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Concept# Multimodal distribution

Summary

In statistics, a multimodal distribution is a probability distribution with more than one mode. These appear as distinct peaks (local maxima) in the probability density function, as shown in Figures 1 and 2. Categorical, continuous, and discrete data can all form multimodal distributions. Among univariate analyses, multimodal distributions are commonly bimodal.
When the two modes are unequal the larger mode is known as the major mode and the other as the minor mode. The least frequent value between the modes is known as the antimode. The difference between the major and minor modes is known as the amplitude. In time series the major mode is called the acrophase and the antimode the batiphase.
Galtung introduced a classification system (AJUS) for distributions:
A: unimodal distribution – peak in the middle
J: unimodal – peak at either end
U: bimodal – peaks at both ends
S: bimodal or multimodal – multiple peaks
This classification has since been modified slightly:
J: (modified) – peak on right
L: unimodal – peak on left
F: no peak (flat)
Under this classification bimodal distributions are classified as type S or U.
Bimodal distributions occur both in mathematics and in the natural sciences.
Important bimodal distributions include the arcsine distribution and the beta distribution (iff both parameters a and b are less than 1). Others include the U-quadratic distribution.
The ratio of two normal distributions is also bimodally distributed. Let
where a and b are constant and x and y are distributed as normal variables with a mean of 0 and a standard deviation of 1. R has a known density that can be expressed as a confluent hypergeometric function.
The distribution of the reciprocal of a t distributed random variable is bimodal when the degrees of freedom are more than one. Similarly the reciprocal of a normally distributed variable is also bimodally distributed.
A t statistic generated from data set drawn from a Cauchy distribution is bimodal.

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Mode (statistics)

The mode is the value that appears most often in a set of data values. 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=argmaxxi P(X = xi)). 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.

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