Here is an exercise you might like to try on a class or individual, when introducing the mean. I have found it interesting and enlightening for all parties, especially those who think they know everything.
Dr Nic: Tell me what a mean is, as if explaining it to someone who doesn’t know about statistics.
Student: It’s an average. Dr Nic: Correct, however you haven’t really increased my understanding with that description. Student: It is what you get when you add all the numbers together and divide by the number of numbers.
Dr Nic: That is a correct description of how to calculate a mean. Still I’m not getting any idea of what it does.
Student: It’s rather like a middle number.
That has merit, though that description works better for a median. I still don’t think you are getting to the essence of it.
This is harder than I thought!
Student: I give up. This is harder than I thought.
Dr Nic: It really is. The idea of a mean is quite tricky. I like to think of it as a way of summarising a whole lot of numbers, in order to make comparisons.
Dr Nic: Say you have a whole lot of numbers that are the times different people take to complete a Rogo puzzle. I can even give you some: 16, 23, 30, 14, 63, 34. Say you want to summarise these numbers in one number, how would you do it?
Student: You could add them up (180 seconds) and say how many there are (six people)– or you could find out what the total is divided by the number of numbers. Which is the mean! (30 seconds per person)
Dr Nic: Very good. However, why would you want to do this? Student: Because then you could say that… on average it took 30 seconds to solve the Rogo?
Dr Nic: Absolutely, but really why would you want to? Mostly we want a mean in order to compare. (Or in Operations Research we may like to use a mean to provide an input to decision-making.) If we had a second group of people who had a mean of 23 seconds for that Rogo, then we can see that on average the second group took less time. Or we could try another Rogo with the first group of people and find that the mean was 47 seconds. We would probably conclude that the second Rogo was more difficult.
Student: Hmm. So a mean is a way to summarise a set of numerical data that can be used for comparisons.
Dr Nic: Fabulous! I couldn’t have put it better myself.
Student: What’s a Rogo puzzle?
Dr Nic (aka Dr Rogo): Funny you should ask – it’s a puzzle we invented some years ago and made an app for, but it has faded away.
Another way to look at a mean is that it is an emergent property of a set of data. One observation can tell us a little bit about a phenomenon, but once you get a set of data, there are emergent properties that can help to explain the phenomenon.
Until I started to think about it, I had thought a mean was a really obvious concept. But it isn’t – and it is worth spending time on to clarify understanding in students. (And unless you wish to baffle them with long words, or have students with a strong mathematical bckground, I’d avoid the terms “measure of central tendency”, and “first moment”, until they have a better grip on the subject.)