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I have a question:

Suppose $5$ players each score an average of $10$ points per game. Then collectively, do they score on average $50$ points per game?

So player 1 scores an average of 10 points per game, player 2 scores an average of 10 points per game, etc...

So as a team they score on average of 50 points per game?

Edit. We want to form a team that averages 50 points per game.

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    @copper.hat: Yes. So you wouldn't be $10*5/5 = 10$?2012-05-17

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If $\sigma_g^p$ is the score of the $p$th player (of $n_p$ players) in the $g$th game, and they all play $n_g$ games, then their average game score is:

$\frac{1}{n_g} \sum_{g=1}^{n_g} \sum_{p=1}^{n_p} \sigma_g^p = \sum_{p=1}^{n_p} \frac{1}{n_g} \sum_{g=1}^{n_g} \sigma_g^p$

The second quantity is just the sum of the per-game averages of each player. So, yes, the team average is the sum of the player averages.

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    There are also many times when I was in a math class and an instructor claimed a step was intuitively obvious. It wasn't to me. In some situations I really think it is obvious to the instructor but in others it could be that he finds the step to be difficult to explain and the statement avoids having to explain. The students are too intimidated to ask for an explanation at that stage in the lecture.2012-05-20
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If $n_i$ is the number of points that player $i$ scores in $m$ games, then you have $\frac{n_i}{m} = 10$. Now the five players together make a total of $n_1 + n_2 + n_3 + n_4+ n_5$ points in $m$ games, so the average is $\begin{align} \frac{n_1 + n_2 + n_3 + n_4 + n_5}{m} &= \frac{n_1}{m} + \frac{n_2}{m} + \frac{n_3}{m}+ \frac{n_4}{m}+ \frac{n_5}{m} \\ &= 10 + 10 + 10 + 10 + 10 \\ &= 50. \end{align}$ So yes you are right. (Assuming that they are all playing the same game of course).