Truncated normal distribution

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The truncated normal distribution shares the same properties as the normal distribution and is determined by the mean (μ) and standard deviation (σ). Additionally, an upper, lower, or double “truncated” range is chosen to limit the distribution. More specifically, the truncated normal distribution arises from bounding a normally distributed random variable either from above, below or both. The truncated normal preserves the main features of the normal distribution while avoiding extreme values.

As an example, the truncated normal distribution is often used in elementary statistics to introduce the normal distribution; it is often truncated at three standard deviations either side of the mean — excluding data that falls four, five, or even 100 standard deviations from the mean. This is to make the analysis more manageable and results in a z-table that is more compact.

The truncated normal distribution is widely used in statistics and econometrics, particularly to model binary outcomes in the probit model and censored data in the tobit model [1]. It can estimate the mean and standard deviation of a population — without considering extremes — and can also be used to test hypotheses about the “bulk” of a population.

Properties of the truncated normal distribution

The truncated normal distribution PDF.
The truncated normal distribution PDF [2].

The truncated normal distribution has four key parameters:

  • μ: mean.
  • σ: standard deviation.
  • a: lower x-value > -∞.
  • b: upper x-value < ∞.

The probability density function (PDF) can be calculated with [3]


  • φ = PDF of the standard normal distribution
  • Φ = CDF of the standard normal distribution
  • μ̄ = mean of the standard normal distribution
  • σ̄ = variance of the standard normal distribution.

The cumulative distribution function (CDF) is:

truncated normal cumulative distribution function

Finding the mean of a truncated distribution can be challenging to do manually (Joel Schneider calls it a “bit of a mess,” as you have to calculate the probability density function (PDF) of the normal distribution first. By hand, the mean can be calculated with the following formula:

truncated normal mean

Schneider’s Excel spreadsheet calculates the mean and standard deviation of a truncated distribution [4] Scroll down under the plot to find the section titled “Truncated Normal Distribution.”


[1] Lecture 7: Models for Censored and Truncated Data — Tobit Model.

[2]  Image: By 018 (talk) via Wikimedia Commons. CC BY 3.0.

[3] Burkardt, J. (2014). The Truncated Normal Distribution. Retrieved May 31, 20223 from:

[4] Schneider, W. (2014). Using the truncated normal distribution. Retrieved June 4, 2014 from:

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