Descriptive Statistics > Unimodal Distribution
What is a unimodal distribution?
A unimodal distribution is any distribution with a single peak, cluster, or high point. It comes from the Latin word uni- (“one”) and Middle French modal (“measure”). The values increase at first, rising to a peak where they then decrease.
The “mode” in “unimodal” doesn’t refer to the most frequent number in a data set (although it’s closely related)—it refers to the local maximum in a graph. Technically they are the same thing: one mode (the most common number) will equal one peak in a graph.
Unimodal distribution examples
The normal distribution, shown above, is an example of a unimodal distribution; The curve has one local maximum (peak).
Other types of distributions in statistics that are unimodal include:
- The uniform distribution.
- The t-distribution.
- The chi-square distribution.
- The Cauchy distribution.
In elementary statistics, you probably will see the first three types of distribution listed above, but not the Cauchy. The Cauchy is a strange type of distribution where the expected value does not exist.
The uniform distribution is a type of probability distribution where the odds of getting any number within the range are the same. For example, if you roll a die, your odds of rolling any number (1, 2, 3, 4, 5, 6) are the same. Although the graph looks like a rectangle, it technically only has one peak, hence it’s classified as unimodal.
Symmetry and the unimodal distribution
Unimodal distributions aren’t necessarily symmetric like the normal distribution. They can be asymmetric, or they could be a skewed distribution.
Data that is both unimodal and symmetrical is often 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, the peak can be to the right of center — called a left-skewed distribution because the left tail is longer than the right.
For example, the chi square distribution is unimodal and skewed.
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 {x ∈ ℝ n : f(x) ≥ c} are contractible for each real number c.”
Hickok et al. [2]
This means that real-valued unimodal functions cannot have disconnected level sets; they can have only one maximal region and no minima.
Other types of distributions
In comparison, a bimodal distribution is not unimodal. “Bi” means two, so there are two local maximums (peaks) in a bimodal distribution.
Multimodal distributions, where there are more than two peaks, are very rare. One example of a multimodal distribution is a trimodal distribution, which has three peaks.
U distributions have a distribution in the shape of the letter U, with large frequencies at the left and right of the distribution and few values in the middle.
Mean-median-mode inequality
The mean, median and mode often occur in alphabetical order (or reverse alphabetical order) on unimodal distributions. The mean-median-mode-inequality 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
- Unimodality and the dip statistic.
- 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