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Half-Normal Distribution Calculator

Enter a non-negative x and the scale σ to evaluate the half-normal distribution.

Result

0.301137
0.546745
1.595769

How it is calculated

For a point x ≥ 0 and a scale σ > 0 the tool returns three numbers with six decimals. Density: f(x) = √2 ÷ (σ√π) × exp(−x² ÷ 2σ²). Cumulative probability: F(x) = erf(x ÷ (σ√2)). Mean: σ × √(2/π). Here σ is the scale of the underlying normal distribution, not the standard deviation of the half-normal itself. The error function is computed with a polynomial approximation accurate to roughly 1e-7, so the sixth decimal of the CDF may differ by one unit from an exact table. Negative x, or σ of zero or less, gives no result. The mean does not depend on x.

Worked example

Take x = 1.5 and σ = 2. Density: √2 ÷ (2 × √π) = 0.398942, and exp(−2.25 ÷ 8) = exp(−0.28125) = 0.754840, so f = 0.398942 × 0.754840 = 0.301137. CDF: z = 1.5 ÷ (2 × √2) = 0.530330, erf(0.530330) = 0.546745, so about 54.7% of the distribution lies below 1.5. Mean: 2 × 0.797885 = 1.595769. The density is not a probability; only the CDF is.

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Common mistakes

  • Entering the standard deviation of the half-normal as σ: the scale is that of the original normal, and the half-normal's own standard deviation is smaller (σ√(1 − 2/π)).
  • Reading the density value as the probability of x: f(x) is a height, and can even exceed 1 when σ is small.
  • Entering a negative x: the distribution has no mass below zero, so the tool refuses it; take the absolute value of the measurement first.
  • Using the mean as the centre of the data: the mean is σ × 0.7979, not σ and not 0.
  • Expecting the upper tail: the CDF is the probability of a value up to x, so the chance of exceeding x is 1 minus the CDF.

About this calculator

The half-normal distribution is what you get by folding a normal distribution centered at zero over onto its positive side — equivalently, it’s the distribution of |Z| when Z follows a normal distribution N(0, σ²). It has a single scale parameter σ, probability density function f(x) = √(2/π)/σ · exp(−x²/2σ²) for x ≥ 0, and mean σ√(2/π) — always positive by construction, unlike the parent normal distribution.

This calculator evaluates the PDF, cumulative probability (CDF), and mean of the half-normal distribution for a chosen x and σ. It shows up wherever a quantity is naturally non-negative but arises from folding or taking the absolute value of an underlying normal process: modeling measurement or manufacturing error magnitudes (where only the size of a deviation matters, not its sign), reliability and survival analysis, and as a common prior distribution for standard-deviation parameters in Bayesian statistics because it’s supported only on positive values.

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Comments (1)

  • Editorial team

    A reassuring check is to set x equal to sigma. With x = 2 and sigma = 2 the cumulative value is 0.682689, the familiar rule that about 68% of a normal curve lies within one sigma, because here the half-normal is simply the absolute value of that curve. The density in the same row, 0.241971, is a height, not a probability. Which quantity are you modelling with it: error magnitudes, waiting times, or a prior?

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