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I am teaching a year 8 class data, and they came across this question which they didn't know how to do.

The question is: Annie averages 15 points per game for the first 3 games of the AFL season. Over the next two weeks she scores 18 and 22 points. Find her average score over the 5 games?

This is my working

Find the average of the next 2 games $$\frac {(18+22)}{2} = 20$$

I then find the average of the 2 averages $$\frac {(15+20)}{2}= 17.5$$

Because you can't have an average of 17.5 points, would you round up or down?

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  • $\begingroup$ So, the question is all about rounding off numbers. $\endgroup$
    – SarGe
    Commented Aug 19, 2020 at 5:05
  • $\begingroup$ You cannot average averages as you do. What if there were 2 million games in the first batch, with average score 0 ? Then playing an additional 2 games (with scores 1 and 1) makes the total average 2/2000002, not 1/2. $\endgroup$
    – Michael
    Commented Aug 19, 2020 at 5:14
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    $\begingroup$ The average of averages isn't the average. Imagine scoring $1$ point a game for $100$ games and than $3$ points in the next game. Is your average $2$? What you need is $\frac{15\cdot 3+18+22}{5}=17=\frac{\textrm{total points}}{\textrm{total games}}$. $\endgroup$
    – user765629
    Commented Aug 19, 2020 at 5:15
  • $\begingroup$ p.s. its good to see a fellow Australian on here $\endgroup$
    – user765629
    Commented Aug 19, 2020 at 5:18
  • $\begingroup$ Also, it makes perfect sense to have a noninteger average. You should not round to an integer. $\endgroup$
    – Michael
    Commented Aug 19, 2020 at 5:22

1 Answer 1

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As mentioned in the comments, taking an average of an average won't provide an accurate picture of the average score over five games.

Let's say Annie scores exactly $15$ points in each of her first three games (providing an average of $15$). Her average score over the five games would be

$$\frac{15+15+15+18+22}{5}=17$$

though it doesn't matter what combination you use to get an average of $15$; you will always have a five-game average of $17$ (i.e. having $14$ points, then $15$, then $16$ would still give you an average score over five games of $17$).

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