6.5 Calculating the median
The next type of average you look at briefly is called the median. Put very simply, the median is the middle number in a set of data and as it is in the middle, it is not affected by abnormally high or low data values. The important thing you need to remember is to put the numbers in size order, smallest to largest, before you begin. Firstly let’s look at two simple examples.
Example: Finding the median 1
Find the median of this data set:
Method
5, 10, 8, 12, 4, 7, 10
Firstly, order the numbers from smallest to largest:
4, 5, 7, 8, 10, 10, 12
Now, find the number that is in the middle:
4, 5, 7, 8, 10, 10, 12
8 is the number in the middle, so the median is 8.
Example: Finding the median 2
Find the median of this data set:
Method
24, 30, 28, 40, 35, 20, 49, 38
Again, you firstly need to order the numbers:
20, 24, 28, 30, 35, 38, 40, 49
And then find the one in the middle:
20, 24, 28, 30, 35, 38, 40, 49
In this example there are actually two numbers that are in the middle. You therefore find the middle of these two numbers by adding them together and then halving the answer:
(30 + 35) ÷ 2
75 ÷ 2 = 37.5
The median for this set of data is 37.5. As there are two numbers that are in the middle, your answer does not appear in the original set of data.
The example below is a little more complicated.
Example: Finding the median 3
Tracy did a survey of the number of cups of coffee her colleagues had drunk during one day. The frequency table shows her results.
| Number of cups of coffee | Frequency |
|---|---|
| 2 | 1 |
| 3 | 5 |
| 4 | 3 |
| 5 | 4 |
| 6 | 6 |
First you need to calculate the number of colleagues by adding together the numbers in the frequency column:
1 + 5 + 3 + 4 + 6 = 19
You then need to work out the median or middle value in the number of cups of coffee. To do this you can list the number of cups of coffee in a line:
2 3 3 3 3 3 4 4 4 5 5 5 5 6 6 6 6 6 6
You know there are 19 colleagues, which is an odd number, so you will be able to find the exact midpoint:
9 + 9 = 18
so count in from either side until you reach the 10th number which can be seen as 5 in the list above.
Another way to do this is to calculate the number of colleagues, which you know is 19. You then find the midpoint by adding the numbers in the frequency table:
1 + 5 + 3 = 9
and the exact midpoint is the 10th colleague which the table shows as 4 in the frequency column.
If you look under the Number of cups of coffee column, you can see that the answer is 5 cups of coffee, so the median is 5.
Activity 14: Calculating the median
Now calculate the median of the following:
The ages of a group of students on a course are:
16, 44, 32, 67, 25, 18, 22
The heights of a group of children in a gymnastics class are:
1.24 m, 1.27 m, 1.20 m, 1.15 m, 1.26 m, 1.17 m
The frequency table below shows the number of televisions that a group of students on a media course have at home.
| Number of televisions | Number of students |
|---|---|
| 0 | 1 |
| 1 | 4 |
| 2 | 8 |
| 3 | 9 |
| 4 | 3 |
Calculate the median number of televisions.
Answer
First you need to list the data in order of size so:
16, 18, 22, 25, 32, 44, 67
Now find the middle value. In this case there are 7 data values so you will be able to find the exact middle.
16, 18, 22, 25, 32, 44, 67
The middle value is 25 so this is the median age of the group of students.
First you need to list the heights in order of size so:
1.15 m, 1.17 m, 1.20 m, 1.24 m, 1.26 m, 1.27 m
Now find the middle value. In this case there are 6 data values so first find the middle two values.
1.15 m, 1.17 m, 1.20 m, 1.24 m, 1.26 m, 1.27 m
Now add together the two middle values:
1.20 m + 1.24 m = 2.44 m
Then halve the answer:
2.44 m ÷ 2 = 1.22 m
So the median height of the students is 1.22 m.
First you need to calculate the number of students.
To do this you add up each number in the frequency table:
1 + 4 + 8 + 9 + 3 = 25 students
You then need to work out the median or middle value in the number of televisions. To do this you can list the number of televisions in a line:
0 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 4 4 4
As you know there are 25 students, you need to find the midpoint which will be 13. Count in until you find the 13th number which is 2 in your list of televisions, so the median is 2.
Another way to do this is to calculate the number of students, which you know is 25. You then find the midpoint as the 13th student using the frequency table. If you count up the ‘Number of students’ column in the frequency table, the 13th value is 2 televisions, so the median is 2.
If you want to see some more examples, or try some for yourself, use the link below:
https://www.mathsisfun.com/ median.html [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)]