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Science, Maths & Technology

Averages

Updated Tuesday 1st May 2007

We often quote averages, but do we really know what they are? Kevin McConway explains the difference between mean, median and mode.

The average man - Cleese, Barker and Corbett on Frost Over England Copyrighted image Icon Copyright: BBC

Did you hear the one about the statistician who had his head in an oven and his feet in a bucket of ice? When asked how he felt, he replied, “On average I feel just fine.”

Well, it’s an old joke, it wasn’t the funniest in the world in the first place, and it never makes jokes funnier if you analyse them — but the joke does draw attention to two facts. First, statisticians do talk about averages rather a lot, and second, averages can be misleading.

What is an average anyway? Well, here we have a difficulty already. There isn’t just one sort of average, there are several. When people talk about an average, they are very often referring to what’s called the mean value of something. You’ve probably come upon this before. Here’s an example. There are four of us working in the same office. The numbers of children each of us has are 0, 2, 4 and 2. The mean number of children per person is the number we’d each have if they were divided out evenly between the four of us. In total there are 8 children, so the mean number is 8 divided by 4 - which is, of course, 2.

What happens if a new colleague joins us, and the new colleague has 3 children? Well, now the total number of children is 11, and the mean number of children per person is 11 divided by 5, which comes to 2.2. Now none of us actually has 2.2 children, and of course it’s not possible for any individual to have 2.2 children; but that doesn’t matter. The mean of a set of numbers is supposed to represent their ‘average’ level in a particular sort of way, and it doesn’t matter that it’s not a whole number when all the original numbers were whole numbers — as long as you realise that this can happen.

"The great majority of people in this country have more than the average number of legs"

In many situations, the average of a set of numbers is somewhere towards the middle of the set, but that doesn’t always happen. For instance, the great majority of people in this country have more than the average number of legs (if ‘average’ means ‘mean’). How can that be? Well, the great majority of people have two legs. Probably nobody has more than two. A minority of people, have only one leg, or no legs at all. So, if all the legs were divided out so that we all had the same number, we’d have a small amount less than two legs each. That is, the mean number of legs per person is a little less than two, and most of us have exactly two, so most of us have more legs than average. (Jokes are never funny if you have to explain them.)

The mean is not the only sort of average. Strictly speaking, the sort of mean I have described is the ‘arithmetic mean’, and there are also ‘harmonic means’ and ‘geometric means’ (and indeed other sorts of mean) that are sometimes used. Then there are the mode and the median.

The mode of a set of numbers is simply the number that appears most commonly on the list, and that’s a kind of ‘average’ because it is a ‘representative’ value. The numbers of children that my colleagues have, after the new recruit has joined us, are 0, 2, 4, 2 and 3. The most common number (just!) is 2, so that is the mode. The mode of the number of legs per person is 2, because most people have two legs.

The median of a set of numbers is another kind of ‘average’, found as follows. First you put the numbers in order. Then you find a point, half way along the ordered list, so that half the numbers are less than this point and the other half are more than it. This middle point is the median. Sorting the numbers of children into order, they are 0, 2, 2, 3, and 4. The one right in the middle is 2, so that is the median. In this case, the median and mode are the same, but the mean is 2.2 and is different.

Though the mean is the most commonly used sort of average, the median is also in common use, and it can be important to know which is being used. For instance, in the UK Government’s major annual survey of earnings (Annual Survey of Hours and Earnings, 2006), the mean gross annual pay of employees was recorded as £24,301, but the median gross annual pay of employees was recorded as £19,496. Half of all employees earned less than the median (because the median is the pay level that half of employees earn less than!). But over 60% of employees earned less than the mean. (So yes, it is true that most of us earn less than average, if ‘average’ means ‘mean’.)

If you think about patterns of pay, it’s not surprising that the mean and the median differ in this way. Nobody can earn, say, £25,000 less than the mean annual pay, because that would be saying they earned less than nothing. However, it’s far from unknown for people to earn £25,000 more than the mean annual pay. A few lucky individuals earn far, far more than that. If the earnings of all employees were put in a pot and divided out equally, the amount we’d all get would be the mean annual pay, and the existence of a few very well paid individuals makes this mean higher than you might expect, or at any rate higher than the median.

If you were involved in negotiating pay, would you use the median or mean in talking about ‘average pay’? Wouldn’t it depend on which side of the negotiation you were on? If someone starts talking about averages, you should be sure to find out what sort of average they are using.

Editor's note: Apologies for a typo which originally gave the mean figure in ninth paragraph as 2.6; and thanks to the reader who spotted our slip.

 

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