Rethinking the truncated normal distribution

A truncated distribution is a conditional distribution which is restricted by the domain of a random variable in special situations. Truncated distributions were introduced more than one hundred years ago, but only a few papers have examined the standardisation of a truncated distribution, especially the standardisation of a truncated normal distribution. However, the traditional truncated standard normal distribution, derived from the truncation of a standard normal distribution, has varying mean and variance, depending on the location of truncation points. As a result, its statistical analysis may not be done on a consistent basis. The contribution of this paper is three-fold. First, we develop a standard truncated normal distribution by standardising a truncated normal distribution. The truncated mean and variance of this distribution are zero and one, respectively, regardless of the location of the truncation points, thereby being more consistent with the well-known standard normal distribution. Second, we develop the cumulative probability table of the standard doubly truncated normal distribution as a set of guidelines for engineers and scientists. Finally, we prove that the variance of the truncated distribution is always smaller than the variance of the original distribution.