MathsDirect

Medians

MathsDirect

The median of a set of data is the value that would fall in the middle, if all the data were written down in order. For example, if we had the following values for a variable x

x = 7,13,4,6,12,10,6

then to determine the median, you would first need to sort the data

x = 4,6,6,7,10,12,13

Now since there are 7 pieces of data, the middle value will be the fourth one. Therefore the median is 7.

Finding the median of grouped data

We saw that a cumulative frequency graph could be used to find the median of a set of grouped data. There are however, obvious limitations to the accuracy of the answer you will obtain from a graph. We will therefore now look at how to calculate the median from grouped data.

x(cm)
f	5	9	10	11	8	7

To find the median, it will still be useful to add another row to our table, for the cumulative frequency.

x(cm)
f	5	9	10	11	8	7
cf	5	14	24	35	43	50

From this we can see that there are 50 results. To find the median add 1 to this number and halve the result.

We are therefore looking for the 25.5th result. This lies in the fourth group, which starts at 24.

We must work out how far though the group we must go. Our result is 1.5 into the group (25.5-24) and the group has a width of 11.

We therefore must go

of the way through the group

The group covers values from 25cm to 30 cm, and therefore has a 5cm range. We need to find a fraction of 5cm

		This is how far through the group the median lies.
Therefore the median is

Although the whole process can appear confusing, each step is straightforward

1 Write out the cumulative frequencies.

2 Find which group the median lies in.

3 Find the location of the median in the group, with the formula:

4 Translate this fraction into a value with the formula:

5 Add this onto the lowest value of the group.

These steps will be illustrated in the next example.

Find the median.

x
f	2	5	9	10	11	6

1 Find the cumulative frequencies

x
f	2	5	9	10	11	6
cf	2	7	16	26	37	43

2 Find which group the median lies in

The position of the median will be

which is in the 4^th group.

3 Find where in the group the median lies

the previous group had a cumulative frequency of 16

and the frequency of our group is 10

4 Translate this fraction to a value

We first need to be clear about the width the group. Because of the way the groups have been given, the lowest value is 11.5 and the greatest value is 15.5. The group therefore has a width of 4.

So we need

5 Add this to the groups lowest value, to find the median

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