A school with 120 Year 12 students and 80 Year 13 students wants 30 opinions. A committee of 30 all comes from Year 12 "because they were easiest to find." What is wrong with the arithmetic AND the method?
Start with a prediction. State what a representative sample requires, and whether a bigger convenience sample would fix it.
Build the model
A population 总体 is the whole group the claim is about; a sample 样本 is the subset measured. A sampling frame 抽样框 is the list the sample is drawn from. Random, systematic, stratified, quota and opportunity sampling are different methods with different bias risks.
A stratified sample 分层抽样 allocates in proportion to group size, then selects randomly within each group. Bias 偏倚 comes from the selection method, not the size: doubling a biased sample repeats the same flaw at scale. Missing data is not zero; check units, dates and variable definitions before comparing.
population: The complete group about which a statistical claim is made; sample: The subset actually measured; sampling frame: The list or population definition the sample is selected from; stratified sample: A sample allocated in proportion to each group\u2019s size; bias: A systematic distortion from an unfair selection method.
Choose evidence that can test it
Name the population, sampling unit and frame before collecting. Stratification is not "whoever is available from each group" — random selection is still needed within each stratum.
A large biased sample remains biased: a voluntary online poll over-represents strong opinions however many respond. AQA large-data-set familiarity needs the actual supplied data and its metadata, not invented values.
Work from known quantities
State the known values and their units. Choose the relation because its assumptions fit this case, then rearrange before substitution.
Known: 120 students in group A and 80 in group B; a proportional stratified sample of 30. Total population = 200. Group A allocation = 30 × 120/200 = 18; group B allocation = 30 × 80/200 = 12. Each group is then sampled randomly at that size.
Example:
A sample includes 45 of 300 people. Find the sample percentage, and the allocation for a stratified sample of 40 from groups of 150 and 150, using the same sequence. (Answer: 45/300 × 100 = 15%. Equal groups split evenly: 20 and 20.)
Check the conclusion and its limits
A proportion applied to a population gives an ESTIMATE with sampling variability, not an exact count. A census measures everyone; any sample, however large, inherits the method's selection effects.
Return to the original observation. Explain what the result supports, which conditions it assumes, and one way to test a competing explanation.
Warn:
Doubling a biased sample automatically removes its selection bias. This claim is false: bias lives in HOW members are chosen, not how many — surveying twice as many volunteers still over-represents people with strong opinions; only changing the method helps.
Key:
Statistical sampling: define the population and frame; stratified sampling allocates proportionally with random selection inside groups; bias comes from the method, not the size, and estimates carry sampling variability.
