Unit 5 Section 2 : Mean, Median, Mode and Range

The mean, median and mode are types of average.
The range gives a measure of the spread of a set of data.

This section revises how to calculate these measures for a simple set of data.
It then goes on to look at how the measures can be calculated for a table of data.

Calculating the Mean, Median, Mode and Range for simple data

The table below shows how to calculate the mean, median, mode and range for two sets of data.
Set A contains the numbers 2, 2, 3, 5, 5, 7, 8 and Set B contains the numbers 2, 3, 3, 4, 6, 7.

Measure Set A
2, 2, 3, 5, 5, 7, 8 Set B
2, 3, 3, 4, 6, 7

The Mean
To find the mean, you
need to add up all the
data, and then divide
this total by the number
of values in the data.
Adding the numbers up gives:
2 + 2 + 3 + 5 + 5 + 7 + 8 = 32
There are 7 values, so you divide
the total by 7:    32 ÷ 7 = 4.57...
So the mean is 4.57 (2 d.p.)
Adding the numbers up gives:
2 + 3 + 3 + 4 + 6 + 7 = 25
There are 6 values, so you divide
the total by 6:    25 ÷ 6 = 4.166...
So the mean is 4.17 (2 d.p.)

The Median
To find the median, you
need to put the values
in order, then find the
middle value. If there are
two values in the middle
then you find the mean
of these two values.
The numbers in order:
2 , 2 , 3 , (5) , 5 , 7 , 8
The middle value is marked in
brackets, and it is 5.

So the median is 5
The numbers in order:
2 , 3 , (3 , 4) , 6 , 7
This time there are two values in
the middle. They have been put
in brackets. The median is found
by calculating the mean of these
two values:    (3 + 4) ÷ 2 = 3.5
So the median is 3.5

The Mode
The mode is the value
which appears the most
often in the data. It is
possible to have more
than one mode if there
is more than one value
which appears the most. The data values:
2 , 2 , 3 , 5 , 5 , 7 , 8
The values which appear most
often are 2 and 5. They both
appear more time than any
of the other data values.

So the modes are 2 and 5
The data values:
2 , 3 , 3 , 4 , 6 , 7
This time there is only one value
which appears most often - the
number 3. It appears more times
than any of the other data values.

So the mode is 3

The Range
To find the range, you
first need to find the
lowest and highest values
in the data. The range is
found by subtracting the
lowest value from the
highest value. The data values:
2 , 2 , 3 , 5 , 5 , 7 , 8
The lowest value is 2 and the
highest value is 8. Subtracting
the lowest from the highest
gives:    8 - 2 = 6
So the range is 6
The data values:
2 , 3 , 3 , 4 , 6 , 7
The lowest value is 2 and the
highest value is 7. Subtracting
the lowest from the highest
gives:    7 - 2 = 5
So the range is 5

Practice Question (for simple data)
Work out the mean, median, mode and range for the simple data set below,
then click on the button marked to see whether you are correct.

A data set contains these 12 values: 3, 5, 9, 4, 5, 11, 10, 5, 7, 7, 8, 10

(a) What is the mean?

(b) What is the median?

(c) What is the mode?

(d) What is the range?

Calculating the Mean, Median, Mode and Range for a table of data

Sometimes we are given the data in a table. The methods for calculating mean, median, mode
and range are exactly the same, but we need to think carefully about how we carry them out.
In this section we will use one set of data in a table and calculate each measure in turn.

Example
A dice was rolled 20 times. On each roll the dice shows a value from 1 to 6.
The results have been recorded in the table below:

Value Frequency

1 3

2 5

3 2

4 4

5 3

6 3

The frequency is the number of times each value occured.
For example, the value 1 was rolled 3 times, the value 2 was rolled 5 times and so on...
When we want to think about calculating the measures for this data set, it can be helpful
to think about what the numbers would look like if we wrote them out in a list:
1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6
We could just calculate the mean, median, mode and range from this list of data, using
the methods described in the first part of this section. The problem is that if there were
hundreds of values in the table then it would take a long time to write out the list of data
and even longer to do the calculations. It would be better if we could work directly from
the table to calculate the measures. The method for doing this is shown below.

Finding the mean from a table of data

Value Frequency

1 3

2 5

3 2

4 4

5 3

6 3

We know that if we write the example data in a list it looks like this:
1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6
Normally we would add up the data and divide the total by the number of values:
The total is 1+1+1 + 2+2+2+2+2 + 3+3 + 4+4+4+4 + 5+5+5 + 6+6+6 = 68
The number of values is 20, so the mean is 68 ÷ 20 = 3.4
We could have found these figures more easily! To get the total, we have added
up 3 lots of "1", 5 lots of "2", 2 lots of "3", 4 lots of "4", 3 lot of "5" and 3 lots of "6".
This is the same calculation as 3×1 + 5×2 + 2×3 + 4×4 + 3×5 + 3×6 = 68.
We have multiplied each value by its frequency and added up the results to get the total
of all the values. We can also get the "number of values" more easily by simply adding
up all the frequencies: 3 + 5 + 2 + 4 + 3 + 3 = 20

So how do we do this in a table?
Firstly, you need to add an extra column in the table:
This is where you multiply each value by its frequency. For example,
the value 5 has a frequency of 3, so we multiply 5 by 3 to get 15.
Secondly, you need to calculate two important totals:
(1) add up the values in the frequency column to find out the
number of data values. In this case there are 20 values.
(2) add up the values in the value × frequency column to find
out the total of all the data values. In this case the total is 68.
Finally, you need to calculate the mean:
To do this, divide the total of all the data values by the number of
data values. In this case you need to divide 68 by 20, giving 3.4.

Value Frequency Value × Frequency

1 3 1 × 3 = 3

2 5 2 × 5 = 10

3 2 3 × 2 = 6

4 4 4 × 4 = 16

5 3 5 × 3 = 15

6 3 6 × 3 = 18

Totals 20 68

This method of calculating the mean for a table of data is exactly the same as the one used with a list of data.
We have still added up all the values and divided by the number of values, but this way is a bit more efficient!

Finding the median from a table of data

Value Frequency

1 3

2 5

3 2

4 4

5 3

6 3

We know that there are 20 data values in our table. If you imagine the 20 values
written out, there would be two values in the middle. These would be the 10th and
11th values, and the median would be the mean of these two "middle values".
From the list below we can see that the "middle values" are 3 and 4:
1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6
The median would therefore be (3+4)÷2 = 3.5
So how do we do this from a table?
Because there are 20 values, we know that we need to find the mean of the 10th and
11th values. To find these values we need to count through the table until we get to them.
Look at the table. The value "1" has a frequency of 3, so the first three values in the table are "1"s.
The value "2" has a frequency of 5, so the next 5 values are all "2"s. This takes us up to the 8th value.
The next 2 values are "3"s, which takes us up to the 10th position in the data, so the 10th value must be a "3".
The next 4 values are "4"s, so the 11th value must be a "4".

We can now see that the 10th and 11th values are a "3" and a "4", so the median is 3.5.

Finding the mode and range from a table of data

Value Frequency

1 3

2 5

3 2

4 4

5 3

6 3

Finding the mode is much easier from a table, because the frequency column
tells us how many times each value occured. We can find the value which
occured the most often by looking for the value with the highest frequency.
In this case we can see that the value with the highest frequency is "2".
The mode of this set of data is therefore 2
Finding the range is also easy from a table. To find the highest and lowest data
values, you simply look for the highest and lowest values in the values column.
In this case the lowest value is "1" and the highest value is "6", and 6 - 1 = 5.
The range of this set of data is therefore 5

Practice Question (for data in a table)
Work out the each of the measures, then click on the button marked

to see whether you are correct.

Value	Frequency	Value × Frequency
0	3
1	7
2	10
3	8
4	1
5	1
Totals

You can fill in the boxes to help you with your working but they will not be marked.

30 couples were asked how many children they have.
The results are shown in the table on the left.

(a) What is the mean number of children?

(b) What is the median number of children?

(c) What is the mode?

(d) What is the range?

Exercises

Work out the answers to the questions below and fill in the boxes. Click on the

button to find out whether you have answered correctly. If you are right then

will appear and you should move on to the next question. If

appears then your answer is wrong. Click on

to clear your original answer and have another go. If you can't work out the right answer then click on

to see the answer.

You may find it helpful to have pencil and paper to do workings for these questions.

You have now completed Unit 5 Section 2

Your overall score for this section is

Correct Answers
You answered questions correctly out of the questions in this section.

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There were questions with wrong answers.
There were questions you didn't attempt.

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		Median =
		Mode =
		Range =
(b)	11, 10, 12, 12, 9, 10, 14, 12, 9	Mean =
		Median =
		Mode =
		Range =
(c)	2, 9, 7, 3, 5, 5, 6, 5, 4, 9	Mean =
		Median =
		Mode =
		Range =