# Mode

One of the numerical characteristics of the probability distribution of a random variable. For a random variable with density \$p(x)\$ (cf. Density of a probability distribution), a mode is any point \$x_0\$ where \$p(x)\$ is maximal. A mode is also defined for discrete distributions: If the values \$x_k\$ of a random variable \$X\$ with distribution \$p_k = \mathsf{P}(X = x_k)\$ are arranged in increasing order, then a point \$x_m\$ is called a mode if \$p_m \ge p_{m-1}\$ and \$p_m \ge p_{m+1}\$.

Distributions with one, two or more modes are called, respectively, unimodal (one-peaked or single-peaked), bimodal (doubly-peaked) or multimodal. The most important in probability theory and mathematical statistics are the unimodal distributions (cf. Unimodal distribution). Along with the mathematical expectation and the median (in statistics) the mode acts as a measure of location of the values of a random variable. For distributions which are unimodal and symmetric with respect to some point \$a\$, the mode is equal to \$a\$ and to the median and to the mathematical expectation, if the latter exists.