How to show that mean and median are the same if the distribution is symmetrical

Question

Given a set of observations denoted by $x_1,...,x_n$, how to show that the mean $\bar{x}$ and the median $\tilde{x}$ are the same if the distribution is symmetrical? My book says so, but I'm having trouble seeing that from the definitions of $\bar{x}$ and $\tilde{x}$. The book doesn't provide a definition for a symmetrical distribution, but I intuit we can define it to mean that $f(\tilde{x}+c)=f(\tilde{x}-c)$ for any number $c$, where $f(x)$ is the absolute frequency of $x$. Is this definition correct? If so, how to proceed from here? Thanks.

Answer 1

Linear transforms (scaling, reflections) of the data produce the same transform on the mean and on the median. In particular, if such a linear transfrom leaves the data invariant, it also leaves mean and median invariant.