< List of probability distributions

A **unimodal distribution** is any distribution with a single peak, cluster, or high point (i.e., global maximum). It comes from the Latin word *uni*– (“one”) and Middle French modal (“*measure*”).

More specifically, the graph of the probability density function (PDF), histogram, or statistic of the distribution has one distinct peak. For example, the PDF of the normal distribution is unimodal: it has one distinct peak.

The values in a unimodal distribution rise at first, reach a maximum, then slowly decrease to resemble the top of an Arabian (one-humped) camel.

Another example is the t-distribution, which tends to be thinner and shorter than the standard normal distribution.

## Unimodal distribution and skewness

Data that is both unimodal and symmetrical is usually described as “normal,” and this idea is an important assumption for many hypothesis tests in statistics. But a unimodal distribution doesn’t have to have one peak exactly in the center: the distribution can be skewed or “off center”. For example, the peak can be to the left of center, in which case it is called a right-skewed distribution because the right tail is longer than the left, or to the right of center — called a left-skewed distribution.

Many other skewed distributions are unimodal, including:

If a distribution has two peaks, it’s called a bimodal distribution; three or more peaks and it’s a multimodal distribution.

## Video Overview

The following video by Prof.Essa gives a useful overview of the unimodal distribution:

## Formal definition of a unimodal distribution

Although we can, in most cases, identify a unimodal distribution by its appearance, unimodality — the property of being unimodal — can be defined more precisely with three requirements [1]

- The function (i.e., the probability density) is nondecreasing on the half-line (−∞,
*b*) for some real*b*; - The function is nonincreasing on the half-line (
*a*, +∞) for some real*a*; - For the largest possible existing
*b*in (1) and the smallest possible existing*a*in (2), we have*a*≤*b*.

Perhaps surprisingly, the uniform distribution falls under this definition although it doesn’t have a classic camel-hump or bell-shaped distribution.

A different way to define these distributions is found in set theory:

“A Unimodal Distribution (which we will refer to as a “unimodal function”) f is a distribution for which the sets {

Hickok et al. [2]x∈ ℝn:f(x)≥c} are contractible for each real numberc.”

This means that real-valued unimodal functions cannot have disconnected level sets; they can have only one maximal region and no minima.

## Mean-median-mode inequality

The **mean-median-mode-inequality **for a unimodal distribution tells us that the mean (μ) median (m) and mode (M) often occur in alphabetical order (or reverse alphabetical) in a unimodal distribution. In other words [3]:

M ≤ m ≤ μ or M ≥ m ≥ μ

Note that while this inequality holds true for many distributions such as the normal distribution, it’s often violated, especially when dealing with unimodal mixture distributions.

## References

Skewed histogram. Audrius Meskauskas, CC BY-SA 3.0 via Wikimedia Commons

[1] Unimodality and the dip statistic.

[2] Hickok, L. et al. Unimodal Category of 2-Dimensional Distributions. Retrieved April 24, 2023 from: https://faculty.math.illinois.edu/~xwang105/unimodal.pdf

[3] Basu, S. & DasGupta, A. (1992). The mean, median and mode of unimodal distributions: a characterization. Department of Statistics, Purdue University. Retrieved April 24, 2023 from: https://www.stat.purdue.edu/docs/research/tech-reports/1992/tr92-40.pdf