Bimodal distribution

In statistics, a bimodal distribution is a continuous probability distribution with two different modes. These appear as distinct peaks (local maxima) in the probability density function, as shown in Figure 1.

More generally, a multimodal distribution is a continuous probability distribution with two or more modes, as illustrated in Figure 3.

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:

This classification has since been modified slightly:

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. 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.

Examples of variables with bimodal distributions include the time between eruptions of certain geysers, the color of galaxies, the size of worker weaver ants, the age of incidence of Hodgkin's lymphoma, the speed of inactivation of the drug isoniazid in US adults, the absolute magnitude of novae, and the circadian activity patterns of those crepuscular animals that are active both in morning and evening twilight. In fishery science multimodal length distributions reflect the different year classes and can thus be used for age distribution- and growth estimates of the fish population. Sediments are usually distributed in a bimodal fashion. Bimodal distributions are also seen in traffic analysis, where traffic peaks in during the AM rush hour and then again in the PM rush hour. This phenomena is also seen in daily water distribution, as water demand, in the form of showers, cooking, and toilet use, generally peak in the morning and evening periods.

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