The meaning of the mean
Re Solutions
28 December 2011
Should students calculate?
6 January 2012Here 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.
Dr Nic:
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.
Student: Huh?
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.
Comment:
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.)
14 Comments
I agree that the concept of the mean is unintuitive even though every student is exposed to it from an early age, at least in terms of how to calculate it and the fact that it’s used to determine their grades. However, your explanation of the mean still uses the word “average” — ‘on average, the second group took less time.’ And a similar explanation would apply to the median as well.
When I discuss the mean in AP Stats, I use the word ‘typical’. The mean is one of many ways to measure what’s typical in a sample or population. A person with a height close to the population mean is typical (we are unsurprised to find such a person); people with heights far from the population mean are atypical in this sense. The mean is sometimes better, sometimes worse than other measures of typical-ness depending on the data. In terms of randomness, a mean gives us a sense of ‘expectation’. From a physical perspective, the mean is the ‘center of mass’ or ‘balancing point’ for a set of data.
It is interesting that, how tricky the mean is – I asked one of my teaching assistants about it this morning and you could see the steam coming out his ears as he tried to work out how to explain what a mean is.
“Typical” is a great word. Thanks for the suggestions.
Interesting discussion. I hope my spring stats classes are open to this kind of engagement. My fall class was pretty quiet.
One way to look at it is that the mean abstracts away the variability in the observations. That is, if you had the same number of observations and the same sum of observation values, the mean is the value that each observation would have to take on if there were no variation allowed among observations. Visually, if you think of a graph with the vertical axis representing observed values and the sequence of observations graphed with a step function where each observation is graphed with a line segment one unit long, the mean is the height of a rectangle with the same area as the graph.
Thanks for your input. That is a very interesting way to look at it. Good luck with getting engagement! Maybe you could bribe them with chocolate. (A tried and true method) Maybe I could do a post on teaching statistics with chocolate. My favourite example to teach about the p-value does involve quite a bit of chocolate. And then there is always Helen with her choconutties.
Maybe a better graph picture: Vertical axis–observation values, horizontal axis–the interval 0-1. The graph is a step function with the width of each step equal to the proportion of observations with that value. Then as above, the mean is the height of the rectangle with equivalent (oreinted) area to the graph.
Another graphical idea: Think of the sum of values as a pie, and each slice as an individual value in the dataset. There will be individual variability in the size of each slice. The mean makes sure that each individual gets an equal sized slice of the pie.
Hi
It’s good to see this issue exposed / discussed. As a teacher, I often encounter colleagues who can’t answer this. As professionals we seem to embed the process to calculate the mean, but somewhere along the way we loose the meaning.
Having sat in on lessons taught by colleagues, I wonder if sometimes we even forget to teach the meaning of the mean and focus on the calculation?
For me, mathematics start with the understanding that deeper questions arrise after the calculations – the “what does that represent?” questions. In that sense, Maths is a toolbox, but knowing how to use a wrench is not enough to tune an engine. Mathematical literacy teaches the tools, statistical / scientific literacy teaches the “what if” questions…..
Or am I alluding to the fact that these artificial demarcations restrict our learners – we put things into boxes all to readily and what we really need to do is to teach learners to think.
Glen
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hi! good to see but what if i say the mean is “normal point of the data”.
Hi Abbas
It wouldn’t mean much to me. I would hesitate to use the word “normal” in statistics other than to refer to the normal distribution. Perhaps the “characteristic point of the data”?
In middle school, we talk about “equal shares” and “balance points”.
If a group of students each gets a handful of the same candies, the mean would be the amount that each person got if the candies were divided equally among all. They would each have an “equal share”.
If you have a teeter-toter and there are 3 kids on it, all having the same mass, the sum of the distances from the fulcrum on one side needs to equal the sum of the distances from the fulcrum on the other side in order for the teeter toter to balance. This can be seen on a ruler or a number line to extend the idea to show the mean as a “balance point”.