TheIBAxolotl
Population and sample
The population is every member of the group you want to know about; a sample is the part you actually measure. Sampling saves time and money, and sometimes testing destroys the item (crash-testing cars), so a census is impossible. The goal is a sample that is representative, so conclusions about it carry over to the population.
Types of data
- Discrete data take separate, countable values: number of siblings, goals scored, shoe sizes.
- Continuous data are measured and can take any value in a range: time, mass, height. They are always rounded when recorded.
The type affects the graph: discrete data suit bar charts and frequency tables of individual values; continuous data are grouped into intervals and shown in histograms.
Bias and reliability
A sample is biased if the method makes some members more likely to be chosen in a way that distorts the result. Asking only people in a gym about exercise habits will overestimate how much the town exercises. Reliability also depends on sample size (larger is usually better), on how questions are worded, and on whether people answer honestly.
Sampling techniques
| Method | How it works | Strength / weakness |
|---|---|---|
| Simple random | Every member has an equal chance; use random numbers | Fair, but needs a full list of the population |
| Systematic | Pick a random start, then every \(k\)th member, \(k = N/n\) | Easy; can be biased if the list has a pattern |
| Stratified | Split into groups (strata) and sample each in proportion | Represents every group; needs group sizes |
| Quota | Fill a fixed number from each group, chosen non-randomly | Quick; interviewer choice can cause bias |
| Convenience | Use whoever is easy to reach | Cheap; usually biased |
Worked example. A school has 480 junior and 720 senior students. A stratified sample of 50 is needed. Juniors are \(\frac{480}{1200} = 40\%\) of the school, so take \(0.4 \times 50 = 20\) juniors and 30 seniors, each chosen randomly within their group.
Stratified versus quota: both fix the numbers per group, but stratified chooses randomly within each group, while quota lets the collector pick.
Outliers
An outlier is a value far from the rest. A common rule flags any value below \(Q_1 - 1.5 \times IQR\) or above \(Q_3 + 1.5 \times IQR\). An outlier may be a recording error (remove it, with justification) or a genuine extreme value (keep it, and say so). Never delete data simply because it is inconvenient.
Common mistakes
- Calling height "discrete" because it was recorded to the nearest cm.
- Describing a convenience sample as random.
- Mixing up stratified and quota sampling.
- Removing an outlier without explaining why.
Exam tips
- When asked to "explain" bias, name who is over- or under-represented and how this changes the result.
- In stratified sampling, show the proportion calculation and round sensibly so the group totals still add to the sample size.
- Use the precise words "population", "sample" and "representative" in written answers.