Math101 · Independent Ontario learning library
Math101learn.math101.caSampling Methods
Sampling methods determine how units enter a study and therefore how credibly sample results can represent a population.
A large sample is not automatically representative; the selection process matters more than size alone.
Census versus sample
A census attempts to measure every population member. A sample measures a subset to save time, cost, or effort.
A census can still suffer nonresponse, measurement error, outdated coverage, and processing mistakes. Sampling introduces sampling variability but can sometimes produce higher-quality measurements with available resources.
Sampling frame
The sampling frame is the practical list or mechanism from which units are selected. If it omits parts of the target population, undercoverage occurs.
Compare the frame with the population definition before drawing the sample.
Simple random sample
In a simple random sample (SRS) of size $n$, every possible sample of that size has an equal chance of selection. Assign identifiers and use a genuine random mechanism.
Choosing whoever is easiest or “randomly looking around” is not an SRS.
Stratified random sampling
Divide the population into meaningful, non-overlapping strata, then take a random sample within each. Strata might reflect grade, region, or program when those groups may differ on the outcome.
Stratification guarantees representation and can improve precision. Analysis may require weights if sampling fractions differ across strata.
Cluster sampling
Divide the population into natural clusters, randomly select clusters, then survey all units or a sample within chosen clusters. Schools, classes, or neighbourhoods can serve as clusters.
Cluster sampling reduces travel or administrative cost, but people within a cluster may be similar, which can reduce effective information.
Systematic sampling
Choose a random start and select every $k$th unit from an ordered frame. This is practical when the list has no pattern aligned with $k$.
A hidden periodic pattern can bias the sample, so inspect how the frame is ordered.
Worked example: school survey
A convenience sample from one lunch period would miss students with different schedules.
Convenience and voluntary response
A convenience sample uses easily reached participants. A voluntary-response sample lets people opt in. Both are quick but often overrepresent accessible or strongly motivated people.
They can support exploratory work but usually cannot justify population-wide estimates without strong assumptions.
Multistage sampling
Large surveys often combine methods: randomly select regions, then schools, then students within schools. The analysis must account for selection probabilities, clustering, and weights.
The design should be documented at every stage.
Sample size and randomness
Larger random samples reduce sampling variability, but increasing a biased sample does not remove selection bias. Ten thousand voluntary online responses can be less representative than a carefully drawn sample of several hundred.
Precision and representativeness are related but distinct.
Nonresponse
Even a well-drawn sample can become biased if selected units do not respond and nonrespondents differ from respondents. Follow-up attempts, accessible formats, and short neutral surveys can improve response.
Report response rates and compare available characteristics of respondents and nonrespondents.
Common mistakes
Calling any varied sample random. Random selection requires a defined chance mechanism.
Assuming a census has no error. Nonresponse and measurement error remain.
Confusing stratified and cluster sampling. Stratified samples from every stratum; cluster samples selected clusters.
Believing size fixes convenience bias. It does not.
Ignoring frame and nonresponse coverage. Selection continues beyond the initial draw.
Quick self-check
- What exact population and sampling frame are used?
- Does every target unit have a known chance of selection?
- Is simple random, stratified, cluster, systematic, or multistage design appropriate?
- Could list order, undercoverage, or self-selection distort results?
- Is sample size sufficient for desired precision without pretending to fix bias?
- Are response rate, weights, and design limitations reported?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
A school randomly samples students separately within each grade so every grade is represented. What method is this?
- Grades are the strata.
- Random sampling occurs inside each stratum.
End of lesson
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