For n distinct elements x$1$; x$2$;:::;xn with positive weights w$1$; w$2$;:::;wn such

that Pn

iD$1$ wi D $1,$ the weighted $($lower$)$ median is the element xk satisfying

X

xi xk

wi

$1$

$2$ :

For example, if the elements are $0$:$1$; $0$:$35$; $0$:$05$; $0$:$1$; $0$:$15$; $0$:$05$; $0$:$2$ and each element equals its weight $($that is$,$ wi D xi for i D $1$; $2$; : : : ; $7),$ then the median is $0$:$1,$

but the weighted median is $0$:$2.$