# MODUS

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Mode of a probability distribution

The mode of a discrete probability distribution is the value x at which its probability mass function takes its maximum value. In other words, it is the value that is most likely to be sampled.

The mode of a continuous probability distribution is the value x at which its probability density function attains its maximum value, so, informally speaking, the mode is at the peak.

As noted above, the mode is not necessarily unique, since the probability mass function or probability density function may achieve its maximum value at several points x1, x2, etc.

The above definition tells us that only global maxima are modes. Slightly confusingly, when a probability density function has multiple local maxima it is common to refer to all of the local maxima as modes of the distribution. Such a continuous distribution is called multimodal (as opposed to unimodal).

In symmetric unimodal distributions, such as the normal (or Gaussian) distribution (the distribution whose density function, when graphed, gives the famous “bell curve”), the mean (if defined), median and mode all coincide. For samples, if it is known that they are drawn from a symmetric distribution, the sample mean can be used as an estimate of the population mode.

## Mode of a sample

The mode of a data 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] the mode is not unique – the dataset may be said to be bimodal, while a set with more than two modes may be described as 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 each value will occur precisely once. 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. 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 code example computes the mode of a sample:

X = sort(x);
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));

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.

## Comparison of mean, median and mode

Comparison of common averages of values { 1, 2, 2, 3, 4, 7, 9 }
Type Description Example Result
Arithmetic mean Sum divided by number of values: (1+2+2+3+4+7+9) / 7 4
Median Middle value separating the greater and lesser halves of a data set 1, 2, 2, 3, 4, 7, 9 3
Mode Most frequent value in a data set 1, 2, 2, 3, 4, 7, 9 2

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Posted on August 11, 2011, in STATISTIKA and tagged , , , , , , , . Bookmark the permalink. 3 Comments.