Samples, populations and justified comparisons · Higher
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| population/ˌpɒpjʊˈleɪʃn/ | 总体 | zǒng tǐ |
A school asks the first twenty students leaving a sports club about exercise. A large share of active answers may describe the club rather than the whole school.
- A school asks the first twenty students leaving a sports club about exercise. A large share of active answers may describe the club rather than the whole school.
- This lesson studies population 总体: The complete group about which a statistical claim is made.
Choose the mathematical structure
- Define the target population and variables before sampling. A sample is a subset; a census includes the whole population. Random selection reduces systematic selection bias but does not eliminate sampling variability or non-response. Primary data is collected for the present investigation; secondary data was collected by another source or purpose. Discrete data is counted; continuous data is measured.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines population?
The complete group about which a statistical claim is made.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
In a random sample of 80 students, 24 walk to school, so the observed proportion is 24/80=0.3. Applied to a school of 600 it suggests about 180 walkers, with sampling uncertainty: it is an estimate rather than an exact count. A sports-club convenience sample may overrepresent active students. A voluntary online poll can miss people who do not respond; adding responses does not necessarily remove that bias. A study should record who could be selected, missing responses and the question wording. Number of siblings is discrete; travel time is continuous even if recorded to whole minutes. A school’s published attendance records are secondary data for a new project; measuring new travel times produces primary data.
Samples, populations and justified comparisons
Define the target population and variables before sampling
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
Find the sample walker proportion for 24 of 80.
24/80=0.3.
Test a tempting shortcut
- Sample size alone cannot fix biased selection. A precise calculated estimate need not be accurate for the target population. Recording continuous measurements as integers does not change the underlying variable type.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A sample result gives an exact population count whenever the arithmetic is correct. This claim is false. Explain which definition or assumption it violates.
Estimate walkers in a population of 600 from this proportion.
600×0.3=180.
A sample result gives an exact population count whenever the arithmetic is correct.
Sample size alone cannot fix biased selection. A precise calculated estimate need not be accurate for the target population. Recording continuous measurements as integers does not change the underlying variable type.
Interpret a new situation
- AQA S1/S4/S5 uses samples to describe populations and recognises limitations/data types. State what the sample supports and what might make extrapolating to the whole population unreliable.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find non-walkers in the sample.
80-24=56.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.6. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The complete group about which a statistical claim is made. Choose the relationship, show the method, check its assumptions and interpret the result.