Bimodal distribution

File:Bimodal.png
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.

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.

A good example is the height of a person. The heights of males form a roughly normal distribution, as do those of females. Each of these distributions is unimodal. However, if we plot a single histogram of the entire population, we see two peaks—one for males and one for females.

Bimodality is a property of many distributions. A bimodal distribution most commonly arises as a mixture of two different unimodal distributions. In other words, the bimodally distributed random variable X is defined as ${\displaystyle Y}$ with probability ${\displaystyle \alpha }$ or ${\displaystyle Z}$ with probability ${\displaystyle (1-\alpha )}$, where Y and Z are unimodal random variables and ${\displaystyle 0<\alpha <1}$ is a mixture coefficient. In the height example, Y would be the height of a random male, Z the height of a random female, and ${\displaystyle \alpha }$ the probability that a random individual is male.

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 very large, even though the deviation of each normal distribution is relatively small.

More generally, a multimodal distribution is a continuous probability distribution with two or more modes, as illustrated in Figure 2. A unimodal distribution has only one mode.

File:Bimodal-bivariate-small.png
Figure 2. A bivariate, multimodal distribution.