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Unimodal and Multimodal Distribution

Definition

A distribution is unimodal if its probability mass function or density function has a single peak (one local maximum), and multimodal if it has several separated peaks; with two peaks it is called bimodal. Here every peak is called a mode, although only the highest peak is a mode in the strict sense of a global maximizer.

Interpretation

Modality counts the peaks of a PMF or density. A unimodal distribution has one peak; a multimodal one has several separated peaks, which often signals that the population consists of several groups, each with its own typical value.

Two panels: a bell-shaped density with a single peak, and a bimodal density with two separate peaks.

The left panel is a bell-shaped density with a single peak, marked in red. The right panel shows two peaks of different heights with a dip between them. Both count as modes in the sense of this page, although only the higher, left peak is a mode in the strict sense of a global maximizer; the right peak is a local maximum.

A mixture of two groups shows two peaks only when the groups are far enough apart compared with their spread. If the group means are close, the two bells overlap and add up to a single, wider peak. For an equal-weight mixture of two normal distributions with the same standard deviation σ\sigma, the density has two peaks exactly when the means are more than 2σ2\sigma apart.

Modality matters for summaries. For a bimodal distribution the mean can fall in the dip between the peaks, a region where few values lie, so a single measure of centre describes it poorly. Bimodal does not require equally high peaks, and a single peak does not imply symmetry: the exponential distribution is unimodal but far from symmetric.

Key Results

  • A unimodal distribution has a single peak, which is then its mode.
  • A multimodal distribution has several peaks; they need not be equally high.
  • Mixtures can create multimodal distributions.

Examples

  • A normal distribution is unimodal.
  • A mixture of weekday and weekend demand distributions may be bimodal.
  • Exam scores from two very different preparation groups can form a multimodal distribution.

Prerequisites

Sources

Aliases: unimodal Keywords: unimodal distribution, multimodal distribution, bimodal distribution Symbols: sigma

Last updated: 27 October 2026 11:52