I read in a publication that the average salary for lawyers in America is $\bar x=\$163,595$. Of these salaries, the average for men is $\bar x_m=\$183,687$, and for women, it is $\bar x_w=\$163,595$. I'm thinking, how is it possible for the average of women's salary be equal the average of the entire set?
Note that $\bar x_w = \bar x < \bar x_m$, and $\bar x_w,\, \bar x,\, \bar x_m > 0$.
Can we prove that this is possible/impossible? Can we find 1 simple example where $\bar x_w = \bar x$ (given $\bar x_w,\, \bar x,\, \bar x_m > 0$, and $\bar x_m > \bar x$) is possible?
This is the calculation behind Ned's comment.
Let $m$ be the number of men, $w$ the number of women and $a_m$, $a_w$ respective averages. Then, the overall average is
$$a = \frac{ma_m + wa_w}{m+w}.$$ If $a = a_w$, from the above we get $ma_m + wa_w = (m+w)a_w$, i.e. $m(a_m-a_w) = 0$. Thus, either $m = 0$ or $a_m = a_w$.
Moreover, if $a_m \geq a_w$, then
$$a_ w = \frac{ma_w + wa_w}{m + w} \leq \frac{ma_m + wa_w}{m + w} \leq \frac{ma_m + wa_m}{m + w} = a_m.$$