# Bimodal distribution

Figure 1. A simple bimodal distribution, in this case a mixture of two normal distributions with the same variance but different means. The figure shows the probability density function (p.d.f.), which is an average of the bell-shaped p.d.f.s of the two normal distributions.
Figure 2. Histogram of body lengths of 300 weaver ant workers.[1]
Figure 3. A bivariate, multimodal 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.

## Terminology

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.

## Examples

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[2]

Important bimodal distributions include the arcsine distribution and the beta distribution. Another bimodal distribution is the U-quadratic distribution.

## Origins

A bimodal distribution most commonly arises as a mixture of two different unimodal distributions (i.e. distributions having only one mode). In other words, the bimodally distributed random variable X is defined as $Y$ with probability $\alpha$ or $Z$ with probability $(1-\alpha),$ where Y and Z are unimodal random variables and $0 < \alpha < 1$ is a mixture coefficient. For example, the bimodal distribution of sizes of weaver ant workers shown in Figure 2 arises due to existence of two distinct classes of workers, namely major workers and minor workers.[1] In this case, Y would be the size of a random major worker, Z the size of a random minor worker, and α the proportion of worker weaver ants that are major workers.

A mixture of two normal distributions has five parameters to estimate: the two means, the two variances and the mixing parameter. A mixture of two normal distributions with equal standard deviations is bimodal only if their means differ by at least twice the common standard deviation.[3] Estimates of the parameters is simplified if the variances can be assumed to be equal (the homoscedastic case).

It is obvious that if the means of the two normal distributions are equal that the combined distribution is unimodal. Conditions for unimodality of the combined distribution were derived by Eisenberger. [4] Necessary and sufficient conditions for a mixture of normal distributions to be bimodal have been identified by Ray and Lindsay.[5]

Mixtures of other distributions require additional parameters to be estimated.

## Properties

A mixture of two unimodal distributions with differing means is not necessarily bimodal. The combined distribution of heights of men and women is sometimes used as an example of a bimodal distribution, but in fact the difference in mean heights of men and women is too small relative to their standard deviations to produce bimodality.[3]

Bimodal distributions have the peculiar property that - unlike the unimodal distributions - the mean may be a more robust sample estimator than the median.[6] This is clearly the case when the distribution is U shaped like the arcsine distribution. It may not be true when the distribution has one or more long tails.

### Moments of mixtures

Let

$f( x ) = p g_1( x ) + ( 1 - p ) g_2( x )$

where gi is a probability distribution and p is the mixing parameter.

The moments of f(x) are[7]

$\mu = p \mu_1 + ( 1 - p ) \mu_2$
$\nu_2 = p[ \sigma_1^2 + \delta_1^2 ] + ( 1 - p )[ \sigma_2^2 + \delta_2^2 ]$
$\nu_3 = p [ S_1 \sigma_1^3 + 3 \delta_1 \sigma_1^2 + \delta_1^3 ] + ( 1 - p )[ S_2 \sigma_2^3 + 3 \delta_2 \sigma_2^2 + \delta_2^3 ]$
$\nu_4 = p[ K_1 \sigma_1^4 + 4 S_1 \delta_1 \sigma_1^3 + 6 \delta_1^2 \sigma_1^2 + \delta_1^4 ] + ( 1 - p )[ K_2 \sigma_2^4 + 4 S_2 \delta_2 \sigma_2^3 + 6 \delta_2^2 \sigma_2^2 + \delta_2^4 ]$

where

$\mu = \int{ x f( x ) dx }$
$\delta_i = \mu_i - \mu$
$\nu_r = \int{ ( x - \mu )^r f( x ) dx }$

and Si and Ki are the skewness and kurtosis of the ith distribution.

## Summary statistics

Bimodal distributions are a commonly used example of how summary statistics such as the mean, median, and standard deviation can be deceptive when used on an arbitrary distribution. For example, in the distribution in Figure 1, the mean and median would be about zero, even though zero is not a typical value. The standard deviation is also larger than deviation of each normal distribution.

### Ashman's D

A statistic that may be useful is Ashman's D:[8]

$D = 2^\frac{ 1 }{ 2 } \frac{ | \mu_1 - \mu_2 | }{ \sqrt{ ( \sigma_1^2 + \sigma_2^2 ) } }$

where μ1, μ2 are the means and σ1 σ2 are the standard deviations.

For a mixture of two normal distributions D > 2 is required for a clean separation of the distributions.

### Bimodality index

The bimodality index assumes that the distribution is a sum of two normal distributions with equal variances but differing means.[9] It is defined as follow:

$\delta = \frac{ \mu_1 - \mu_2 }{ \sigma }$

where μ1, μ2 are the means and σ is the common standard deviation.

$BI = \delta \sqrt{ p( 1 - p ) }$

where p is the mixing parameter.

### Bimodality coefficient

Sarle's bimodality coefficient b is[10]

$\beta = \frac{ \gamma^2 + 1 }{ \kappa }$

where γ is the skewness and κ is the kurtosis. The kurtosis is here defined to be the standardised fourth moment around the mean. The value of b lies between 0 and 1.[11] The logic behind this coefficient is that a bimodal distribution will have very low kurtosis, an asymmetric character, or both - all of which increase this coefficient.

The formula for a finite sample is[12]

$b = \frac{ g^2 + 1 }{ k + \frac{ 3( n - 1 )^2 }{ ( n - 2 )( n - 3 ) } }$

where n is the number of items in the sample, g is the sample skewness and k is the sample excess kurtosis.

The value of b for the uniform distribution is 5/9. This is also its value for the exponential distribution. Values greater than 5/9 may indicate a bimodal or multimodal distribution. The maximum value (1.0) is reached only by a Bernoulli distribution with only two distinct values or the sum of two different Dirac delta functions.

The distribution of this statistic is unknown. It is related to a statistic proposed earlier by Pearson - the difference between the kurtosis and the square of the skewness (vide infra).

## Statistical tests

### Unimodal vs. bimodal distribution

A necessary but not sufficient condition for a symmetrical distribution to be bimodal is that the kurtosis be less than three.[13][14] Here the kurtosis is defined to be the standardised fourth moment around the mean. The reference given prefers to use the excess kurtosis - the kurtosis less 3.

Pearson in 1894 was the first to devise a procedure to test whether a distribution could be resolved into two normal distributions.[15] This method required the solution of a ninth order polynomial. In a subsequent paper Pearson reported that for any distribution skewness2 + 1 < kurtosis.[11] Later Pearson showed that[16]

$b_2 - b_1 \ge 1$

where b2 is the kurtosis and b1 is the square of the skewness. Equality holds only for the two point Bernoulli distribution or the sum of two different Dirac delta functions. These are the most extreme cases of bimodality possible. The kurtosis in both these cases is 1. Since they are both symmetrical their skewness is 0 and the difference is 1.

Baker proposed a transformation to convert a bimodal to a unimodal distribution.[17]

Several tests of unimodality versus bimodality have been proposed: Haldane suggested one based on second central differences.[18] Larkin later introduced a test based on the F test;[19] Benett created one based on the G test.[20] Tokeshi has proposed fourth test.[21][22] A test based on a likelihood ratio has been proposed by Holzmann and Vollmer.[23]

### Antimode

Statistical tests for the antimode are known.[24]

### General tests

To test if a distribution is other than unimodal, several additional tests have been devised: the bandwidth test,[25] the dip test,[26] the excess mass test,[27] the MAP test,[28] the mode existence test,[29] the runt test,[30][31] the span test,[32] and the saddle test.

## References

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2. ^ Introduction to tropical fish stock assessment
3. ^ a b Schilling, Mark F.; Watkins, Ann E.; Watkins, William (2002). "Is Human Height Bimodal?". The American Statistician 56 (3): 223–229. doi:10.1198/00031300265.
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24. ^ Hartigan JA (2000) Testing for antimodes. Studies in Classification, Data Analysis, and Knowledge Organization 169-181
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26. ^ Hartigan JA, Hartigan PM (1985) The dip test of unimodality. Ann Statist 13 (1) 70-84
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