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

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