Split Normal Distribution

Basic Concepts

The split normal distribution (aka the two-piece normal distribution) splices two normal distributions with the same mode but possibly different variances together at the mode.

The probability density function (pdf) of this distribution, denoted N(μ, σ₁², σ₂²), is

An alternative expression for the pdf is

where φ(z) = the pdf of the standard normal distribution at z. Using Excel, we see that φ(z) = NORM.S.DIST(z,FALSE) and φ((x-μ)/σ) = NORM.DIST(x, μ, σ, FALSE).

The cumulative distribution function (cdf) is

where Φ(z) = the cdf of the standard normal distribution at z. Using Excel, we see that Φ(z) = NORM.S.DIST(z,TRUE) and Φ((x-μ)/σ) = NORM.DIST(x, μ, σ, TRUE).

The inverse of the cdf is

where σ = (σ₁+σ₂)/2.

Properties

Key properties of the split normal distribution are

Worksheet Functions

Real Statistics Functions: The Real Statistics Resource Pack provides the following functions.

SNORM_DIST(x, μ, σ1, σ2, cum) = the probability density function value f(x) for the split normal distribution when cum = FALSE and the corresponding cumulative distribution function F(x) when cum = TRUE.

SNORM_INV(p, μ, σ1, σ2) = the value x such that SNORM_DIST(x, μ, σ1, σ2, TRUE) = p, i.e. inverse of SNORM_DIST(x, μ, σ1, σ2, TRUE).

References

Julio, J. M. (2007) The fan chart: the technical details of the new implementation
https://www.banrep.gov.co/docum/ftp/borra468.pdf

Wikipedia (2022) Split normal distribution
https://en.wikipedia.org/wiki/Split_normal_distribution