Teaching the median with terms of absolute value, differentiability, and optimization

Abstract

The verbal definition of the sample median sounds a bit strange in the early Statistics courses in the context of being non-mathematical or non-functional. It is also interesting that estimators based on an analytical calculation such as the sample mean are equally strange, but not seen as strange by the students as the median estimator. In this study, we have expanded the studies on teaching the sample median with its optimization definitions. We have also shown that such definitions provide a natural way of understanding the sample median in multivariate case and regression analysis. Seeing that statistical estimators, from the simplest to the most complex, are obtained as a solution to an optimization problem can pave the way for other types of insights.

Keywords

• Teaching
• statistics
• absolute value
• median
• optimization

References

1. 1. H. Fritz, P. Filzmoser, & C. Croux. A comparison of algorithms for the multivariate L 1-median. Computational Statistics, 27-3 (2012): 393-410.
2. 2. J.P. Paolino. "Teaching univariate measures of location‐using loss functions." Teaching Statistics 40.1 (2018): 16-23.
3. 3. R Core Team (2021). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/.
4. 4. P. Rousseeuw. "Least median of squares regression." Journal of the American statistical association 79.388 (1984): 871-880.
5. 5. A. Tali. "Minimizing the Sum of Absolute Deviations." Teaching Statistics 7.3 (1985): 88-89.
6. 6. L. Yong, L. Sanyang, and Z. Shemin. "Smoothing Newton method for absolute value equations based on aggregate function." International Journal of Physical Sciences 6.23 (2011): 5399-5405.
7. 7. Chen, K., Ying, Z., Zhang, H., & Zhao, L. (2008). Analysis of least absolute deviation. Biometrika, 95(1), 107-122.
8. 8. Charnes, A., Cooper, W. W., & Ferguson, R. O. (1955). Optimal estimation of executive compensation by linear programming. Management science, 1(2), 138-151.