< 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 mode (or peak), the mean is close to the median, and the majority of data points cluster close to the distribution’s center.

The normal distribution is the “classic” bell-shaped distribution and is sometimes called *the *bell-shaped distribution. However, it isn’t the only type of bell-shaped distribution: many other types of probability distributions have a bell curve shape, including the logistic distribution, t-distribution family and the Cauchy distribution. These curves are either narrower than the normal distribution — with more outliers in heavier tails or flatter with fewer outliers in thinner tails.

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.

## Advantages of working with a bell-shaped distribution

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.

## 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 Cauchy distribution can take on a variety of shapes including tall and thin or flat and wide.
- 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].

## Other types of distribution shapes

In addition to bell-shaped, we can also describe distributions as *skewed*, symmetric, or *uniform*.

- Skewed distribution: is a type of distribution in which one tail is longer than the other. The “bell” is off-center and has a squashed appearance.
- Symmetric distribution: a curve where the left side of the plot mirrors the right side. These do not have to be bell-shaped. They can also be circular or triangular shaped.
- Uniform distribution: A distribution shaped like a rectangle.

## References

Top Image: Melikamp, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons

Cauchy Image: https://creativecommons.org/licenses/by-sa/3.0/

Logistic image: [1] No machine-readable author provided. Anarkman~commonswiki assumed (based on copyright claims)., CC BY-SA 3.0 http://creativecommons.org/licenses/by-sa/3.0/, 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.