Truncated Normal Distribution Calculator

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.