Short on time? This page is the recap. For explanations, figures and worked examples, use the full study guide.
Recap
- The five-number summary is minimum, Q1, median, Q3, maximum.
- A boxplot draws it on a labelled number line. The box runs from Q1 to Q3 (the middle 50%), with a line at the median.
- Whiskers run from the box towards the minimum and maximum. If there are outliers, each whisker stops at the most extreme value that is not an outlier, and each outlier gets its own symbol.
- Each of the four sections holds about 25% of the data. A longer section means more spread, not more values.
- Shape and centre: the mean is pulled towards a long tail; the median is not.
Key relationships
| Feature | What it shows or means |
|---|---|
| Box length | IQR = Q3 − Q1, the spread of the middle 50% |
| Whole width (min to max) | Range |
| Fences (not drawn) | Q1 − 1.5 × IQR and Q3 + 1.5 × IQR |
| Roughly symmetric | Mean and median close together; halves of the boxplot similar in length |
| Skewed right | Right half longer; mean usually greater than median |
| Skewed left | Left half longer; mean usually less than median |
| Not shown by a boxplot | n, gaps, clusters, number of peaks, the mean (unless added) |
Assumptions and conventions
- Quartiles: split at the median; when n is odd, leave the median out of both halves. Show your halves.
- Outliers are judged with the 1.5 × IQR rule unless a question says otherwise.
- The mean–median link is a tendency. Say “usually” or “suggests”, not “always”.
Mistakes to avoid
- Running the whisker to an outlier or to the fence. It ends at a real data value that is not an outlier.
- Calling the line in the box the mean. It is the median.
- Saying a longer section has more data. Every section has about a quarter.
- Getting the direction wrong. The mean moves towards the tail: right tail, mean larger.
- Reading a symmetric boxplot as mound-shaped. It could hide two clusters.
- No scale or no label. Name the variable and the units on the axis.
Quick self-check
- Data (minutes): 3, 4, 4, 5, 6, 7, 8, 15. Where does the upper whisker end? (Q1 = 4, Q3 = 7.5, IQR = 3.5, upper fence = 12.75; 15 is an outlier, so the whisker ends at 8 minutes.)
- A boxplot has Q1 = 22 and Q3 = 30. About what share of the data lies between 22 and 30? (About 50%.)
- Mean 41, median 52. What shape is suggested? (Skewed left: the mean is pulled below the median.)
Next: practice questions.