Mode (statistics)

In statistics, the mode is the value that appears most often in a set of data values.^[1] If X is a discrete random variable, the mode is the value x at which the probability mass function takes its maximum value (i.e., x=argmax_xi P(X = x_i)). In other words, it is the value that is most likely to be sampled.

Like the statistical mean and median, the mode is a way of expressing, in a (usually) single number, important information about a random variable or a population. The numerical value of the mode is the same as that of the mean and median in a normal distribution, and it may be very different in highly skewed distributions.

The mode is not necessarily unique in a given discrete distribution since the probability mass function may take the same maximum value at several points x₁, x₂, etc. The most extreme case occurs in uniform distributions, where all values occur equally frequently.

A mode of a continuous probability distribution is often considered to be any value x at which its probability density function has a locally maximum value.^[2] When the probability density function of a continuous distribution has multiple local maxima it is common to refer to all of the local maxima as modes of the distribution, so any peak is a mode. Such a continuous distribution is called multimodal (as opposed to unimodal).

In symmetric unimodal distributions, such as the normal distribution, the mean (if defined), median and mode all coincide. For samples, if it is known that they are drawn from a symmetric unimodal distribution, the sample mean can be used as an estimate of the population mode.