Mode of a Continuous Variable

A mode is a value of \(x\) at which the given density \(\mathrm{f}(x)\) attains its greatest value on its support.

From a sketch, identify the highest point or points of the density. The mode is an input value, not the height of the density.

Using calculus, solve \(\mathrm{f}^{\prime}(x)=0\) on each differentiable piece, then compare density values at admissible stationary points, endpoints and joins.

A stationary point need not be a maximum, and a maximum need not be stationary. Retain every value attaining the greatest density; if the density has a flat highest section, every value in that section is a mode.