# Unimodal Distribution

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 

• 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 {x ∈ ℝ n : f(x)c} are contractible for each real number c.”

Hickok et al. 

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 :

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

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

 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

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