Half-Normal Distribution

Basic Concepts

If x ∼ N(0, σ2) then |x| has a half-normal distribution, which is essentially the right half of the normal distribution with mean 0 and variance σ2.

The pdf of this distribution is

Half-normal pdf

for x ≥ 0. This is equivalent to

f(x) = 2/σ ⋅ φ(x/σ)

where φ(z) is the pdf of the standard normal distribution. In Excel, this means that

f(x) = 2 * NORM.S.DIST(x/σ,FALSE) / σ

The cdf is

Half-normal cdf

where Φ(z) is the cdf of the standard normal distribution at z. Thus

Excel formula for cdf

Since F(x) = 2Φ(x/σ)-1, it follows that the inverse of the cdf is

Half-normal inverse function

Properties

Some key statistical properties of the half-normal distribution are:

Key properties half-normal

Worksheet Functions

Real Statistics Functions: The Real Statistics Resource Pack provides the following functions regarding the half-normal distribution:

HNORM_DIST(x, σ, cum) = the probability density function value f(x) for the half-normal distribution with parameter σ when cum = FALSE and the corresponding cumulative distribution function F(x) when cum = TRUE

HNORM_INV(p, σ) = the value x such that HNORM_DIST(x, σ, TRUE) = p; i.e. the inverse of HNORM_DIST(x, σ, TRUE).

Reference

Wikipedia (2021) Half-normal distribution
https://en.wikipedia.org/wiki/Half-normal_distribution

4 thoughts on “Half-Normal Distribution”

  1. Dear Prof. Charles,

    How are you?
    First, thanks for providing this tool.
    May I ask your help, please?
    I am trying to apply the half-normal distribution of my data set using Excel.
    The normal function (=NORM.DIST) is working quite fine, since I have a mean and a standard deviation value. However, I am unsure how to the =HNORM_DIST. I am using as my X value my number of interest, the sigma value as the standard deviation, and cumulative as TRUE. I am unfortunately stuck over there, since my final result is always “1”, no matter my X value from my dataset. Would you know where I am doing wrong, please?

    Thank you a lot,
    Humberto

    Reply

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