Geometry of deviation measures for triangular distributions

• Published in: Frontiers in Applied Mathematics and Statistics
• Main Authors: Yuhe Wang, Eugene Pinsky
• Format: Article
• Language: English
• Published: Frontiers Media S.A. 2023-12-01
• Subjects: triangular distribution; mean absolute deviation; quartile deviations; probability distributions; geometry of deviations
• Online Access: https://www.frontiersin.org/articles/10.3389/fams.2023.1274787/full

Triangular distributions are widely used in many applications with limited sample data, business simulations, and project management. As with other distributions, a standard way to measure deviations is to compute the standard deviation. However, the standard deviation is sensitive to outliers. In this paper, we consider and compare other deviation metrics, namely the mean absolute deviation from the mean, the median, and the quantile-based deviation. We show the simple geometric interpretations for these deviation measures and how to construct them using a compass and a straightedge. The explicit formula of mean absolute deviation from the median for triangular distribution is derived in this paper for the first time. It has a simple geometric interpretation. It is the least volatile and is always better than the standard or mean absolute deviation from the mean. Although greater than the quantile deviation, it is easier to compute with limited sample data. We present a new procedure to estimate the parameters of this distribution in terms of this deviation. This procedure is computationally simple and may be superior to other methods when dealing with limited sample data, as is often the case with triangle distributions.