# Folded Normal Distribution & Half-Normal Distribution

Probability Distributions > Folded Normal Distribution Contents:

## What is a Folded Normal Distribution?

A folded normal distribution is a distribution of the absolute values of a normal distribution. It is used when you’re only interested in the size of a random variable (i.e. 2 standard deviations away from the mean) and not the direction or sign (either positive or negative). This happens in many practical situations where only the magnitude of a random variable is recorded. It’s called a “folded” normal distribution because, quite literally, the probability mass values on the left half of the distribution have been folded over to the right half; the absolute values are taken from the left half and added to the right half.
The mean( μ) and variance 2) of X in the original normal distribution becomes the location parameter (μ) and scale parameter (σ) of Y in the folded distribution. A more formal definition uses these two facts:
If Y is a normally distributed random variable with mean μ (the location parameter) and variance σ2 (the scale parameter), that is, if Y ∼ N μ,σ2, then the random variable X = |Y | has a folded normal distribution.

## Folded Normal Distribution Calculator

This calculator on the University of Alabama in Huntsville website allows you to create a CDF of the folded normal distribution and change the parameters of the function. You can also calculate the median and the first and third quartiles. To use the calculator, select “folded normal distribution” from the drop down menu and set the view to CDF.

## The Half-Normal Distribution

The half normal distribution is the distribution of the absolute value of a normally distributed random variable. It is a special case of the folded normal and truncated normal distributions. The probability density function is
Where the mean is zero and the standard deviation is 1/θ [2]. The folded normal distribution is also defined as the distribution of the absolute value of a normally distributed random variable, which means that the folded normal distribution only considers the positive values of the normal distribution. However, the half-normal distribution is a special case where the mean (μ) of the normal distribution is zero; in other words, when μ = 0, the folded normal distribution becomes the half-normal distribution. Thus, it could be more aptly called the “half standard normal” distribution. The half normal distribution has some useful applications, such as modeling measurement and lifetime data. In fact, this is one of the most important variations of the folded normal, because you’re more likely to be interested in normal distributions with a mean of 0 (i.e. a standard normal). For example, the half-normal distribution models Brownian motion — the random movement of microscopic particles suspended in a liquid or gas.

## Applications of the half normal Distribution

One application for which this type of distribution can be used is in modeling measurement data. Measurement data can often be skewed or contain outliers that make it difficult to model accurately. By using a half-normal distribution, however, one can easily identify these anomalies and create an accurate model for the data that excludes these outliers. The half-normal distribution can also be used to model lifetime data, such as when analyzing failure rates in components or products over time. By using this type of distribution, one can determine how many failures are expected over a certain period of time and use this information to plan accordingly. The half-normal distribution also models Brownian motion — the random movement of microscopic particles suspended in a liquid or gas.

## References

[1] Graph created with Desmos. [2] Half-normal distribution. Retrieved September 6, 2023 from: https://archive.lib.msu.edu/crcmath/math/math/h/h026.htm