Sampling and selection bias

Our fictional school

300 students

Whole school50%Club member

Selected: 0

Each figure is one studentClub memberNot a memberIn the sample

Sampling frame300/300

Club members in this frame50%

No chance of selection0

Latest samplen = 30

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Draw a sample to see who was selected.

Sampling methodSample size30

Changing the method or sample size starts a new series; the pinned result is kept.

Each draw starts from the original frame. A student appears at most once in each sample.

Select a group name to survey only that group.

0 samples

Distribution of sample rates

0 samples

0%50%100%

Blue line: school 50% · Dark bar: latest sample.

Latest sample minus whole school

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Interpreting results

Every student in the school has an equal chance of selection. Samples still vary; a larger sample usually reduces this variation.

One sample close to 50% does not establish that the method is unbiased.

Each bar counts samples within 5 percentage points; the last includes 100%. The blue marker shows the true 50% rate.

SD describes variation across samples; one sample cannot establish variability.

Model notes

The fictional school contains 300 students in three equal groups with club membership rates of 20%, 50% and 80%, making 50% overall.
Students are drawn without replacement within each sample. Independent draws reuse the initial sampling frame with a reproducible pseudorandom seed.
The convenience frame contains 150 members and 10 nonmembers per group. This illustrates coverage bias, not every convenience sample. Nonresponse and measurement error are not modeled.
A single sample difference combines chance variation with the effects of selection. Larger samples cannot fix an unrepresentative frame.
SD is descriptive (divided by the number of samples), not a standard error or confidence interval. At most 300 samples are retained.

Mathematics

Sampling and selection bias

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