Editing Mode (statistics) (section)

==Mode of a sample==

The mode of a sample is the element that occurs most often in the collection. For example, the mode of the sample [1, 3, 6, 6, 6, 6, 7, 7, 12, 12, 17] is 6. Given the list of data [1, 1, 2, 4, 4] its mode is not unique. A dataset, in such a case, is said to be [[bimodal]], while a set with more than two modes may be described as [[Multimodal distribution|multimodal]].

For a sample from a continuous distribution, such as [0.935..., 1.211..., 2.430..., 3.668..., 3.874...], the concept is unusable in its raw form, since no two values will be exactly the same, so each value will occur precisely once. In order to estimate the mode of the underlying distribution, the usual practice is to discretize the data by assigning frequency values to [[interval (mathematics)|interval]]s of equal distance, as for making a [[histogram]], effectively replacing the values by the midpoints of the
intervals they are assigned to. The mode is then the value where the histogram reaches its peak. 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. An alternate approach is [[kernel density estimation]], which essentially blurs point samples to produce a continuous estimate of the probability density function which can provide an estimate of the mode.

The following [[MATLAB]] (or [[GNU Octave|Octave]]) code example computes the mode of a sample:

<syntaxhighlight lang="matlab">
X = sort(x);                               % x is a column vector dataset
indices   =  find(diff([X, realmax]) > 0); % indices where repeated values change
[modeL,i] =  max (diff([0, indices]));     % longest persistence length of repeated values
mode      =  X(indices(i));
</syntaxhighlight>

The algorithm requires as a first step to sort the sample in ascending order. It then computes the discrete derivative of the sorted list and finds the indices where this derivative is positive. 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.