Mode (statistics)

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 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.

[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.

In order to estimate the mode of the underlying distribution, the usual practice is to discretize the data by assigning frequency values to intervals of equal distance, as for making a histogram, effectively replacing the values by the midpoints of the intervals they are assigned to.

For small or middle-sized samples the outcome of this procedure is sensitive to the choice of interval width if chosen too narrow or too wide; typically one should have a sizable fraction of the data concentrated in a relatively small number of intervals (5 to 10), while the fraction of the data falling outside these intervals is also sizable.

Next it computes the discrete derivative of this set of indices, locating the maximum of this derivative of indices, and finally evaluates the sorted sample at the point where that maximum occurs, which corresponds to the last member of the stretch of repeated values.

Unlike mean and median, the concept of mode also makes sense for "nominal data" (i.e., not consisting of numerical values in the case of mean, or even of ordered values in the case of median).

In any voting system where a plurality determines victory, a single modal value determines the victor, while a multi-modal outcome would require some tie-breaking procedure to take place.

Unlike median, the concept of mode makes sense for any random variable assuming values from a vector space, including the real numbers (a one-dimensional vector space) and the integers (which can be considered embedded in the reals).

For example, a distribution of points in the plane will typically have a mean and a mode, but the concept of median does not apply.

For some probability distributions, the expected value may be infinite or undefined, but if defined, it is unique.

Taking the mean μ of X to be 0, the median of Y will be 1, independent of the standard deviation σ of X.

When X has a larger standard deviation, σ = 1, the distribution of Y is strongly skewed.

In a footnote he says, "I have found it convenient to use the term mode for the abscissa corresponding to the ordinate of maximum frequency."