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A unimodal distribution, under the density or mass-function convention, has probabilities that rise toward one central region and then fall away. For an absolutely continuous probability distribution, this means it has a probability density function that is nondecreasing up to a mode and nonincreasing afterward. This is the single-maximum meaning of unimodality; the chosen density is a one-dimensional quasiconcave function. A plateau of maximizing points is allowed.

For a discrete original example on outcomes0,1,2and3, choose probabilities0.1,0.4,0.4and0.1. The probability mass function rises, has two adjacent maximizing values, then falls. It is unimodal under the weak monotonicity definition despite not having a unique mode. In contrast, separated high-probability regions with a genuine dip between them are not one mode plateau.

Unimodality does not imply symmetry or equality of arithmetic mean, median and mode. For the exponential distribution with density $p(x)=\lambda e^{-\lambda x}$ on $x\ge 0$, $\lambda >0$, the mode is0, the mean is $1/\lambda$ and the median is $\log (2)/\lambda$. The density decreases from its boundary maximum. The NIST handbook gives the family and these summary characteristics, using a scale parameter reciprocal to this rate parameter.^[ref]

Nor is unimodality generally preserved by a mixture distribution. Equal mixing of uniform distributions on $[0,1]$ and $[3,4]$ gives density1/2 on those two intervals and zero in between. Each component has a flat unimodal density, while the mixture has separated peaks.

A density is defined only up to changes on sets of measure zero. Thus the distributional statement concerns existence of a representative with the required shape, rather than artificial isolated spikes added to one displayed formula. In higher dimensions, several notions of unimodality are used; one convex-level-set formulation requires convex superlevel sets, as in the quasiconcave function definition.^[ref] The shape convention should be specified before extending a one-dimensional conclusion to another setting.

Unimodality, in the single-maximum convention, means that a real-valued function rises toward one mode or plateau and then falls away. More precisely, for a function $f$ on an interval, there is a point $m$ such that $f$ is nondecreasing to the left of $m$ and nonincreasing to its right. These inequalities are weak: a flat top is allowed, and more than one point may achieve the maximum.

For an original example,is zero throughout $[-1,1]$ and decreases as one moves outward beyond that plateau. It is unimodal despite lacking a unique maximizing point. In contrast, two separated peaks with a genuine dip between them violate the rise-then-fall requirement.

A useful characterization is that a unimodal function in this convention has interval-shaped superlevel sets $\{x:f(x)\ge c\}$. For a function attaining its maximum, the converse holds too: if all those sets are intervals, a decline followed by a later rise on the same side of a maximizer would create a disconnected superlevel set. In one dimension this is the quasiconcave function viewpoint. Boyd and Vandenberghe define quasiconcavity through convex superlevel sets.^[ref]

The convention must be stated. In minimization, unimodal is also used for a function that first decreases and then increases; that corresponds to a quasiconvex function and interval-shaped sublevel sets. A single minimum and a single maximum are opposite shape conditions.

Unimodality also differs from concavity. The function $g(x)=e^{-x^2}$ has derivative $-2x{e}^{-x^2}$, so it rises up to zero and falls afterward. However,so it is not a concave function on the whole real line. Unimodality constrains the ordering of levels rather than the sign of curvature everywhere.

For a probability density, this shape defines one common meaning of a unimodal distribution. Adding symmetry around zero strengthens the conclusion: the centred interval maximizes mass for an even unimodal density lemma shows why a fixed-width observation window collects most probability when centered on the mode.