Unimodal Distribution


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

A selection of Normal Distribution Probability Density Functions (PDFs). Both the mean, μ, and variance, σ², are varied. The key is given on the graph.

The values 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 is thinner and shorter than the normal distribution:

Student’s T distribution

Unimodal distribution and skewness

Unimodal distributions do not need to be symmetric; they can be off-center (skewed).

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 in the center: the distribution can be skewed.

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).

A unimodal right-skewed distribution, where the right tail is longer than the left.

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:

References

Skewed histogram. Audrius Meskauskas, CC BY-SA 3.0 <http://creativecommons.org/licenses/by-sa/3.0/>, via Wikimedia Commons