Box Plots and Interquartile Range [H]
Higher Tier — This topic is assessed at Higher tier only (grades 4–9).
Box Plots and Interquartile Range
A box plot (box-and-whisker diagram) gives a compact visual summary of a dataset using five key values. It is particularly useful for comparing two distributions side by side.
The Five-Number Summary
| Value | Position in ordered data (\( n \) values) |
|---|---|
| Minimum | Smallest value |
| Lower quartile \( Q_1 \) | Median of the lower half (below the median) |
| Median \( Q_2 \) | Middle value — position \( (n+1)/2 \) |
| Upper quartile \( Q_3 \) | Median of the upper half (above the median) |
| Maximum | Largest value |
Box Plot Structure
Interquartile Range
\[ \text{IQR} = Q_3 - Q_1 \]The IQR is the width of the box. It measures the spread of the middle 50% of the data and is resistant to outliers.
Reading and Comparing Box Plots
When comparing two box plots on the same scale: compare medians (typical value), compare IQRs (consistency), comment on skewness (position of median within the box — near the left edge indicates positive skew).
Key Takeaways
- Five-number summary: min, Q1, median, Q3, max.
- IQR = Q3 − Q1 — width of the box; spread of the middle 50%.
- Whiskers extend to the min and max (or to outlier boundaries in more advanced treatments).
- To compare box plots: median (average), IQR (consistency), skewness, overlap.
- Median inside the box: if near the left edge → positive skew; near the right edge → negative skew.