What it is
A box plot draws a box from the first quartile to the third, so the box holds the middle half of the values. A line inside marks the median, whiskers extend to the furthest values that are not outliers, and points beyond them are drawn individually. Chart Buffet also marks the mean with a cross.
Because each group needs only a narrow column, a box plot can compare a dozen distributions in the space of one histogram.
✓ Use it when
- You have many values per group — waiting times by day, scores by class — and want to compare their typical values and spread.
- You want to see outliers at a glance.
✗ Avoid it when
- The audience is unfamiliar with quartiles — add a short explanation, or use a histogram.
- A distribution has two peaks; a box plot hides that shape.
- You have only a handful of values per group — plot the points themselves.
Tips for a better chart
- Sort groups by median when they have no natural order.
- Mention the number of values in each group if it differs a lot.
- Let the outliers show; they are often the most interesting points.
The maths
The quartiles Q1 and Q3 cut off the lowest and highest quarters of the sorted data, and the median Q2 splits it in half. The interquartile range is IQR = Q3 − Q1.
Tukey’s convention draws the whiskers to the most extreme values within 1.5 × IQR of the box; anything beyond Q1 − 1.5·IQR or Q3 + 1.5·IQR is plotted as an outlier (Tukey 1977). For normally distributed data this flags roughly 0.7% of values.
McGill, Tukey and Larsen added notches and variable widths: a notch shows an approximate confidence interval for the median, and width can reflect group size (McGill, Tukey & Larsen 1978).
A little history
Mary Eleanor Spear drew a “range bar” — a box for the middle half with a line at the median — in her 1952 handbook Charting Statistics (Spear 1952). John Tukey developed it into the box-and-whisker plot with its outlier rule, presented in the early 1970s and published in Exploratory Data Analysis (Tukey 1977).
Questions
What do the whiskers show?
The most extreme values that are not outliers — those within one and a half box-lengths of the box. Points beyond them are drawn separately.
How should my data be laid out?
Either one column of group labels and one of values (one row per observation), or one column of values per group.
References
- Tukey, J. W. (1977). Exploratory Data Analysis. Reading, MA: Addison-Wesley.
- McGill, R., Tukey, J. W. & Larsen, W. A. (1978). Variations of box plots. The American Statistician 32(1), 12–16.
- Spear, M. E. (1952). Charting Statistics. New York: McGraw-Hill.