8 Averages
An average is a middle, or ‘typical’, value. Sometimes it’s easier to present data numerically rather than graphically, and to find one number to represent a collection of data instead of lots of numbers. You can do this by finding the arithmetical average: ‘arithmetical’ means ‘doing sums’, and the ‘average’ is the representative value of all our data. So working out the arithmetical average means working out a representative value for your data with mathematical calculations. The arithmetical average is better known as the mean average. There are other types of average, but we will focus on the mean here.
Note: With data we talk about ‘data sets’, or sets of data. ‘Sets’ is just another word for ‘group’. So if we carried out a survey, we would have a data set.
You’ll be familiar with the word ‘average’. Outside maths, it is used to mean ‘not special’ or ‘just OK’. But in maths, ‘average’ means we can have one typical value that is representative of all our data and that uses all our data.
Where do we find averages in real life?
- If you look at a holiday brochure you will see that it will talk about the ‘average’ hours of sunshine in a day.
- A teacher might work out the average marks for students in a class.
- When you go on a journey you might talk about our average speed.
- The average goals scored per game over a season by your football team.
Example: Mean test scores
The arithmetical, or mean, average is not difficult to work out.
Look at the following example based on the first example in the list above:
| Student | Score |
|---|---|
| Sara | 11 |
| Ceri | 13 |
| Sian | 14 |
| Dylan | 15 |
| Aled | 17 |
| Ewan | 17 |
| Paul | 15 |
| Elisa | 20 |
| Bea | 20 |
| Gwyn | 18 |
In order to calculate the mean average, you need to do the following:
Add up all of your data to a total (let’s call this total ‘A’).
In this example the data is the students’ test scores, so we need to add:
11 + 13 + 14 + 15 + 17 + 17 + 15 + 20 + 20 + 18 = 160
- Add up the number of categories that your data falls into. This would be the number of students (let’s call this amount ‘B’). In this case there are 10 students.
To calculate the mean average you divide the total of your data (A) by the number of bits of data (B). So:
A ÷ B = the average
In the example above the scores added to a total of 160, divided by 10 (the number of students):
160 ÷ 10 = 16
So the mean average score would be 16.
Have a look at the example below, where you will be looking at the mean average hours of sunshine.
Example: Mean average hours of sunshine
The hours of sunshine per day during one week’s holiday to Barmouth in June was recorded as follows:
| Day | Hours of sunshine |
|---|---|
| Sunday | 6 |
| Monday | 1 |
| Tuesday | 7 |
| Wednesday | 8 |
| Thursday | 5 |
| Friday | 2 |
| Saturday | 6 |
You could draw a bar chart or a line graph to present this data. However – as you might expect from the British weather – the amount of sunshine varied a lot from day to day.
It might be more useful to find out the mean average amount of sunshine per day. This would give you one value, which you could use as a guide as to how much sunshine to expect per day.
Method
To work out this mean value you need to:
- add up the amount of sunshine for each day
- divide this by the number of days you have the data for.
With this example we have:
- 6 + 1 + 7 + 8 + 5 + 2 + 6 = 35 hours of sunshine for the week
and seven days of data. So, the mean is:
- 35 ÷ 7 = 5 hours
Note: You must remember what units you are working in and write in these units after your average value – otherwise, it won’t make sense.
So from this data you can see that, on average, there were five hours of sunshine per day in a week in June in Barmouth. You could then use that information to help choose your next holiday: if you wanted more than five hours of sunshine a day for a holiday in June, you would choose somewhere hotter (like Spain, perhaps).
Method summary
- Add up all of your data.
- Find out the number of categories that your data falls into (how many bits of data you have).
- Divide the total of your data by the number of categories of data to give the mean average.
- Don’t forget to put what units you are working in, for example hours, goals, people, etc.
Now try the following activity. Remember to refer to the example if you get stuck and to check your answers once you have completed the questions.
Activity 15: Finding the mean average
Calculate the answers to the following problems without using a calculator. You may double-check your answers with a calculator if you need to. Remember to check your answers once you have completed the questions.
- The ages of four children in a family are 4, 6, 8 and 10 years. What is the mean average age?
- Find the average of the following data sets:
- a.4, 6, 11
- b.3, 7, 8, 4, 8
- c.8, 9, 10, 9, 4, 2
- d.11, 12, 13, 14, 15, 16
- The number of goals scored by a football team in recent matches were as follows:
| 2 | 3 | 0 | 1 | 3 |
| 2 | 3 | 2 | 1 | 3 |
- Work out the mean number of goals per match.
Hint: Notice how it is important to include the zero in the calculations.
Answer
Check your answers with the answers below.
- First, add all of the ages together:
- 4 + 6 + 8 + 10 = 28
Then divide this total by the amount of data given:
- 28 ÷ 4 = 7
The average age is 7.
- You will find the following answers using the same calculation you used for question 1:
- a.Add all the numbers (4 + 6 + 11 = 21) and then divide this answer by the amount of data given (21 ÷ 3 = 7). The answer is 7.
- b.Add all the numbers (3 + 7 + 8 + 4 + 8 = 30) and then divide this answer by the amount of data given (30 ÷ 5 = 6). The answer is 6.
- c.Add all the numbers (8 + 9 + 10 + 9 + 4 + 2 = 42) and then divide this answer by the amount of data given (42 ÷ 6 = 7). The answer is 7.
- d.Add all the numbers (11 + 12 + 13 + 14 + 15 + 16 = 81) and then divide this answer by the amount of data given (81 ÷ 6 = 13.5). The answer is 13.5. Note that the mean average may not be a whole number.
- The average number of goals per match is 2:
2 + 3 + 0 + 1 + 3 + 2 + 3 + 2 + 1 + 3 = 20
20 ÷ 10 = 2
Now have a go at another activity to check your knowledge.
Activity 16: The maths test
As before, calculate the answers to the following problems without using a calculator. You may double-check your answers with a calculator if you need to. Remember to check your answers once you have completed the questions.
- In a maths class the scores for a test (out of 10) were as follows:
| 5 | 6 | 6 | 4 | 4 |
| 7 | 3 | 5 | 6 | 7 |
| 8 | 6 | 2 | 8 | 5 |
| 4 | 5 | 6 | 5 | 6 |
- What is the mean score?
- Some of the students felt that the teacher had been too harsh with their marks. The tests were remarked and the new results were as follows:
| 4 | 6 | 6 | 4 | 4 |
| 6 | 1 | 5 | 6 | 6 |
| 7 | 6 | 1 | 9 | 5 |
| 3 | 5 | 6 | 5 | 5 |
- Work out the mean score for these new results. Which set of results gave the best marks? Was the teacher harsh with the first marking?
Answer
First, add up the total number of marks:
- 5 + 6 + 6 + 4 + 4 + 7 + 3 + 5 + 6 + 7 + 8 + 6 + 2 + 8 + 5 + 4 + 5 + 6 + 5 + 6 = 108
Then divide this by the number of scores (or the number of students), which is 20:
- 108 ÷ 20 = 5.4
So the average score is 5.4 out of 10.
Again, first add up the total number of marks:
- 4 + 6 + 6 + 4 + 4 + 6 + 1 + 5 + 6 + 6 + 7 + 6 + 1 + 9 + 5 + 3 + 5 + 6 + 5 + 5 = 100
Then divide this total by 20:
- 100 ÷ 20 = 5
The best set of results was the first set. The teacher had not been marking it harshly.
What are the advantages and disadvantages of using the mean average?
Ever heard of families with 2.4 children? This is the national average but it means nothing – because you can’t have 0.4 of a child! This highlights one of the problems with averages: the value you get may not be a real value in terms of what you are talking about.
Another problem is that the mean value will be affected by values that are much higher or much lower than the others in the data set. For example, your football team could be having a really bad season, scoring nothing in nine games. The mean number of goals scored per game in these nine games would be zero (total goals = 0 and matches played = 9, so the mean would be 0 ÷ 9 = 0). Then, suddenly, they start to play very well and in the next match score ten goals. This would increase the mean average goals scored to one goal per match (total goals = 10 and matches played = 10, so the mean would be 10 ÷ 10 = 1), which would make it look as though they’d scored a goal in every match when they hadn’t.
The mean is a good way of calculating the average, however, because it isn’t too complicated to work out (compared to some other statistical calculations) and it uses all the available data.
Summary
In this section you have:
- learned that the mean is one sort of average
- learned that the mean is worked out by adding up the items and dividing by the number of items
- understood that the mean can give a ‘distorted average’ if one or two values are much higher or lower than the other values.