Professor leaves Princeton to go to Stanford and increases the quality of both departments An old joke is that a certain professor left Princeton to go to Stanford and thereby improved the average quality of both departments. Is this mathematically possible? 
I found it in statistics course while I am studying. It sounds like ordered statistics. How it is possible? Thanks.
 A: Sure, it is possible. For this to occur, it is necessary and sufficient that:
$$
\text{average quality at Princeton}
> \text{quality of professor } X > \text{average quality at Stanford}.
$$
Indeed, suppose $p_1, p_2, \ldots, p_m$ are the qualities of other professors at Princeton, and $s_1, s_2, \ldots, s_n$ are the qualities of the other professors at Stanford, and $x$ is the quality of the professor who is leaving Princeton to go to Stanford.
Assuming that
$$
\frac{\sum_{i=1}^m p_i}{m} > x > \frac{\sum_{i=1}^n s_i}{n}
$$
we can prove that
$$
\underbrace{\frac{x + \sum_{i=1}^m p_i}{m+1}}_{\text{quality at Princeton before}} <
\underbrace{\frac{\sum_{i=1}^m p_i}{m}}_{\text{quality at Princeton after}}
$$
and additionally that
$$
\underbrace{\frac{\sum_{i=1}^n s_i}{n}}_{\text{quality at Stanford before}} <
\underbrace{\frac{x + \sum_{i=1}^n s_i}{n+1}}_{\text{quality at Stanford after}}.
$$
This relies on a simple key fact: the average of a set of numbers always increases when you add a number higher than the average to the set, and decreases when you add a number lower than the average to the set.
A: Let's assign a real-number measure to the "quality" of a professor. (For example,
$Q_n$ is the number of  grad students professor $n$ can drive insane per semester.)  
Before the move, Princton had $m$ professors and
$$
\sum_n Q_n = P
$$
for an average quality $P/m$.
After the move of professor $k$ the quality became $\frac{P-Q_k}{m-1}$ which will be more than $P/m$ precisely if $Q_k < P/m$.
Now the professor comes to Stanford, who had an $t$ professors and an average quality $S/t < P/m$.
As long as $Q_k > S/t$, the addition of this professor has raised the quality at Stanford, thus accomplishing the conditions of the statement.
