Everyone has an instrumental concept of the mean: it's what you get when you add up all the numbers and divide by the number of numbers. Fine. And people have some sense that it represents the centre of the data.
But the actual concept of the mean is a difficult one. What is the mean (as opposed to how do you calculate it)? And how do you interpret it.
It's easier to say what the mean isn't than what it is!
I struggle with teaching the mean. I teach the median as a measure of centrality that everyone understands in the same way. But I don't start talking about the mean until we start doing prediction models.
How do others explain the mean? Do you explain it?
Skemp, in his seminal work on mathematics learning, (http://eledu.net/rrcusrn_data/Skemp%20article.pdf) makes a distinction between instrumental understanding (knowing how to) and relational understanding (understanding the relationships implied in the concept).
I have always tried to teach relational understanding. I teach the mean as the best guess – the guess that produces least error in the long run (error is also a concept that needs to be understood here). But this means that I don't start talking about the mean really until I start regression. Up until then, I concentrate on the erroneous ideas people have about it, and on promoting the use of quantiles as more flexible and easy to understand. I introduce the mean as a null model (I don't call it that, of course!) The mean is the model that says "When we know nothing that will help us guess, then guess the mean to make least error")
José: you make some important points, which fall into two categories, I think: first, the mean can mislead people where the distribution is asymmetric, and second the mean can distract attention from more important features of the data such as as the variability. This latter is often a problem with social surveys. If you measure something on a Likert scale, giving the mean is only half the story. There is a difference between a situation in which there is a small net support for a controversial policy and little variation and a situation where there is a very considerable spread of variation. In the latter case, opinion is polarised, and there will be a lot of controversy. In fact, pollsters have an index called the potential for conflict index.
More detail here: http://warnercnr.colostate.edu/~jerryv/PCI2/references/Manfredo_etal_(2003)_The_potential_for_conflict_index.pdf
Andy Field's book has some interesting pages about the mean. https://books.google.co.in/books/about/Discovering_Statistics_Using_SPSS.html?id=5253SAL5nDgC&hl=en
Andy views mean as a model of the data. And I kind of likee
It may possibly help to first start with explaining the concept of 'central tendency', and then introduce the 'mean' as one of several measures of central tendency.
Dr. Ronan,
Your question, in my opinion, opened an excellent and philosophical discussion in mathematics and thus in statistics. I believe that teachers worldwide explain the level of responses based on a scale of certain quantity used to evaluate a specific problem divided by number of participants in the experiment. Mathematically, the mean is an operation that involves sum within a quantity and the division by participants. Statistically, it has been a dangerous measurement of central tendency. It hidden outliers and induce consultants to mistakes. Generally, it is used politically by large corporations with several interests. One crucial example is about global warming. How to believe that mean increases about 1 OC is capable of changing the whole world? The real fact that is affecting the world is the variability in temperature, and the activity of its counterparts along the seasons. Under field conditions we are seeing high variability in the weather conditions. The division of a cake in two equal parts does not mean that either participant can get the same weight. In a car crash, the equipment measuring the velocity shows the speed at the moment of the accident. In this case, the distribution of the velocity along the road is more important, and a distribution of these data can have normal or asymmetric shape. The use of the word mean in popular talks is just for shorten the conversation and the expression have made difficult the comprehension of the measurement. In statistics, as you know, is possible to calculate the expected value of the mean. Could be the expected value the best explanation? Currently, I am discussing in my classroom all the data taken from a specific treatment using the possibility offered by the box-and-whisker plots. Well, as I told you: it is a great question. I will discussion it with my partners: one agronomist and one statistician. I will be back.
José Walter Carneiro.
Agronomist.
Skemp, in his seminal work on mathematics learning, (http://eledu.net/rrcusrn_data/Skemp%20article.pdf) makes a distinction between instrumental understanding (knowing how to) and relational understanding (understanding the relationships implied in the concept).
I have always tried to teach relational understanding. I teach the mean as the best guess – the guess that produces least error in the long run (error is also a concept that needs to be understood here). But this means that I don't start talking about the mean really until I start regression. Up until then, I concentrate on the erroneous ideas people have about it, and on promoting the use of quantiles as more flexible and easy to understand. I introduce the mean as a null model (I don't call it that, of course!) The mean is the model that says "When we know nothing that will help us guess, then guess the mean to make least error")
José: you make some important points, which fall into two categories, I think: first, the mean can mislead people where the distribution is asymmetric, and second the mean can distract attention from more important features of the data such as as the variability. This latter is often a problem with social surveys. If you measure something on a Likert scale, giving the mean is only half the story. There is a difference between a situation in which there is a small net support for a controversial policy and little variation and a situation where there is a very considerable spread of variation. In the latter case, opinion is polarised, and there will be a lot of controversy. In fact, pollsters have an index called the potential for conflict index.
More detail here: http://warnercnr.colostate.edu/~jerryv/PCI2/references/Manfredo_etal_(2003)_The_potential_for_conflict_index.pdf
Hi Ronán. Interesting question.
The way I introduce the mean in an introductory course is shown starting at about slide 34 in the Chapter 02 slides at the link below. I start by saying that it is the same thing that most people think of as the average--i.e., the sum of the numbers divided by the number of numbers.
But then, I go on to say that it is the exact balancing point of the distribution. (I try to illustrate this graphically, using an approach similar to what you can see on this page: http://onlinestatbook.com/2/summarizing_distributions/what_is_ct.html.) Another way of saying the same thing is that the sum of the deviations about the mean (for any distribution) = 0.
After discussing the median & mode, I then talk about how skewness affects the mean, median & mode (e.g., the mean gets pulled in the direction of the skewness, and is more sensitive to it that the median, etc.).
A bit later, when discussing measures of dispersion, I get to the variance and then the SD. When introducing variance, I remind students that the sum of the deviations about the mean = 0 for any distribution (or set of scores). Therefore, the sum of the deviations about the mean is not terribly useful. But for reasons I don't go into at that point, the sum of the squared deviations about the mean is very useful to statisticians, and that if they take that SS and divide by (n-1), they get a measure of dispersion called the variance. And SQRT(variance) = SD, etc.
It occurs to me now, as I review those slides, that I could insert another slide at around this point saying that not only is the sum of the deviations about the mean = 0, but the sum of the squared deviations about the mean is a minimum. I used to have an Excel file demonstration of this point, but can't find it right now. However, I just cobbled together an SPSS syntax file to demonstrate--see attached. (I imagine there are online demos somewhere too.)
Like you, Ronán, I come back to the mean after introducing OLS linear regression. In my chapter on regression, I have a slide that says the mean is the fitted value of Y for an intercept-only OLS linear regression model. Therefore, the sum of the squared deviations about the mean is a minimum.
Cheers,
Bruce
https://sites.google.com/a/lakeheadu.ca/bweaver/Home/statistics/introductory-biostatistics
Thank you Dr.Ronan
So we should let the data talk without affecting it's variability. So we need to predict the reality. The mean could affect the data quality and replacing the missing data by the mean will decrease the variability which in turn could make some relations not significant.
Warm regards,
Mu'taman Jarrar
Dear Ronán
The mean is a measure of central tendency that only has a useful meaning when there is no extreme values, in which case, should not be used. In fact, it is a measure that should always be supported by a measure to assess its representativeness with respect to the data set that averages. This gives some indication of its fragility, really: whether it should be supported by a measure of dispersion is not always "served", but this is rarely indicated, unless teachers must teach the mean, of course.
I hope I was helpful.
Best regards!
Montserrat: What always startles me is what happens when I ask students What do you mean when you say "the mean is 42"?
They become nervous and insecure, and I have to reassure them that this is not a test, but a demonstration. And if they don't really feel they understand the mean, then they should not assume that anyone else does either, and should look for other ways of summarising their data.
My motto is "if you don't understand it, don't use it"
It's like the rule I would like to apply to the US - If they can't pronounce the name of a country correctly, they can't invade it. My reckoning is that wars would rapidly die out.
A problem with the mean is that people grasp on to a hard number. Once they are given a mean, it is a number, a value, a point of comparison. A suggestion is to give the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. Ask for the mean.
The number 5.5 has little meaning in this case. Yet, many use this number as though it were the data. It is not data. It is the mean of a distribution. When speaking of a mean, we are speaking of a distribution. The mean will have little meaning if the distribution does not fit the data or when do not understand the distribution.
Dear Roman,
With your permission, and always takin, from now on I think I often say that about ... "If you do not understand it, do not use it"
It's just great.
But when you've really made me smile it is when you say that if If they can not pronounce the name of a country correctly, they can not invade it. My reckoning is rapidly that wars would die out.
Thank you for your nice, but true answer. You made me much more enjoyable day.
Best regards!
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