Uses and abuses of statistics
How surveys are designed, how samples and graphs can mislead, and how to judge a statistical claim. It is a short reasoning unit tested in Paper 2 and in "explain" parts.
What you will learn
- 18.1 Survey sampling and questionnaire design:
- 18.2 Uses and abuses of statistical methods in daily life.
- 18.3 Judge statistical claims in news, reports and other sources.
You need first: Geometry and data basics, Measures of central tendency, Measures of dispersion
Key terms
| English | 中文 |
|---|---|
| population, sample | 總體，樣本 |
| probability / non-probability sampling | 概率抽樣／非概率抽樣 |
| simple random sampling | 簡單隨機抽樣 |
| systematic sampling | 系統抽樣 |
| stratified sampling | 分層抽樣 |
| cluster sampling | 整群抽樣 |
| convenience sampling, quota sampling | 便利抽樣，配額抽樣 |
| questionnaire | 問卷 |
| bias | 偏差 |
| validity, reliability | 有效性，可靠性 |
Standard steps
Name the sampling method
| Method | How it works | Type |
|---|---|---|
| simple random | every member has an equal chance (draw lots, random numbers) | probability |
| systematic | pick every kth member from a list after a random start | probability |
| stratified | split into groups (strata), sample from each in proportion | probability |
| cluster | split into groups, randomly pick whole groups | probability |
| convenience | whoever is easy to reach | non-probability |
| quota | fill a fixed number per group, not randomly | non-probability |
Check a survey or claim: four questions
- Who was asked? Is the sample representative of the population, or biased (for example, only people at one MTR station at one time)?
- How many? Is the sample large enough?
- How was it asked? Leading wording ("Don't you agree that…?"), overlapping options, or missing options.
- How is it shown? A graph's vertical axis that doesn't start at 0, unequal picture sizes, the mean used when extreme values make the median fairer.
"Explain" answers
Name the problem, then say what effect it has: "The sample includes only students in the library, so it may overestimate how long students study."
Worked examples
No. It is a convenience sample of students who already use the canteen, so it may be biased towards favourable views and misses students who avoid the canteen. A better plan is stratified random sampling from each form's class lists.
stratified random sampling (the forms are the strata)
total = 400 + 300 + 300 = 1000
S1: 50 × 400/1000 = 20
S2: 50 × 300/1000 = 15
S3: 50 × 300/1000 = 15bar heights above 90: A 9, B 3 → looks 3 times as many
true comparison: 99 ÷ 93 ≈ 1.06 → about 6% moreYes. The axis does not start at 0, so a small difference looks large.
mean = 200 000 ÷ 5 = $40 000
median = $17 000The one extreme value ($134 000) pulls the mean up, so the mean is higher than what four of the five workers earn. The median, $17 000, is a fairer average here.
Common mistakes
- Confusing stratified (sample from every group) with cluster (sample whole groups).
- Saying "the sample is too small" when the real problem is that it isn't random.
- Describing a misleading graph without saying how it misleads.
How the DSE tests it
- Paper 2: identify the sampling method; spot the problem with a question or graph.
- Paper 1: an "explain" part in a statistics question.
Questions you might not ask in class
Why not just ask everyone?
That's a census. It's often too slow or expensive, so a good sample gives a reliable estimate for much less effort.
Is 'random' the same as 'without a plan'?
No. Random sampling is carefully planned so that every member has a known chance of being chosen.