## What is a Parabolic Distribution?

A **parabolic distribution** is any distribution that has the shape of a parabola. Many different probability distributions from a wide variety of families can be parabolic in shape.

## Examples of Parabolic Distribution

In general, if a probability distribution can be modeled with a quadratic function (creating a U or upside down U), then that distribution is parabolic. For example, the following distribution [1], which is a quadratic function, is parabolic:

f(x_{i}) = 6(x_{i} – A)(B – x_{i}) / (B – A)^{2}.

The beta distribution can take on many different shapes, from u-shaped curves to bell curves. However, when α = β = 2, the distribution becomes parabolic.

The following image shows two examples of a parabolic distribution: a = 5 (red) and a = 10 (blue). The x-axis is a random variable (T), and the y-axis shows the probability density of T [2].

If you have a set of data that makes a parabola and you need to find the best fit equation, use quadratic regression.

Parabolic distributions of order 2 are members of the location-scale family of distributions.

## References

Graph created with Desmos.com.

[1] Meloun, M. & Militky, J. Errors in instrumental measurements.

[2] Image: Motohisa Osaka via Researchgate. CC by 4.0.

**CITE THIS AS:**

**Stephanie Glen**. "Parabolic Distribution" From

**StatisticsHowTo.com**: Elementary Statistics for the rest of us! https://www.statisticshowto.com/parabolic-distribution/

**Need help with a homework or test question?** With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!

**Comments? Need to post a correction?** Please ** Contact Us**.