Five Numbers, One Glyph , Tukey's Distribution Sketch Figure 22.1 , Five Numbers, One Glyph , Tukey's Distribution Sketch

A box and whisker plot is the chart John Tukey introduced in Exploratory Data Analysis (1977) to summarise a univariate distribution as a single compact glyph. The glyph carries five numbers: the minimum (or lower fence), the first quartile (Q1), the median, the third quartile (Q3), and the maximum (or upper fence). The box spans Q1 to Q3 , the interquartile range, IQR , and contains the middle 50 percent of observations.

Read the median line first. Its position tells you the center and, by its offset within the box, the direction and degree of skew: a median pushed toward the bottom of the box indicates a right-skewed distribution; pushed toward the top, left-skewed; centered, roughly symmetric. Read the box width second. A wide box means high variance in the middle 50 percent; a narrow box means tight clustering. Read the whisker lengths third. Asymmetric whiskers reveal tail asymmetry that the box itself cannot show. Read the outlier dots last.

Tukey set the whisker length at 1.5 times IQR by deliberate calibration. Under a normal distribution, the rule flags roughly 0.7 percent of observations as potential outliers , small enough that a typical sample produces few flagged points, large enough that genuinely anomalous values are not absorbed into the whisker silently. The rule is robust: because IQR uses the middle 50 percent of the data, it is insensitive to the outliers it is trying to detect.

A bar chart of group means with error bars , the substitute most commonly chosen by audiences trained on parametric statistics , assumes the distribution is roughly symmetric and approximately normal. The moment one group is skewed, the error bar either crosses zero (when zero is not a realistic value) or implies symmetric tails that do not exist in the data. The viewer reads the chart as if the groups differ in average; the real story is in skew or tail behavior the encoding has erased.

FRAMEWORK FT Visual Vocabulary category: Distribution , "Show the range of values in a dataset and how they are distributed." Abela quadrant: Distribution (single variable, multiple groups, comparison of shape). Tufte principle: the box plot is data-ink near 1.0 , every pixel of every glyph encodes a quantile, the median, a fence, or an outlier. The one design decision worth knowing: 1.5 times IQR for the whisker fence, set by Tukey in 1977, is the convention the chart's interpretive grammar depends on.

Paste this into Claude Code to generate a working version of this chart, plus its data file. The result will not be a perfect replica , the goal is that the reader can run the prompt, get a chart of this type, and read its source. Generate a complete, self-contained box-and-whisker plot in D3 v7. Two files:

The ideas in this chapter didn't appear from nowhere. Andreas Buja has worked across visualization, computational statistics, and exploratory data analysis since the 1980s , including foundational papers on interactive box-whisker variants and on the inferential interpretation of visual patterns. Andreas Buja, circa 1995. AI-generated portrait based on a public domain photograph (Wikimedia Commons).

The ideas in this chapter didn't appear from nowhere. Andreas Buja has worked across visualization, computational statistics, and exploratory data analysis since the 1980s , including foundational papers on interactive box-whisker variants and on the inferential interpretation of visual patterns. Andreas Buja, circa 1995. AI-generated portrait based on a public domain photograph (Wikimedia Commons).