# Right Truncated Normal

• Nick T. Thomopoulos
Chapter

## Abstract

The right-truncated normal distribution (RTN) takes on many shapes, from a normal to an inverted exponential-like. The distribution has one parameter, k, called the right-location parameter of the standard normal distribution. The distribution includes all the standard normal values smaller than k, and thereby the density skews towards the left-tail. The important statistics are readily computed and are the following: mean, standard deviation, coefficient-of-variation, cumulative probability, and a variety of percent-points. Table values are provided to allow the analyst to apply the distribution to sample data. Also included is a series of plots that show the analyst how the density is shaped with respect to the right-location parameter. The percent-points are all negative values since the distribution starts with zero at the right-truncated location parameter, and subsequently moves downward to the left. Hence, the average and coefficient-of-variation are also negative values. When sample data is available, the average, standard deviation and coefficient-of variation are easily computed, and the analyst can apply these to identify the right-truncated normal distribution that best fits the sample data. This also allows the analyst to estimate any probabilities needed on the sample data, without having to always assume the normal distribution. Examples are provided to help the user on the application of the distribution. In Chap. , (Truncated Normal Spread Ratio), another statistic is introduced and allows the analyst to easily identify the type of distribution (left-truncated, right-truncated, normal) that best fits sample data. This chapter pertains to a right-truncated normal distribution; while the prior chapter describes the left-truncated normal. Recall from Chap. , (Continuous Distributions), only the beta distribution offers a choice of shapes that skew to the left, and thereby this right-truncated normal distribution may well be a welcome alternative. Furthermore, the percent-points of the right-truncated normal are far easier to compute that those of the beta distribution.

## References

1. 1.
Johnson, A. C. (2001). On the truncated normal distribution. Doctoral Dissertation. Chicago: Stuart School of Business, Illinois Institute of Technology.Google Scholar
2. 2.
Johnson A. C., & Thomopoulos, N. T. (2001). Tables on the left and right truncated normal distribution. Proceedings of the Decision Sciences Institute, Chicago.Google Scholar