###### Re Solutions
28 December 2011 ###### Should students calculate?
6 January 2012

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 (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.) ##### Dr Nic

1. Tao Wang (@MathLaoshi) says:

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.

• Dr Nic says:

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.

2. Matthew Saltzman says:

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.

• Dr Nic says:

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.

• Matthew Saltzman says:

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.

• Justin O says:

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.

3. Glen Gilchrist says:

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

4. […] The meaning of the mean – it is trickier than you think […]

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6. […] summary of a set of data. The whole idea of the mean is actually quite tricky, as you can read in one of my early posts about explaining what the mean is. Generally the summary value is used to compare with another sample or […]

7. […] summary statistics tend to be the mean and the standard deviation. I’ve written previously about what a difficult concept a mean is, and then another post about why the median is often preferable to the mean. In that one I promised […]

8. Abbas says:

hi! good to see but what if i say the mean is “normal point of the data”.

• Dr Nic says:

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”?

9. Dory Witzeling says:

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”.