Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- Compare distributions of the same variable for two or more groups, drawn on the same scale.
- A full comparison covers shape, centre, variability and unusual features (outliers, clusters, gaps).
- Each sentence needs a comparative word, values for both groups, units and context.
- Any summary statistic can be used as evidence. For skewed data or outliers, prefer the median and IQR.
- A z-score tells you how many standard deviations a value is above (+) or below (−) the mean.
Key relationships
| Idea | What to remember |
|---|---|
| Dot plots, back-to-back stemplots, histograms | Show centre, variability, shape, outliers, clusters and gaps |
| Parallel boxplots | Show centre, variability, outliers and skewness or symmetry; hide clusters, gaps and n |
| Groups of different sizes | Compare relative frequencies (proportions), not counts |
| Skewness on a boxplot | Median near one end of the box and a long whisker on the other side |
| z-score with parameters | z = (x − μ) / σ |
| z-score from a sample | z = (x − x̄) / s |
| Value from a z-score | x = μ + zσ |
| Units of z | None, so values on different scales can be compared |
Assumptions and conventions
- The groups measure the same variable in the same units.
- Use μ and σ when they are given for the whole population; otherwise use x̄ and s.
- A larger z-score means “further above the mean”, which is not always “better”. For times, a lower value (more negative z) can be the better result.
- A difference between groups describes these data. Whether it generalises, or shows cause, depends on how the data were collected.
Mistakes to avoid
- Listing two sets of statistics without saying which is greater.
- No context or units (“A is bigger than B”).
- Reading box width as sample size. A wider box means more spread.
- Claiming a boxplot shows no gaps. It cannot show gaps at all.
- Comparing counts for groups of different sizes.
- Sign errors in z: always calculate x − μ, not μ − x.
- Comparing raw values from different scales instead of z-scores.
Quick self-check
- Population mean 50 kg, σ = 4 kg. What is the z-score of 61 kg? (z = 11 ÷ 4 = 2.75)
- μ = 200 points, σ = 20 points. Which value has z = −1.5? (x = 200 − 1.5 × 20 = 170 points)
- Sana scored 78 in a quiz with mean 70 and σ = 4, and 88 in a test with mean 75 and σ = 10. Where did she do relatively better? (Quiz: z = 2.00 against 1.30 in the test)
- Name two features a boxplot cannot show. (Any two of: clusters, gaps, number of values)
Next: practice questions.