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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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    Do they all play all games?2012-05-17
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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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    Seems like a lot of mathematics to state the obvious.2012-05-18
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    @MichaelChernick, Actually, it is not a lot of mathematics: it is just notation.2012-05-19
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    Well both answer seem to elaborate on something that is rather obvious and I would think could be told more suscinctly.2012-05-19
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    I'm sure your suCcinct contribution would be welcomed. Perhaps having a PhD in Statistics might bias your notion of obvious?2012-05-19
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    @copper.hat you sound offended. I don't mean to sound like a snob if that is the way you took it. It would seem obvious to me even if i didn't have a PhD. I meant it lightheartedly. The answers are good. I just thought that it would be easier to say it in short sentence rather than through mathematical symbols which could be confusing to some who read it. I would have just said sums of averages are averages of sums.2012-05-19
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    @MichaelChernick: I wasn't bothered by any comment on my answer, just that describing the question/answer as obvious belittles the 'asker'. Anyway, I'm over it :-).2012-05-19
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    I apologize for that. I really went to the trouble of explaaining because I didn't intend to offend. Sometimes I do offend people when it is not intended and I don't want to leave a bad impression even if it was unintended. Personally I do not think there is anything that could be called a stupid question. What is obvious to you and me was not obvious to the OP. He thought he new the answer but was unsure. He is certainly entitled to ask for confirmation. Many times I learned things that were not obvious to me but became obvious later after an explanation.2012-05-20
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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).