Half normal distribution

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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 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.

Overview of the Half Normal Distribution

Probability density function (PDF) of the half normal distribution [1].

The half-normal distribution is used when you’re interested in the magnitude (i.e., absolute values) of normally distributed variables. For example, you might be interested in the size of a random variable (such as three standard deviations from the mean) and not the direction (positive or negative standard deviations).

The mean(μ) and variance 2) of the random variable X in the original normal distribution becomes the location parameter (μ) and scale parameter (σ) of Y in the folded and half-normal distributions. This leads to a more formal definition for the half normal distribution:

If Y is a normally distributed random variable with mean μ = 0 (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.

The probability density function (PDF) for the half normal is given by [2]

PDF of half normal distribution

Its cumulative distribution function (CDF) is

cdf of half normal distribution

Where t is the scale parameter and erf() is the error function.

The mean of the half normal distribution is 1 / θ [2]

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.


[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

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