# Left Truncated Normal

• Nick T. Thomopoulos
Chapter

## Abstract

The left-truncated normal distribution, (LTN), takes on many shapes from normal to exponential-like. It has one parameter, k, the left-location parameter of the standard normal distribution. The shape of the distribution follows the normal for all standard normal z values larger than k, and thereby the distribution skews to the right. The statistical measures of this distribution are readily computed and include the mean, standard deviation, coefficient-of-variation, cumulative probability, and all percent-points needed. Three sets of table values are listed in the chapter allowing the user easy access and use to the distribution choice. One of the tables includes a range of percent-points denoted as tα where the probability of t less or equal to tα is α. The percent-points start with a value of zero and this is located at the left-location parameter, and since all subsequent values are larger, they thereby are all positive quantities. In addition to the tables, plots of the distribution are provided to observe the various shapes and relation to k. When sample data is available, the average, standard deviation and coefficient-of variation are easily computed, and the analyst applies these to identify the left-truncated normal distribution that best fits the sample data. This 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 that further aids the analyst to identify the type of distribution (left-truncated, right-truncated, normal) that best fits sample data, and also provides an estimate of the low limit for the left-truncated normal, and the high-limit for the right truncated normal. This chapter pertains to a left-truncated normal, while the next chapter describes the right-truncated normal.

## References

1. 1.
Thomopoulos, N. T. (1980). Applied forecasting methods (pp. 318–324). Englewood Cliffs: Prentice Hall.Google Scholar
2. 2.
Johnson, A. C. (2001). On the truncated normal distribution. Doctoral Dissertation. Stuart School of Business: Illinois Institute of Technology.Google Scholar
3. 3.
Johnson A. C., & Thomopoulos, N. T. (2002). Characteristics and tables of the left-truncated normal distribution. In Proceedings of the Midwest Decision Sciences Institute (pp. 133–139).Google Scholar
4. 4.
Thomopoulos, N. T. (2016). Demand forecasting for inventory control. New York: Springer.Google Scholar