 Investigation: The problem with "average"

Lesson

Mean scores are often called "averages" in everyday life. Averages are used a lot, both inside and outside the classroom. For example, when your teacher gives you your test marks back, they often tell you the "average" mark so that you can get a sense of how you performed compared to the rest of your class. However, you have to be careful when comparing scores to the mean because they can sometimes be misleading.

Activity 1

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

Student Score
Abdi $70$70
Bao $65$65
Charlie $60$60
Dahlia $55$55
Eden $50$50
Faith $61$61
Gabriel $57$57
Hakim $48$48
Isobel $72$72
You $63$63

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$10.

 $Mean$Mean $=$= $\frac{70+65+60+55+50+61+57+48+72+63}{10}$70+65+60+55+50+61+57+48+72+6310​ $=$= $60.1%$60.1%

Since the class average was around $60%$60%, and your mark is $63%$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 (陶哲軒), an Australian-American mathematician who has been described as the smartest person in the world. When Terence was $8$8 years old, he was teaching calculus to high school students, and he started university when he was just $14$14 years old. In 2014, he won a $\$3$$3 million prize for his groundbreaking discoveries in mathematics. Needless to say, Terence would find your test fairly easy. Let's say everyone gets the same score in the next test, except Terence, who gets 100%100%. Now how would your classes scores look? Student Score Abdi 7070 Bao 6565 Charlie 6060 Dahlia 5555 Eden 5050 Faith 6161 Gabriel 5757 Hakim 4848 Isobel 7272 You 6363 Terence 100100 Now what is the average? Let's add them up and see.  MeanMean == \frac{70+65+60+55+50+63+57+48+72+100}{11}70+65+60+55+50+63+57+48+72+10011​ == 64.6%64.6% Now your score of 63%63% is below average. Oh no! But really, if we think about it, it doesn't make sense to be disappointed. You did just as well in the test, and just because there is a genius in same class as you, it shouldn't change how you look at your mark. These kinds of situations, where one abnormally high result changes the mean significantly, are called "outliers". They are one of the biggest problems with using the mean as an average. Discussion questions 1. What would the median be for your original class of 1010 people? 2. What would the median be for the class with Terence in it? 3. How does the change in the median when Terrance was added compare with the change in the mean? 4. If a plastic mannequin joined your class scored 0%0% on the test, what would be the mean for this class of 1212? 5. How many additional mannequins would you need to make all the human student's scores be above the class mean? 6. How many mannequins do you think you would need for this to work in your real class at school? 7. Would adding mannequins work if your teacher uses the median as the average? Activity 2 Imagine a country where the average income is over \1000000$$1000000 (US) per year. Would you want to live there? Sounds pretty good, doesn't it!

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$1000000 people. $999900$999900 of these people are incredibly poor. Since the UN defines poverty as anyone living on less than $\$1$$1 per day, which is \365$$365 per year, we'll say that these people all make $\$300$$300 per year. The other 100100 of these people are all as rich as Bill Gates, one of the richest people in the world, having around \96500000000$$96500000000 (that's $96.5$96.5 billion US dollars) each.

Discussion questions

1. What would the mean income for the country above be?
2. What is the median of this country?
3. Would you still want to live there?
4. Is the mean, median or mode the best measure of center? Explain your choice.
5. Do you think there are any actual countries like this?
6. Is there anyone who actually has the mean as their income?

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 "humps" 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"!