The choice of the formula for standard deviation is explained in elementary statistics textbooks in various ways. We give an explanation for this formula by representing the data as a vector in $\mathbb R^n$ and considering its distance from a central tendency vector. In this setting the "standard" formula represents a shortest distance in the standard metric. We also show that different metrics lead to different measures of central tendency.
"What is "Standard" About the Standard Deviation." Missouri J. Math. Sci. 22 (2) 86 - 90, May 2010. https://doi.org/10.35834/mjms/1312233137