The concept

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.

Common real-world examples: traffic volume (AM peak + PM peak), geyser eruption intervals, the size of weaver ant workers (two distinct castes), sediment grain sizes in geology, and galaxy colour distributions in astronomy.

Why summary statistics mislead

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. The standard statistical summary (mean ± SD) is categorically wrong for multimodal data.

How to detect bimodality — the three statistics shown

Ashman's D measures the separation between two Gaussian components relative to their spread: D = |μ₁ − μ₂| / √(½(σ₁² + σ₂²)). 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 β uses the skewness (γ) and kurtosis (κ) of the combined distribution: β = (γ² + 1) / κ. The key threshold is β > 5/9 ≈ 0.555. Values above this suggest bimodal or multimodal structure; below it suggests unimodal. The logic: a bimodal distribution has low kurtosis (flat top, light tails between peaks) and/or high skewness — both increase β.

KDE (kernel density estimation) is the visual detection method: the smoothed density curve directly shows the number and position of modes as local maxima. The bandwidth is critical — too narrow and noise creates false modes; too wide and real modes merge into apparent unimodality. This chart uses Silverman's rule of thumb: h = 0.9 × min(σ, IQR/1.34) × n^(−1/5).

The chart design

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.

Vertical lines mark the overall mean (obsidian, solid) and the antimode (muted, dashed) — the exact location that a naive summary statistic would report as the "centre" of a distribution that has no observations near it.

What the alternative would break

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.