Is a half-normal distribution just a normal distribution with negative values removed?

A centered normal conditioned to be nonnegative gives the same normalized distribution as its absolute value.

Can it model any positive quantity?

It is a candidate only when its shape and assumptions suit the quantity.

Does increasing sigma shift the lower boundary?

No. It stretches the distribution; the basic boundary stays at zero.

AI Glossary / Statistics and probability / Intermediate

Half-normal distribution

Definition

The half-normal distribution describes the absolute value of a normally distributed variable centered at zero. If Z is standard normal and sigma is positive, X = sigma*|Z| has a half-normal distribution with support from zero upward. It is useful for modeling certain nonnegative magnitudes and as a prior distribution for positive scale parameters. Nonnegativity alone does not establish that observed data follow a half-normal model.

What it means

A centered normal error can be positive or negative. Taking its absolute value discards the direction while retaining its size. Negative and positive errors of the same magnitude contribute to the same location in the resulting distribution.

The half-normal density is largest at zero and declines as magnitude increases. Most values are relatively small, with a tail extending to larger values. Its mean is positive even though the signed normal variable from which it is formed has mean zero.

The parameter sigma is the standard deviation of the underlying centered normal variable. It is a scale parameter for the half-normal distribution, but is not the standard deviation of the half-normal variable itself.

How it works

For x at least zero, the density is f(x) = sqrt(2/pi)/sigma * exp[−x^2/(2*sigma^2)]. It is zero for negative x. The cumulative probability is F(x) = 2*Phi(x/sigma) − 1, where Phi is the standard normal cumulative distribution function.

The mean is sigma*sqrt(2/pi), and the variance is sigma^2*(1 − 2/pi). The factor of two in the density and cumulative calculation accounts for both sides of the original symmetric normal distribution.

Some software supports a shifted version X = location + sigma*|Z|. Its lower boundary is the location. That shift is not the same as taking the absolute value of a normal variable with a nonzero mean, which generally gives a folded-normal distribution. Check parameter definitions before interpreting fitted results.

Three practical examples

A measurement team models absolute calibration errors when the signed errors are plausibly centered normal.
A Bayesian analyst considers a half-normal prior for a nonnegative standard-deviation parameter and checks the implications of its scale.
A simulation developer generates nonnegative error magnitudes by taking absolute values of centered normal draws, while retaining signs separately when direction matters.

Think of it this way

Imagine folding a symmetric paper bell curve along its vertical centerline. The left side lands on the right, combining the probability associated with equal negative and positive distances. This captures the absolute-value construction. Simply cutting away half the curve without renormalizing would leave only half the total probability and would not define the same probability density.

Worked example

Suppose signed measurement errors E follow a normal distribution with mean zero and standard deviation two units. Their magnitudes X = |E| then have a half-normal distribution with sigma = 2.

The expected magnitude is 2*sqrt(2/pi) = 1.5958 units. Its variance is 4*(1 − 2/pi) = 1.4535 squared units, so the half-normal standard deviation is about 1.2056 units, not two.

The probability of a magnitude at most three is P(X ≤ 3) = 2*Phi(1.5) − 1 = 0.8664. The probability of exceeding four is 2*[1 − Phi(2)] = 0.0455. Independent evaluation through the error function confirms the probabilities and moment calculations.

Thus the model predicts about 4.55% of magnitudes above four units in repeated draws. It does not promise that exactly that fraction appears in a finite sample, and the result depends on the centered-normal assumption for the signed error.

Illustrative case study

In a fictional sensor company, an analyst fits a half-normal model to absolute deviations from a reference value. The fit looks reasonable near zero, but a review of the original signed errors reveals a persistent positive offset after a firmware change.

The team realizes that absolute values have hidden the direction of the problem. It investigates and corrects the calibration offset, then reassesses the signed-error model before modeling magnitudes again. It also checks whether unusually large deviations occur more often than the half-normal tail predicts. The scenario illustrates model checking and information loss; it does not claim a verified reliability improvement for a real sensor.

Common misconceptions and mistakes

Do not confuse the distribution's scale with its actual standard deviation. Do not treat every absolute normal variable as half-normal: the normal variable must be centered at zero for the basic construction. A shifted half-normal and a general folded normal have different meanings.

A probability density value is not the probability of an exact magnitude, which is zero under a continuous model. The mode at zero does not imply a positive probability mass at exactly zero. A large pile of recorded zeros may instead reflect rounding, censoring, or a separate zero-generating process that needs explicit treatment.

Business use

The half-normal model can help translate an error scale into expected magnitudes and exceedance probabilities. For operational use, verify the measurement reference, units, rounding process, and behavior of the signed data. Compare predicted tail frequencies with relevant validation observations rather than relying only on a visually plausible histogram.

When used as a Bayesian prior, choose sigma through plausible parameter values and prior predictive checks. A prior that concentrates near zero can materially influence a weakly identified scale parameter. Label that use clearly so readers do not confuse a modeling assumption with observed evidence.

Check your understanding

With sigma equal to two, is the mean magnitude two?

Answer: No; it is approximately 1.5958.

Why does the exceedance calculation include both normal tails?

Answer: Large positive and large negative signed errors both become large absolute magnitudes.

Related terms

Normal distribution
Absolute error
folded-normal distribution
truncated distribution
scale parameter
probability density
prior predictive check

Learn next

Next, compare signed errors and absolute errors from the same dataset. Then distinguish half-normal, folded-normal, and truncated-normal models, paying particular attention to parameter meanings and the information lost by discarding signs.

Sources

Sources: SciPy, scipy.stats.halfnorm documentation, https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.halfnorm.html; MathWorks, “Half-Normal Distribution,” https://www.mathworks.com/help/stats/half-normal-distribution.html. Official documentation was opened and the density, location-scale convention, moments, and cumulative distribution checked. The numerical example and sensor scenario are original. Mathematical support is stated explicitly: the basic density is zero below zero.

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