Investigation: The problem with "average"

Let's look at one example of this. Let's say that your math class has 10 students in it, and these are everyone's test scores:

Let's calculate the average mark for this class, using the mean. To do this, we add all the scores up, and then divide by 10.

Since the class average was around 60, and your score is 63, you'd probably be feeling pretty good. You're above average.

However, the next day let's say a new student comes to your class. That student is Terence Tao, an Australian-American mathematician who has been described as the smartest person in the world.

When Terence was 8 years old, he was teaching calculus to high school students, and he started university when he was just 14 years old. In 2014, he won a \$3 million prize for his groundbreaking discoveries in mathematics. Needless to say, Terence would find your test fairly easy.

Let's say everyone gets the same score in the next test, except Terence, who gets 100. Now how would your classes scores look?

Now what is the average? Let's add them up and see.

Now your score of 63 is below average. Oh no. But really, if we think about it, it doesn't make sense to be disappointed. You did just as well in the test, and just because there is a genius in same class as you, it shouldn't change how you look at your mark.

These kinds of situations, where one abnormally high result changes the mean significantly, are called "outliers". They are one of the biggest problems with using the mean as an average.

However, if this "average" is the mean, you have to be careful with this statistic. For example, let's say we have a country made up of 1000000 people. 999900 of these people are incredibly poor. Since the UN defines poverty as anyone living on less than \$1 per day, which is \$365 per year, we'll say that these people all make \$300 per year.

The other 100 of these people are all as rich as Bill Gates, one of the richest people in the world, having around \$96500000000 (that's 96.5 billion US dollars) each.

Normally, what you expect is a nice smooth distribution where there are a few people with very low incomes, lots of people in the middle and a few people with very high incomes, where the mean lies in the middle, like this:

However, in a bimodal distribution, you end up with two "peaks" of people, like this:

In this case, the mean lies in-between the two groups. Since in this case no-one actually has the "mean" as their income, it seems silly to call it "average".