A box and whisker diagram is constructed from the five-number summary: minimum, lower quartile, median, upper quartile, and maximum. It is interpreted by comparing centre, spread, symmetry, and possible outliers.
This is SL 4.2: Presentation of data, studied by both Mathematics: Applications and Interpretation SL and HL students.
Construction
First, order the data or use your GDC to find the lower quartile , median , and upper quartile . Calculate the interquartile range:
Check for outliers using the boundaries
A value outside these boundaries is an outlier. Draw the box from to , place a line at the median, and extend whiskers to the smallest and largest non-outlier values. Plot each outlier separately with a cross.
| Feature | Meaning |
|---|---|
| Median line | Centre of the distribution |
| Box length | Middle of the data, measured by the IQR |
| Whiskers | Spread outside the middle , excluding outliers |
| Crosses | Values classified as outliers |
Interpretation
When comparing distributions, discuss at least two relevant features. A higher median indicates a higher typical value, while a larger IQR indicates greater variability in the central half of the data.
If the median is near the centre of the box and the whiskers have similar lengths, the distribution may be approximately symmetric. An off-centre median or one noticeably longer whisker suggests skewness. However, a box plot does not show individual frequencies or the precise shape of the distribution.
A common misconception is that whiskers must always reach the overall minimum and maximum. They reach the most extreme non-outlier values when outliers are present.
Exam Technique
For a command such as compare, refer to both distributions throughout and support statements with values, for example: “Group A has a higher median, but Group B has a larger IQR.” Keep full GDC precision when finding quartiles, and follow the quartile convention produced by the approved technology or information supplied in the question.