without phones.
Non-response bias: the people who refuse or can't be reached differ from those who answer.
Response bias: people give inaccurate answers, for example under-reporting smoking or exaggerating study hours, often because of how the question is worded.

A bigger sample reduces random error but does not fix bias. A huge biased online poll can be less accurate than a small random sample.

Sampling Error and Sample Size

Even a perfect random sample will not match the population exactly; the difference is sampling error. For the sample mean, the typical size of this error is the standard error, σ/√n, where σ is the population standard deviation and n the sample size. Because of the square root, quadrupling the sample size only halves the error. This is why opinion polls of about 1,000 people are common: they give a margin of error of about ±3 percentage points for a proportion, and going much bigger costs a lot for little extra precision.

Choosing a Sampling Method

A complete list and a fairly uniform population: simple random or systematic sampling.
Important subgroups that must all be represented: stratified sampling.
A large, scattered population with no list of individuals: cluster or multistage sampling.
A pilot study or a hidden population: a non-probability method, with its limits stated clearly.

Summary

A sample is used to estimate population parameters; the sampling frame should cover the whole population.
Probability sampling methods (simple random, systematic, stratified, cluster, multistage) give every member a known chance of selection.
Stratified samples take members from every group; cluster samples take whole groups.
Non-probability methods are convenient but cannot be generalised with confidence.
Larger samples reduce random error (in proportion to 1/√n) but never remove bias.