# Trapezoidal Distribution

A trapezoidal distribution is so-named because the graph of its probability density function (PDF) has a quadrilateral shape defined by two parallel and two non-parallel sides. It provides a suitable model for data that undergoes fast growth, a stagnation period, and a rapid decline, all of which can be described by linear functions. Specifically, the function starts with positive slope, remains constant during the plateau, and then ends with a negative slope. Although the distribution is simple to understand and use, it does result in conservative estimates for analyses [1].

## Properties of the trapezoidal distribution

The PDF for the trapezoidal distribution is

• a = minimum value for the random variable,
• b = lower mode (where the constant stage starts),
• c = upper mode (where the constant stage ends),
• d = maximum value for the random variable.

The lower bound or minimum a and upper bound or maximum value d define the range of possible values or events in a trapezoidal distribution. Values beyond this range have a probability of zero. This distribution also has two non-differentiable discontinuities, or sharp bending points, denoted as b and c, which occur between a and d; b and c satisfy the conditions abcd.

Note that the distribution’s mode is not unique, rather it can assume a value between the lower mode c and the upper mode d.

The cumulative distribution function (CDF) is 0 for x < a and 1 for xd. Otherwise, it is linear between b and c, and quadratic for ab and cd:

## Uses for the trapezoidal distribution

The literature commonly discusses the uniform, triangular, Irwin-Hall, Bates, Poisson, normal, bimodal, and multimodal distributions more often than the trapezoidal distribution, likely due to their more frequent occurrences in nature. In particular, the normal distribution is prevalent in nature, as expected by the central limit theorem. However, the trapezoidal distribution is seen in many natural processes that occur over time — with a slow build up, a plateau, and a slow decline.

The trapezoidal distribution is best for modeling data with a clear beginning, middle, and end. For instance, it can approximate the mean, median, and mode of a dataset and determine the probability of a value falling within a specified range. Using the trapezoidal distribution for data analysis has several benefits. They have been advocated for use in risk analysis by Pouliquen [3] and Powell and Wilson [4]. They have also been applied as membership functions in fuzzy set theory [5] and as models for observed axial distributions for burnup credit calculations in nuclear engineering [6].

This simple distribution can also be used to estimate production time, project expenses, and patient volume — assuming that the data follows the shape of the distribution. As a result, it can aid in scheduling, budgeting, and staffing. For example:

• In finance, the trapezoidal distribution can model the amount spent on a project. This helps with budgeting for the project as well as tracking expenses.
• In healthcare, hospitals use the trapezoidal distribution to model the number of patients who may require admission on any given day. This information is key for hospitals to plan their staffing and delivery of care. The distribution has also been used in cancer screening and detection [7].
• In healthcare, hospitals use the trapezoidal distribution to model the number of patients who may require admission on any given day. This information is key for hospitals to plan their staffing and delivery of care.

## Similar distributions

Special cases include the uniform distribution and the triangular distribution. There is no growth or decay stage in a uniform distribution, implying that the minimum (a) equals the lower mode (c), and the maximum (d) equals the upper mode (b). The triangular distribution lacks the constant stage, resulting in the lower mode (b) equaling the upper mode (c).

## References

[1] Neuber, JC (2000). Axial Burnup Profile Modeling and Evaluation. Second PIRT Meeting on BUC, Washington D.C./jcn2000-08-14.

[2] Image credit: Fuzzyrandom, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons

[3] Pouliquen LY (1970). Risk Analysis in Project Appraisal, World Bank Staff Occasional Papers 1. John Hopkins University Press, Baltimore Md.

[4] Powell MR, Wilson JD (1997). Risk Assessment for National Natural Resource Conservation Programs, Discussion Paper 97-49. Resources for the Future, Washington D.C.

[5] Chen SJ, Hwang CL (1992). Fuzzy Multiple Attribute Decision-Making: Methods and
Applications, Springer-Verlag, Berlin, New York.

[6] Wagner JC, DeHart MD (2000). Review of Axial Burnup Distributions for Burnup Credit
Calculations. Oak Ridge National Laboratory, ORNL/TM-1999/246, Oak Ridge,
Tennessee.

[7] Brown SL (1999). An SAB Report: Estimating Uncertainties in Radiogenic Cancer Risk, Science and Advisory Board, United States Environmental Protection Agency, EPASAB-RAC-99-008, Washington D.C..

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