Range? Median? Mode? What does that Mean?

The different types of average is one aspect of the mathematics component of Secondary School Entrance Examinations (11+) and Common Entrance Examinations that many children find a stumbling block. The main issue that confronts pupils is remembering which form of average is which when faced with the mathematical meaning of the terms ‘range’, ‘mode’, ‘median’ and ‘mean’. Many children know how to calculate each form, but have trouble recalling the differences between the terms.

Before I explore the different types of average, let’s have a reminder of what ‘average’ means.

An average is a number that represents the central or most common value in a set of numerical data. There are three different types of average: the mode, the median and the mean. Each of the three types of average gives a slightly different interpretation of the data.

For the purpose of this blog post it is not necessary to expand on when it is most appropriate to use each type of average, as this will not form part of a question at 11+ level (the level at which this post is targeted). What I will do next is provide simple ways to recognise the difference between the steps needed to calculate the mean, the median and the mode. But, before doing that, let me turn to the range – a very quick and easy calculation as long as you understand the meaning of the term ‘range’.

The ‘range’ is simply the difference between the highest and lowest values in a set of numbers. For example, if the highest score in a maths test was 10 and the lowest score was 7 then the range is 3 (highest score 10 subtract lowest score 7) – easy! All you have to do is remember the meaning of the word ‘range’. One way is to think of the ‘range’ as the length of the data; another way is to think how much space the numbers cover. Often questions on the range are presented in a bar chart or pictogram format, so just look how far it is from the lowest number to the highest number.

Now let’s look at easy ways to remember and calculate the ‘mode’, ‘median’ and ‘mean’. For all three methods let’s use the same set of data – the scores of seven children in a spelling test: 5, 6, 7, 7, 7, 8, 9.

Mode – this is the number that appears most often. One way to remember this is that the words ‘mode’ and ‘most’ both have four letters, with the first two being the same. To find the mode just look at the numbers (putting them in ascending order helps) to see which one is there the most. So, using the data above, the mode is 7.

Median – this is the number that is in the middle when all the numbers are put in ascending order. One way to remember this is that both the words ‘median’ and ‘middle’ have the same number of letters. To find the median simply put the numbers in order and see which one is in the middle of the set. This works easily when the total of numbers in the set is odd, as in the case of the data above where the median is 7, but when the total of numbers in the set is even all you need to do is find the two middle values and divide by two.

Mean – to find the mean all you need to do is find the total of the numbers in the set (49 in the example above), then divide by the amount of numbers (7 in the example above), which gives a mean of 7. It is worth remembering that the mean, median and mode will not necessarily be the same for every set of data, however they are in these examples as I like to keep things simple!

To conclude, there is a popular rhyme that helps to remember the differences in the calculations needed for the mode, median and mean – it goes like this:

Hey diddle diddle, the median’s the middle

You add then divide for the mean

The mode is the one that you see the most

And the range is the difference between.

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