Calculation of range and median along with Box-and-whisker plots and Cumulative frequency tables are effective ways to compare distributions and to summarise their characteristics.

It’s very easy to calculate the **range** of a set of data by subtracting the smallest value from the largest.

For example, with the following data:

3, 5, 7, 8, 9, 13, 15

The range is 15 - 3 = 12.

A range is important to us because it tells us something about the **spread** of the data and when we have large group of numbers to handle this can be a quick way of saying something about the data set as a whole.

One limitation of the range is that it is affected by outliers. Consider the data below:

1, 1, 3, 12, 2, 4, 5, 2, 1, 1, 6, 3, 4

Currently the range is 11 (12-1). However, this does not tell us anything useful about the data. The reason for this is that the one large value (12) is distorting the range. Without this one value the range would be 5 (6-1). The range has been more than halved. There are other measures of spread that are not so affected by outliers.

The median of a set of data is the **middle value** when the data is arranged in size order. When two middle values are present the median is the mean of the two.

For example, from the above data:

3, 5, 7, 8, 9, 13, 15

There are seven values, so the middle value will be the fourth. The median of the above data is therefore eight.

Sometimes it is said that if we have \({n}\) pieces of data, the median is the \(\frac{(n+1)}{2}\) value. Using the above example again we see that we have seven values (n=7) so the \(\frac{(n+1)}{2}\) value would be \(\frac{(7+1)}{2}\) = the 4th value, which as mentioned previously is eight.

Let’s look at another example, what is the median of the following data?

12, 14, 13, 20, 17, 15

First we must arrange this into size order:

12, 13, 14, 15, 17, 20

Now we notice that there are two **middle numbers** (14 and 15) so we must find the mean of these. The mean of two numbers is simply the number halfway between them \([\frac{(14+15)}{2}]\).

So the median is 14.5.

The median is a measure of **central tendency**. It tells us something valuable about the data – roughly what values we can expect in the middle. The mean is another measure of central tendency, but like the range it can be affected by outliers or extreme values.

- Question
What is the range of the following numbers?

13, 3, 8, 16, 12

The largest number is 16 and the smallest is 3.

So the range is 16 - 3 = 13.

- Question
Jim records in his phone how long, in minutes, he exercises per day.

What is the median of the following numbers?

19, 42, 45, 36, 23, 27

Writing the numbers in order gives:

19, 23, 27, 36, 42, 45

So the median will be the mean of the middle two numbers.

\(\frac{{27+36}}{2}={31.5}\) minutes

Be careful when inputting this into a calculator, if you type in 27 + 36 ÷ 2 your answer would be 45 which is the wrong answer. Your calculator works things out in order (using BIDMAS) of division then the addition. Put brackets around the addition (27 + 36) ÷ 2