Averages

We sometimes hear strange statistics quoted.  For instance, it was once stated that the average number of children in a British household was two point four (2.4).   Yet there were no households which actually had exactly 2.4 children in it.  Another example: I read that every child in the UK has above the average number of arms. How can this be?

What is an ‘average’?

It is said that a well-known politician was shocked to hear that 50% of the children in his country’s schools were ‘below average’.  Yet the most frequent use of the word ‘average’ is the word known in mathematics as the mean.

The mean

The mean can be calculated by simple arithmetic, and is the result of adding up a series of numbers – let us say n of them – and then dividing by n.

So perhaps you might want to know the average number of cars that pass your house each morning between 10am and 10.30am.  First you need to collect the data – or information – which means sitting down outside your home with a pencil and paper, and counting, over several days. Let’s say that you do this for twenty days, and record the following information:

15, 14, 17, 21, 17, 18, 19, 15, 13, 13, 19, 15, 15, 16, 18, 13, 19, 14, 12, 17.

Next you need to add all these numbers up (it’s easiest to use a calculator).  It comes to 320.   Since you collected data over twenty days, you have twenty numbers, so in the calculation of the mean, n=20. That means that you need to divide 320 by 20 to get the mean – and the answer is 16.

So you can say that on average, 16 cars pass by your house between those times each day.

Now let’s say you want to know the mean number of cats in your neighbours’ homes.  So you go around and ask 10 different people how many cats they have.   You find that three home have no cats, five homes have one cat, one home has three cats, and one home has seven cats. So the numbers you have are:  0, 0, 0, 1, 1, 1, 1, 1, 3, 7.  There are ten different numbers, so n = 10 in the calculation of the mean.

Adding up the ten numbers:  0+0+0+1+1+1+1+1+3+7 = 15.  Now divide 14 by 10 and you get 1.4, which is one and a half.

So you could say that the average number of cats in your neighbourhood is 1.5 per home, which doesn’t make much sense.  It’s a bit  like the first example given at the beginning of this page, when the average number of children in a household was given as 2.4. It’s mathematically correct, but not actually very helpful.  None of the homes actually have one-and-a-half cats in them.

So it may be more useful to look at another kind of average, the mode.

The mode

The mode is simply the result that occurs most often in a set of data.  With the cats, there are five households that have one cat, so that is the mode.  It makes more sense to say that the majority of cat-owning households have one cat, or even that ‘on average’ your neighbourhood has one cat per home.

With the car data, there were 12 cars on one day, 13 cars on three days, 14 cars on two days, 15 cars on four days, 16 cars on one day, 17 cars on three days, 18 cars on two days, 19 cars on three days, and 21 cars on one day.  So the number that appears on more days than any other is 15 – it is the mode.  The mode is often close to the mean, as it is in this instance.

There’s one more form of average that is sometimes used, and that is the median.

The median

The median is the middle result of a set of data.  To find it, the numbers have to be arranged in numerical order, so that you can find the mid-point.  If your data is:  1, 1, 3, 5, 8 then the median is 3.

When, as in the two examples given above, n is an even number, the median is half-way between the two middle results (if they are different).  With the cats, the data is already arranged in numerical order; the fifth and sixth results are both 1, so the median – like the mode – is 1.

With the cars, putting the results in numeric order, we see that the 10th result is 15 and the 11th result is 16. These are the two middle places out of twenty, so the median is 15.5 (fifteen and a half).

Another useful piece of information you can use is the range of data.

A range

The range of a set of data is the difference between the highest and the lowest values.  In the example with the cars, the lowest number is 12 and the highest is 21.  So the range is 21-12 which is 9.  In the example of the cats, the lowest number is 0 and the highest number is 7.  The range, then, is 7.

The range can help you determine how useful your averages are.  Let’s suppose you went around another ten households, and found again that about half of them had one cat, a third of them had no cats, and the rest had two cats, then the range of data would be 2.   The mean would then be much closer to 1.

In practical terms, perhaps you might consider buying a new computer, and are considering two different options.  You read up the information, and learn that both models have an ‘average’ (mean) life of five years. However, further reading tells you that one model has a range of ten years, while the second has a range of four. The first model has a few that last as long as eleven years, but it has others which break down after just a year, whereas the second lasts anything from three to seven years.

Of course, if you like to take a gamble, you might opt for the computer with a higher range, in the hope of having one that lasts a long time. But if you prefer to be secure, and confident that your computer will be reliable for a reasonable amount of time, you would be more likely to opt for the one with the smaller range of lifespans.

Other introductory maths pages on this site:

Multiplication tricks
Prime numbers
Number bases
Understanding angles