# 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 A2, 2, 3, 5, 5, 7, 8 Set B2, 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.

Question 1
Calculate the mean, median, mode and range for each set of data below:
 (a) 3, 6, 3, 7, 4, 3, 9 Mean =  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 =  Question 2
Several seaside hotels were rated between "no stars" and "five stars" by the tourist board.
The table below shows how many hotels got each number of stars.
Work out the mean number of stars by filling in the gaps in the table below.
You can check your answers to each part as you go along.
Question 3
The price of a litre of petrol was recorded at several garages, and the results are displayed in the table below.
 Price Frequency 74p 1 75p 2 76p 8 77p 10 78p 2 79p 1 80p 1
Calculate:
(a) the mean
pence  Question 4
Professor Baker keeps a record of his golf scores, as shown in the table below:
 Score Frequency 70 3 71 4 72 4 73 4 74 3 75 2
Calculate his mean score:  Question 5
A class collected data on their shoe sizes and presented it in the table below:
 Shoe Size Frequency 3 2 4 7 5 6 6 5 7 3 8 2
Calculate:
(a) the mean  (d) the range  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. Incorrect Answers There were questions where you used the Tell Me button. There were questions with wrong answers. There were questions you didn't attempt.
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