Mean
The mean is the most commonly used measure of .
To calculate the mean, add the values together and divide the total by the number of values.
Question
Five friends compare their marks in a French test. See the table below:
| Name | Mark |
|---|---|
| Shabana | 41 |
| Bea | 54 |
| Liam | 79 |
| Dave | 26 |
| Ed | 65 |
What is the mean mark?
Answer
Mean \(=(41 + 54 + 79 + 26 + 65) \div 5\)
\(= 265 \div 5\)
\(= 53\)
Notice that the mean does not have to be any of the given data values.
Key point
Check your answer by asking yourself if it looks correct.
If you had forgotten to divide by 5, then the answer would be 265, which is incorrect.
Question
A die is thrown \(10\) times. These are the results:
\(3,~5,~1,~2,~6,~4,~2,~5,~6,~1\)
What is the mean score?
Answer
To find the answer, add the values together and divide the total by the number of values:
Mean \(= (3 + 5 + 1 + 2 + 6 + 4 + 2 + 5 + 6 + 1) \div 10\)
\(35 \div 10 = 3.5\)
Median
If you place a set of numbers in order, the median number is the middle one.
If there are two middle numbers, the median is the mean of those two numbers.
Question
Find the median of each of the following sets of numbers:
a) \(2,~4,~7,~1,~9,~3,~11\)
b) \(4,~1,~3,~10,~6,~9\)
Answer
a) Place these numbers in order:
\(1,~2,~3,~4,~7,~9,~11\)
The middle number is \(4\).
Therefore the median is \(4\).
b) Place these numbers in order:
\(1,~3,~4,~6,~9,~10\)
There are two middle numbers (\(4\) and \(6\)), so we find the mean of these two numbers.
The median is therefore:
\((4 + 6) \div 2 = 5\)
Generally, when there are \(n\) numbers, the median will be the \(\frac{(n + 1)} {2} th\) number.
For example, if there are \(3\) numbers, the median will be the \((3+1)\div{2}={2}^{nd}\) number.
If there are \(4\) numbers, \((4 + 1) \div 2 = 2 {\frac{1}{2}}\).
This refers to the value halfway between the \({2}^{nd}\) and \({3}^{rd}\) numbers.
Find the mean of the \({2}^{nd}\) and \({3}^{rd}\) values.
Question
Rachel records the number of goals scored by her five-a-side team in their first \(20\) matches
The results are shown in the frequency table below:
| Number of goals | Frequency |
|---|---|
| 0 | 7 |
| 1 | 5 |
| 2 | 6 |
| 3 | 0 |
| 4 | 2 |
| 5 | 0 |
What is the median number of goals scored?
What is the median number of goals scored?
Answer
\(20\) matches were played, \({(20 + 1)} \div {2} = {10}\frac{1}{2}\).
The median will be the mean of the \({10}^{th}\) and the \({11}^{th}\) values.
\(0\) goals were scored in \(7\) of the matches, and \(1\) goal was scored in \(5\) of the matches.
The \({10}^{th}\) and \({11}^{th}\) value lies in the '\({1}\) goal' category.
Mode
The mode is the value that occurs most often.
The mode is the only average that can have no value, one value or more than one value.
When finding the mode, it helps to order the numbers first.
Question
Find the mode of each of the following sets of numbers:
a) \(3,~7,~1,~3,~4,~8,~3\)
b) \(2,~7,~2,~1,~4,~7,~3\)
c) \(1,~4,~2,~5,~3,~6\)
Answer
a) Start by placing the numbers in order:
\(1,~3,~3,~3,~4,~7,~8\)
The number \(3\) occurs most often so the mode is \(3\).
b) Start by placing the numbers in order:
\(1,~2,~2,~3,~4,~7,~7\)
The numbers \(2\) and \(7\) occur more often so the modes are \(2\) and \(7\).
c) Start by placing the numbers in order:
\(1,~2,~3,~4,~5,~6\)
Since each value occurs only once in the data set, there is no mode for this set of data.
Example
In this frequency table, the mode is the value with the highest frequency:
| Shoe size | 5 | 6 | 7 | 8 | 9 |
| Frequency | 2 | 5 | 11 | 4 | 1 |
The modal shoe size is \({7}\) because more people wear size \({7}\) than any other size.
Range
The range is the difference between the highest and lowest values in a set of numbers.
To find it, subtract the lowest number in the distribution from the highest.
Question
Find the range of the following set of numbers:
a) \(23,~27,~40,~18,~25\)
b) \(25,~26,~57,~15,~47\)
Answer
a) The largest value is \(40\) and the smallest value is \(18\).
Therefore, the range is \(40 - 18 = 22\).
b) The largest value is \(57\) and the smallest value is \(15\).
Therefore, the range is \(57 - 15 = 42\).
Comparing two sets of data
Averages can be used to compare two sets of data and draw conclusions about the information.
It is important to choose the average which best compares the data.
Question
The following data shows percentage scores in English and Maths tests for a set of ten students.
| English | 93 | 61 | 71 | 69 | 96 | 85 | 66 | 75 | 96 | 68 |
| Maths | 81 | 78 | 76 | 83 | 74 | 73 | 78 | 78 | 82 | 77 |
Complete the following table:
| Mean | Median | Mode | |
|---|---|---|---|
| English | 78 | ||
| Maths | 78 |
Answer
| Mean | Median | Mode | |
|---|---|---|---|
| English | 78 | 73 | 96 |
| Maths | 78 | 78 | 78 |
If you were to compare the scores in the two subjects, which measure of average would you use and why?
Mean
The mean score in each subject is \(78\).
This suggests that the scores of the students are similar in English and Maths.
However, looking at the actual scores, you can see that this is not the case.
Median
The medians, \(73\) and \(78\) suggest that the students generally scored less well in English.
This is partly true, but there are also some much higher scores.
The median is only a measure of the middle value, as there will be the same number of values above and below this middle value.
Mode
The modal score for each subject \(96\) and \(78\) suggests that the students did better in English however this is only considering the two top marks in English and you have no information about the scores of the other students.
So, which is best?
It depends on the context in which the result is to be used.
The mean is usually the best measure of the average, as it takes into account all of the data values.
However, in order to highlight the differences in the marks scored and to give maximum information, a combination of the median and the range would be best.
Range
The range is not an average, but a measure of the spread of the values (or marks in this case).
The range of scores in English is \(35\).
This is far greater than the range of scores in the Maths which is \(10\).
In summary, both English and Maths have a mean score of \(78\) however English has a median score of \(71\) and a range of \(35\) as some students scored much higher than others.
Maths has a median score of \(78\) and a range of \(10\) so all the results were close to the mean and the median.
Calculating the mean from a frequency table
Sara wanted to know the ages of the children on the school bus.
She conducted a survey and her results are shown below:
| 13 | 14 | 11 | 12 | 12 | 15 |
| 13 | 14 | 12 | 16 | 15 | 11 |
| 11 | 12 | 11 | 12 | 14 | 16 |
| 14 | 15 | 14 | 14 | 13 | 12 |
| 13 | 11 | 11 | 14 | 12 | 13 |
To find the mean add all the ages together and divide by the total number of children.
If you type all those ages into a calculator it is easy to make an error.
It can be helpful to see these results displayed in a frequency table:
| Age | Frequency |
|---|---|
| 11 | 6 |
| 12 | 7 |
| 13 | 5 |
| 14 | 7 |
| 15 | 3 |
| 16 | 2 |
The frequency table shows us that there are six children aged \({11}\), seven children aged \({12}\), five children aged \({13}\)… etc.
To find the sum of their ages, calculate:
\((6 \times 11) + (7 \times 12) + (5 \times 13)\)
\(+ (7 \times 14) + (3 \times 15) + (2 \times 16) = 390\)
The total number of children is \(6 + 7 + 5 + 7 + 3 + 2 = 30\)
So, the mean age is \(390 \div 30 = 13\)
You could also put this information into a table like the one below:
| Age | Frequency | Age x Frequency |
|---|---|---|
| 11 | 6 | 66 |
| 12 | 7 | 84 |
| 13 | 5 | 65 |
| 14 | 7 | 98 |
| 15 | 3 | 45 |
| 16 | 2 | 32 |
Mean = \(390 \div 30 = 13\)
Question
Find the mean of the shoe sizes of pupils in Mrs Harris’ class:
| Shoe size | Frequency |
|---|---|
| 3 | 2 |
| 4 | 4 |
| 5 | 3 |
| 6 | 7 |
| 7 | 3 |
| 8 | 1 |
Answer
Mean \(= 108 \div 20 = 5.4\)
Notice that the mean value does not have to be a whole number or an actual shoe size.
Calculating the mean and modal class from grouped data
Calculating the mean and modal class for grouped data is very similar to finding the mean from an ungrouped frequency table, although you do not have all the information about the data within the groups so you can only estimate the mean.
Question
The following table shows the weights of children in a class.Using this information:
a) Estimate the mean weight
b) Find the modal class
Answer
To estimate the mean weight, you know that \({7}\) children are between \({30~kg}\) and \({40~kg}\).
As you do not know exactly how much they weigh, assume that they all weigh \({35~kg}\) (the midpoint of the group).
Find the mean for all the other groups:
a) Estimation of mean = \({1,215}\div{25} = {48.6}~{kg}\)
b) The modal class is the class that has the highest frequency. In this case the modal class is:
\(50 \leq m \textless 60\)
Test section
Question 1
What is the mean of these numbers: \({3}\), \({5}\), \({7}\), \({9}\)?
Answer
Add the numbers and divide by \({4}\) to get the correct answer, \({6}\)
Question 2
Here's a list of pocket money that a class of children get.
What's the mean?
\(\pounds{3.50}\), \(\pounds{2.40}\), \(\pounds{5}\), \(\pounds{1.50}\), \(\pounds{2}\), \(\pounds{3.30}\), \(\pounds{2.80}\), \(\pounds{5}\), \(\pounds{6}\), \(\pounds{2.50}\)
Answer
You need to add all of the money together and then divide the total by \({10}\) to get the mean.
So the correct answer is \(\pounds{3.40}\)
Question 3
The mean height of \({5}\) children is \({1.63}~{m}\).
What's the mean if they're joined by another child who's \({1.75}~{m}\) tall?
Answer
You need to multiply \({1.63}\) by \({5}\) and then add \({1.75}\) before dividing the total by \({6}\) to get the new mean - which is \({1.65}~{m}\).
Question 4
Here's the number of goals scored in Premier League fixtures one Saturday: \({1}\), \({5}\), \({3}\), \({5}\), \({1}\), \({3}\), \({4}\), \({1}\), \({2}\).
What's the median?
Answer
You need to place the \({9}\) numbers in order: \({1}\), \({1}\), \({1}\), \({2}\), \(\textbf{3}\), \({3}\), \({4}\), \({5}\), \({5}\).
Then it's easy to see that \({3}\) is the middle number.
Question 5
Here's the number of goals scored in Premier League fixtures one Saturday: \({1}\), \({5}\), \({3}\), \({5}\), \({1}\), \({3}\), \({4}\), \({1}\), \({2}\).
What's the mode?
Answer
It's useful to put the numbers in order: \({1}\), \({1}\), \({1}\), \({2}\), \({3}\), \({3}\), \({4}\), \({5}\), \({5}\) or construct a frequency table to see which number occurs most often.
In this instance, it is \({1}\)
Question 6
Here's a set of marks for a maths class: \({12}\), \({45}\), \({78}\), \({66}\), \({39}\), \({98}\), \({25}\), \({48}\), \({66}\), \({41}\).
What's the range?
Answer
The range is the difference between the highest and lowest values.
In this case, \({98}-{12}={86}\).
Question 7
The table shows the ages of children in a class of \({30}\).
What's the median of the ages?
Answer
The \({15}^{th}\) and \({16}^{th}\) children are \({13}\) years old.
So the \({15.5}^{th}\) value which is the median must be \({13}\).
Question 8
The table shows the number of goals scored by Helen's \({5}\)-a-side team:
What is the mean number of goals scored per game?
Answer
You need to multiply the number of goals by the frequency on each row and then add the answers to get the total of \({25}\).
Dividing \({25}\) by \({20}\) (total frequency) we get \({1.25}\).
Question 9
Here are the results Owen got by throwing a dice a number of times: \({1}\), \({5}\), \({3}\), \({4}\), \({2}\), \({5}\), \({5}\), \({3}\), \({6}\), \({2}\), \({1}\), \({3}\), \({2}\), \({5}\), \({4}\), \({3}\), \({3}\), \({5}\).
What's the mode?
Answer
\({3}\) appears more than every other number apart from \({5}\), which appears the same number of times.
So there are two modes, \({3}\) and \({5}\).
Question 10
The table shows the weights of a class of children.
Estimate the mean weight.
Answer
You need to multiply the middle value of each group by the frequency before going on to calculate the mean.
In this case, the mean is \({48.6}\).