Mean median and mode - working out averages made simple

At times in your life, people might have asked you questions like "What's your average score over all your games?" or "How long does it take you to get to work in the morning on average?" Usually, what people understand by "average" is what's known in mathematics as the mean. To calculate the mean, you add up all the items you've got, and then divide by the number of items. So in the first example, you add up all your scores and divide by the number of games you played. In the second, you total up the times taken for each journey and then divide by the number of journeys.

However, the mean isn't the only type of average you can get. Sometimes it's more appropriate to use one of the two other types of average: the median or the mode.

The mode

The mode is the number that crops up most often in a set of values. Suppose you work in a women's shoe shop. The manager wants you to find out the average customer shoe size. She tells you that the reason she's asking is because she wants to know which size to get more of next time she orders fresh stock. In other words, she wants to know the most popular shoe size; it would be more useful to her if you found the mode rather than the mean! (It would also be a lot easier too LOL, as the example below shows.)

Let's say you sold twenty pairs of shoes in one day:

1 pair was a size 3
2 pairs were size 4
3 pairs were size 5
6 pairs were size 6
5 pairs were size 7
2 pairs were size 8
1 pair was a size 9

You can see straight away that the most popular size (the mode) is a 6, i.e. that's what the manager needs to order more of when she next buys in more stock. BTW, it's possible to have more than one mode in a series of numbers. If you'd sold six pairs of size 7's instead of five, then you'd have two mode sizes (i.e. sizes 6 and 7).

Just out of interest, how would you work out the mean of these sizes? Firstly, you'd need to add the size of each individual pair of shoes together:

3 + 4 + 4 + 5 + 5 + 5 + 6 + 6 + 6 + 6 + 6 + 6 + 7 + 7 + 7 + 7 + 7 + 8 + 8 + 9

which can be written more simply as

(1 x 3) + (2 x 4) + (3 x 5) + (6 x 6) + (5 x 7) + (2 x 8) + (1 x 9)

= 3 + 8 + 15 + 36 + 35 + 16 + 9

= 122

Then divide by the number of pairs of shoes, i.e. 20. This would give an answer for the mean of 6.1.

The median

You don't come across this one very often in everyday life, but it does occasionally crop up. The median is the so-called "middle value". In my shoes example above, I listed all the pairs of shoes in order like this:

3 4 4 5 5 5 6 6 6 6 6 6 7 7 7 7 7 8 8 9

There are twenty pairs of shoes, so the "middle" would come half way between the 10th and the 11th pair, like this:

3 4 4 5 5 5 6 6 6 6 | 6 6 7 7 7 7 7 8 8 9

To work out the median, you would just work out the mean of the sizes of the 10th and the 11th pair. It wouldn't need any calculation in this case, because since the 10th and the 11th pairs are both size 6, then the mean of the two must also be 6! Hence the median is 6.

Suppose you had fifteen pairs of shoes, with the following sizes:

3 4 5 5 6 6 6 7 7 7 7 8 8 8 9

The mid-point occurs at the eighth pair of shoes, which is a size 7. Hence the median size is 7. As luck would have it, the mode is also 7 as well.

That's all very well, but when would you use the median in preference to the mean or mode? Not very often, I admit, although it does sometimes come in handy. For example, when talking about people's incomes there's often a very good reason to use the median to describe average income, rather than the mean or mode. (One of my other hubs, about the use and misuse of statistics, discusses why this should be the case. If you want to read more, click here).

The range

This is actually pretty easy.  The range is simply the difference between the highest and lowest values.  So in my shoe example above, the range of sizes is 9 - 3, i.e. 6.

 How do I remember all this?

I'm a great fan of mnemonics and little ditties as a way to remember things, and one of my ex-colleagues when I taught basic skills maths came up with a great one to help remember the four terms mean, median, mode and range.  It's based on a well-known nursery rhyme, and goes like this:

Hey diddle diddle, the median's in the middle
You add then divide for the mean
The mode is the one you use the most
And the range is the difference between!