Folded normal distribution

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The folded normal distribution is defined as the absolute value of the normal distribution [1]. It can be used to describe what happens when only the magnitude of a random variable is recorded, but not its sign. The name comes from the fact that the negative half of the density of the normal distribution on (−∞,0] is folded over to the positive half on [0,∞).

Let’s take a closer look at what this means and how it works.

What is the Folded Normal Distribution?

Given a normally distributed random variable X with mean μ and variance σ2, the random variable Y = |X| has a folded normal distribution. This means that if we take the absolute value of X (the magnitude of X without considering its sign), then Y will have a folded normal distribution. This can happen if only the magnitude of some variable is of interest, but not its sign.

The probability density function (PDF) is given as follows: Suppose that a random variable Y ~ N (μ, σ2) is a normally distributed with mean μ and variance σ2. Let X = |Y|. Then X has a folded normal distribution with PDF [2]

Where σ2 and μ are the scale parameter and location parameter for the parent normal distribution (they become the shape parameters for the folded normal distribution).

Alternatively, the PDF can be expressed as a function of the hyperbolic cosine function cosh by taking z as z = (y – μ)/σ

The cumulative distribution function (CDF) can be written as

Taking z as z = (y – μ)/σ in the above integral, the CDF can also be expressed as

z ≥ 0, |μ| < ∞, σ > 0.

Graph of the folded normal

The graph below shows an example of how this continuous distribution looks. The probability mass to the left of x = 0 is folded over by taking the absolute value of x (represented in green).

Probability density function (pdf) of a folded-normal random variable with μ = 1, σ = 1.

Note that although it’s called a folded normal distribution, it is not what you would get if you drew a normal distribution on a sheet of paper and folded it in half — although its close! That’s because we’re interested in taking the absolute values on the left half of the distribution (less than x = 0) and folding those values over to the right — so it’s more like folding in cooking — where two components (such as eggs and flour) are combined. 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.

Ahsanullah et al. [1]

The shape of this graph reflects some interesting properties about how data behaves in relation to each other in real-world scenarios – for example, in many cases it may be more useful to know only whether a given value exceeds some threshold, rather than both its direction and extent relative to that threshold (e.g., when measuring changes in stock prices). By folding over at zero, we no longer need to consider negative values separately from positive values – instead they are combined into one single value which can be compared against other data points or thresholds more easily.

Furthermore, by folding over at zero we also shift our focus away from any particular point on the graph – instead we are looking at how much total area lies above or below any given point on the graph (i.e., what percentage of all possible outcomes lie above or below that point). This allows us to make statements about likelihoods based on these areas under curves – for example, “what is the probability that my random variable exceeds some threshold?”


In conclusion, understanding concepts like the folded normal distribution can give us better insight into real-world scenarios where probabilities play an important role – such as stock markets or weather forecasting. Taking an absolute value transforms our original data into something more tangible and easier to understand; by folding over at zero we no longer need to consider negative values separately from positive values, allowing us to make comparisons between different data sets more easily without having to worry about signs or directions relative to certain thresholds or points on graphs. By understanding concepts like this one we can gain deeper insights into how data relates and interacts in everyday life.


[1] Kumbhaker, S. et al. (2015). A Practitioner’s Guide to Stochastic Frontier Analysis Using Stata. Cambridge University Press.

[2] Ahsanullah, M. et al. (2014). Normal and Student ́s T Distributions and Their Applications. Atlantis Press.

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