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.
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 .
- 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 .
- The logistic distribution has slightly heavier tails than the normal distribution. It appears in logistic regression and feedforward neural networks .
Image: Melikamp, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons
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 M. J. Fischer, Generalized Hyperbolic Secant Distributions, 1
SpringerBriefs in Statistics, DOI: 10.1007/978-3-642-45138-6_1
 Ross, G. Probability/Density Distributions.