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Truncated Normal Distribution Calculator

Enter the point, mean, standard deviation and bounds to get the truncated density.

How it is calculated

First the point is standardised, z = (x − μ) / σ, and the standard normal density φ(z) = e−z²/2 / √(2π) is found. The normalisation factor is the probability the untruncated normal puts inside the bounds, Z = Φ((b − μ)/σ) − Φ((a − μ)/σ). The truncated density is f(x) = (φ(z) / σ) / Z. The cumulative function Φ is computed with a rational approximation of the error function (Abramowitz and Stegun 7.1.26, error below about 1.5e-7), so the last digits of Z are approximate. The tool shows f(x), φ(z) (not divided by σ) and Z with six decimals. It needs σ > 0 and a < b, and returns nothing if x lies outside [a, b], where the true density is zero.

Worked example

Let μ = 50, σ = 10, bounds a = 40 and b = 70, and ask for the density at x = 60. Standardise: z = (60 − 50) / 10 = 1, so φ(1) = 0.241971. The bounds are at z = -1 and z = 2, so Z = Φ(2) − Φ(-1) = 0.97725 − 0.158655 = 0.818595. Then f(60) = (0.241971 / 10) / 0.818595 = 0.0241971 / 0.818595 = 0.029559. Without truncation the density would be 0.0241971, so cutting off the tails raises it by about 22%.

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Common mistakes

  • Reading the output as a probability. It is a density, so it can exceed 1 when σ is small; probabilities come from areas under the curve, not from f(x) alone.
  • Entering the variance instead of the standard deviation. The field is σ, so a variance of 100 must be typed as σ = 10.
  • Choosing a point outside the bounds and expecting 0. The tool returns no result there, because the density is simply not defined for the retained part.
  • Putting the bounds in z-score units when the mean and σ are in real units. a and b are on the same scale as x and μ.
  • Comparing the 'Standard PDF' line with f(x). It is the untruncated standard value φ(z), before dividing by σ and by the normalisation factor.

About this calculator

A truncated normal distribution is an ordinary normal distribution that has been restricted to lie within a lower bound a and an upper bound b, with all probability outside that range discarded and the remaining density rescaled so it still integrates to 1. Its probability density function is f(x) = φ((x−μ)/σ) / [σ · (Φ((b−μ)/σ) − Φ((a−μ)/σ))] for a ≤ x ≤ b (and zero elsewhere), where φ is the standard normal PDF and Φ is the standard normal CDF — the denominator is simply the probability mass of the untruncated normal that falls inside [a, b], and dividing by it renormalizes the density.

Truncated normals appear whenever a naturally normal-looking quantity has a hard physical or logical floor and/or ceiling — measurement instruments with detection limits, reliability and lifetime data bounded below by zero, exam scores capped at a maximum, or Bayesian priors placed on parameters that can’t be negative. This calculator evaluates the truncated density at a chosen point given the mean, standard deviation, and the two bounds, reproducing exactly the renormalized PDF used in these applications.

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Comments (1)

  • Editorial team

    Compare the first two results to read the effect of truncation: the truncated density is simply the ordinary bell-curve density divided by the normalization factor. With the preset bounds of -2 and 2 the factor is 0.9545, so every point is raised by about 4.8%. The value is a density, not a probability, so it can exceed 1 when the window is narrow. What kind of bounded quantity are you modelling, such as test scores or sensor limits?

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