Being Careful When You Average an Average: A Basketball Problem
Alan and Brian are two young basketball players competing to have the highest points per game average on their team.
For the first half of the season, Alan has the higher average of 10 points per game compared to Brian's 9 points per game. In the second half of the season, Alan again has the higher average, this time averaging 16 points per game to Brian's 15. Imagine Alan's surprise when he then discovers that it is Brian who has ended up with the highest average for the whole season.
How has this happened? How can Brian's average for the season be higher, when it is Alan's averages that were higher in each individual half? The problem comes from trying to average averages.
For the first half of the season, Alan has the higher average of 10 points per game compared to Brian's 9 points per game. In the second half of the season, Alan again has the higher average, this time averaging 16 points per game to Brian's 15. Imagine Alan's surprise when he then discovers that it is Brian who has ended up with the highest average for the whole season.
How has this happened? How can Brian's average for the season be higher, when it is Alan's averages that were higher in each individual half? The problem comes from trying to average averages.
What Average Are We Using?
Before we start, it will be useful to clarify what we mean when we say 'average'. In this article, we are using the mean average (as opposed to using the median or mode). The mean average of a group of numbers is calculated by adding together the numbers and then dividing by how many numbers there are.
E.g. To calculate the mean average of the group (2, 5, 7, 3, 3) we add together 2 + 5 + 7 + 3 + 3 = 20 and then divide by five as there are five numbers in the group. The mean average = 20 ÷ 5 = 4.
The mean has the effect of sharing the values out and redistributing them evenly, so that if our group of (2, 5, 7, 3, 3) was how many sweets each of five children had, the mean of four would be how many sweets each child would get if they all shared equally.
E.g. To calculate the mean average of the group (2, 5, 7, 3, 3) we add together 2 + 5 + 7 + 3 + 3 = 20 and then divide by five as there are five numbers in the group. The mean average = 20 ÷ 5 = 4.
The mean has the effect of sharing the values out and redistributing them evenly, so that if our group of (2, 5, 7, 3, 3) was how many sweets each of five children had, the mean of four would be how many sweets each child would get if they all shared equally.
Why Has Averaging the Averages Not Worked Here?
To work out how this apparent contradiction has happened we should look at the numbers involved and the calculations used.
The table below shows the points scored by each player in each game during the first half of the season.
The table below shows the points scored by each player in each game during the first half of the season.
Here's a look at Alan and Brian's scores through the first half of the season.
Points Scored in the First Half of the Season
We can see from the table that Alan scored a total of 80 points across 8 games and so has a mean average of 80 ÷ 8 = 10 points per game.
Brian meanwhile scored 72 points across 8 games and so has a mean average of 72 ÷ 8 = 9 points per game.
So far, so good.
Brian meanwhile scored 72 points across 8 games and so has a mean average of 72 ÷ 8 = 9 points per game.
So far, so good.
Here's a look at Alan and Brian's scores through the second half of the season
Points Scored in the Second Half of the Season
In the second half of the season there were another eight games. Unfortunately for Alan, he suffered some injuries and only managed to play in three games. In these games he scored a total of 48 points and so a mean average of 48 ÷ 3 = 16 points per game.
Brian, however, played in all eight games, scoring a total of 120 points and having a mean average of 120 ÷ 8 = 15 points per game.
Brian, however, played in all eight games, scoring a total of 120 points and having a mean average of 120 ÷ 8 = 15 points per game.
Bringing Our Data Together
So far our calculations have backed up everything we were told. Alan did indeed have the higher points per game average in both the first half of the season and the second half. It is when we put the two halves together that things get interesting.
Over the whole of the season, Alan scored 128 points over 11 games for an average of 128 ÷ 11 = 11.63 points per game.
Brian scored 192 points over 16 games for an average of 192 ÷ 16 = 12 points per game.
Brian's average is higher!
Over the whole of the season, Alan scored 128 points over 11 games for an average of 128 ÷ 11 = 11.63 points per game.
Brian scored 192 points over 16 games for an average of 192 ÷ 16 = 12 points per game.
Brian's average is higher!
How Has This Happened?
The problem here is that we were comparing averages using two different parameters. Our original calculations were for average points per game, whereas when assuming that we could average the two season halves we were trying to average two time periods. The fact that Alan played fewer games than Brian in the second half of the season means that these two things are not equivalent, hence why we ended with a false conclusion.
Note that if Alan and Brian had each played the full eight games in each half of the season, our method would have been fine.
Our problem was compounded by the difference in scoring records of the two players in each half. During the first half of the season, when both players played the full eight games, they were both fairly low scoring. During the second half of the season both players average per game improved significantly. During this period, Brian had more games in which to capitalise on the higher average compared to Alan. Over the course of the whole season, this was a significant factor in Brian's overall average being higher.
Note that if Alan and Brian had each played the full eight games in each half of the season, our method would have been fine.
Our problem was compounded by the difference in scoring records of the two players in each half. During the first half of the season, when both players played the full eight games, they were both fairly low scoring. During the second half of the season both players average per game improved significantly. During this period, Brian had more games in which to capitalise on the higher average compared to Alan. Over the course of the whole season, this was a significant factor in Brian's overall average being higher.
Be Careful and Don't Just Assume
In conclusion, we need to be extremely careful when trying to average mean averages. Generally, the best advice is to total all of the numbers and recalculate the average, don't just assume like Alan did.
Comments
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