Bell-shaped distribution

< List of probability distributions

A bell-shaped distribution is – perhaps not surprisingly – any distribution that looks like the shape of a bell when plotted as a graph.

These distributions have one peak in the center (i.e., they are unimodal distributions) and are symmetric: if you draw a vertical line down the center of the graph, the left half will mirror the right.

Bell-shaped distributions have many advantages including the fact that their spread is relatively easy to describe with standard deviations — which can be thought of as roughly the average distance data points fall from the mean. Thus, the empirical rule can be used to calculate probabilities.

bell-shaped distribution
This histogram demonstrates the empirical rule, also known as the 68–95–99.7 rule.

Types of bell-shaped distribution

The most well-known bell-shaped distribution is the normal distribution. Others include:

  • The Cauchy distribution, which has an undefined mean and variance. It occurs as the ratio of two independent standard normal random variables [1].
  • The Gaussian mixture model, made up of multiple weighted multivariate Gaussian (normal) distributions. The model is an overlapping of bell-shaped curves.
  • The hyperbolic secant distribution: a symmetric member of the exponential family of distributions. It is like the normal distribution — both in shape and in symmetry — but has heavier tails [2].
  • The logistic distribution has slightly heavier tails than the normal distribution. It appears in logistic regression and feedforward neural networks [3].


Image: Melikamp, CC BY-SA 4.0, via Wikimedia Commons

[1] Cunningham, A. Probability Playground: The Cauchy Distribution.

[2] M. J. Fischer, Generalized Hyperbolic Secant Distributions, 1
SpringerBriefs in Statistics, DOI: 10.1007/978-3-642-45138-6_1

[3] Ross, G. Probability/Density Distributions.