Mode¶

Definition¶

Let $X$ be a random variable. A mode of $X$ is a value at which the probability mass function or probability density function attains its maximum.

Interpretation¶

A mode is the most likely value of a discrete distribution, or the value where a density is highest, that is, where values are most concentrated. On a graph it is the location of the highest peak.

Two panels: a skewed density with its peak, the mode, marked at 1, and the bar chart of a Poisson(3) distribution whose two highest bars, at 2 and 3, are both modes.

In the left panel the density $f(x)=x\mathrm{e}^{-x}$ peaks at $x=1$ $1$ , where $f'(x)=(1-x)\mathrm{e}^{-x}$ is zero. For this density the median is about $1.68$ and the mean is $2$ , so the three centres differ. In the right panel the $\operatorname{Pois}(3)$ PMF has two equally high bars, $\mathbb{P}(X=2)=\mathbb{P}(X=3)=\tfrac{9}{2}\mathrm{e}^{-3}\approx0.224$ $0.224$ , so both $2$ and $3$ are modes.

For the lognormal density, change $\sigma$ and watch mode, median and mean drift apart. For the Poisson distribution, set $\lambda$ to a whole number to get two modes.

For a density, finding the mode is a maximization problem from calculus: look for a stationary point or an endpoint of the support. A mode of a continuous distribution has probability $0$ like every value; it is where short intervals of a fixed length carry the most probability. For the Poisson distribution with a whole-number rate $\lambda$ , the ratio $\mathbb{P}(X=k)/\mathbb{P}(X=k-1)=\lambda/k$ equals $1$ at $k=\lambda$ $λ$ , which explains the two modes $\lambda-1$ and $\lambda$ .

A mode need not be unique or lie near the middle. Every face of a fair die is a mode, and the exponential density has its mode at $0$ , the smallest possible value. A density with several local peaks is called multimodal, even when only the highest peak is a mode in the strict sense.

Key Results¶

A mode is a maximizer of the mass function or density function.
A distribution may have more than one mode.
For continuous distributions, density height is not itself probability.

Examples¶

A fair die has every face as a mode because all faces are equally likely.
A normal distribution has one mode at its centre.
A mixture of two separated normal distributions can be bimodal, with two peaks; both peaks are modes in the strict sense only when they have equal height.

Prerequisites¶

Summary Measures of a Distribution

Sources¶

Probability Theory for EOR · Week 6

Aliases: modal value Keywords: mode of a distribution Symbols: lambda, sigma

Last updated: 20 October 2026 03:02