Figure 45.1 , Multimodal Distribution

A multimodal distribution is a probability distribution with more than one local maximum (mode) in its probability density function. The most common case , bimodal , has exactly two peaks. The peaks appear because the data is a mixture of two or more distinct sub-populations, each with its own central tendency. The valley between peaks is called the antimode ; the larger peak is the major mode ; the smaller is the minor mode .

This is the critical insight the Wikipedia article emphasises, and the reason this visualisation is built the way it is. For a strongly bimodal distribution, the mean and median both fall in the antimode , the least-populated valley between the two peaks. A single reported mean of 0 for a distribution with peaks at −2.5 and +2.0 is not just imprecise; it actively misrepresents the data. The standard deviation compounds the problem by spanning both peaks, appearing large without revealing the two-peak structure.

Ashman's D measures the separation between two Gaussian components relative to their spread: D = |mu₁ − mu₂| / √(½(sigma₁² + sigma₂²)) . A value of D > 2 indicates clean separation between the modes , the two sub-populations are distinct enough to be treated separately. Values below 2 suggest overlap that may or may not produce visible bimodality depending on the mixing weights. Sarle's bimodality coefficient beta uses the skewness (gamma) and kurtosis (κ) of the combined distribution: beta = (gamma² + 1) / κ .

Three layers are drawn simultaneously: (1) a histogram of sampled data (walnut fill, low opacity) showing the empirical distribution; (2) a KDE curve (solid walnut line) showing the smoothed density estimate; (3) individual component curves (dashed, colour-coded) showing each Gaussian component's contribution to the mixture. Seeing all three simultaneously reveals how the KDE's peaks emerge from the component overlap , a pedagogically essential view that is lost when only the aggregate is shown.

A box plot of bimodal data shows a wide IQR centred on the antimode, with the median in the gap between peaks , it is structurally incapable of revealing multimodality. A bar chart of means ± SD is even worse: it conveys false precision about a central value that is meaningless. The KDE curve is the correct primary representation for any dataset where the shape of the distribution is the message.

Framework , FT Visual Vocabulary FT Visual Vocabulary category: Distribution , "How values in a dataset are distributed across a range." Abela quadrant: Distribution . Tufte principle applied: the three-layer chart (histogram + KDE + components) adds ink only when each layer reveals structure the others cannot , the histogram shows empirical frequency, the KDE shows continuous shape, and the component curves show the generative mechanism. Each layer earns its place.

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 multimodal distribution in D3 v7. Two files:

The ideas in this chapter didn't appear from nowhere. Maurice Kendall developed the rank-correlation statistic that bears his name (Kendall's tau) and co-wrote the standard 20th-century reference on advanced statistics. His careful work on bimodal and multimodal distributions shapes how we read the curves in this chapter. Maurice Kendall, circa 1955. AI-generated portrait based on a public domain photograph (Wikimedia Commons).

This is the critical insight the Wikipedia article emphasises, and the reason this visualisation is built the way it is. For a strongly bimodal distribution, the mean and median both fall in the antimode , the least-populated valley between the two peaks. A single reported mean of 0 for a distribution with peaks at −2.5 and +2.0 is not just imprecise; it actively misrepresents the data. The standard deviation compounds the problem by spanning both peaks, appearing large without revealing the two-peak structure.