Averages
Averages can be very useful to us if they are used in appropriate situations.
They can also be very misleading if they are used in the wrong situations. We
will spend most of the time looking at useful averages, but at the end of the
section we will look at how averages can be unhelpful.
The idea of an average is to ‘level out’ a series of numbers. We will start with
small groups of numbers in order to make the idea of an average make sense.
Later, we will look at much larger groups of numbers.
,The Average of 2 Numbers
Q: Jill and Bill each have some money. Jill has £80 and Bill has £50.
What is the average of the two amounts of money?
Basically, what the question is asking is ‘if they both had the same amount of
money, how much would they each have?’
Between them they have £130 (80 + 50)
If they shared that £130 equally, then 130 ÷ 2 = 65
They would each have £65 if they shared the money equally.
So the average of £80 and £50 is £65.
And that’s what an average is. It levels out the differences between all the
amounts and makes it so that every amount is the same.
The total amount of money is the same (£130) but the average tells us what
each person would have if their shares were levelled out.
,Let’s look at another example.
Q: Sue and Lou have measured their heights.
Sue is 145cm tall and Lou is 153cm tall.
What is their average height?
The two people are different sizes. If we add up their total height (145
+ 153) and then share it between both of them, that will tell us the
average.
145 + 153 = 298cm
If they stood on top of each other, they would reach a height of 298cm!
If we now split the 198cm between the two people (298 ÷ 2 = 149) their
average height is 149cm.
149cm is a bit taller than Sue and a bit shorter than Lou – somewhere
in-between.
The average will always be somewhere in between. It can never be smaller
than the smallest number in the list, and it can never be bigger than the
biggest number in the list. But it isn’t always exactly in the middle in big lists
of numbers, as we will see later.
So let’s recap. When we find the average of 2 numbers, we add them up and
divide the answer by 2 to share the total equally.
, What happens if we need to find the average of 3 numbers?
Q: Wayne, Shane and Jane have some money.
Wayne has £20, Shane has £80 and Jane has £110
What is the average amount of money?
As we did before, we start by adding up all the money to see how much
there is altogether.
20 + 80 + 110 = £210
They have a total of £210 between them. The average is the amount each
person would have if the £210 was shared equally.
This time we have to share it by 3, because there are 3 people.
210 ÷ 3 = £70
If the money was shared equally between the three people, they would
each have £70 (and it would still add up to £210)
So the average of £20, £80 and £110 is £70.
Have you noticed that £70 is not midway between the smallest and the
biggest number? 70 is more than half way between 20 and 110 but it is the
average of the three numbers.
Why is that?
In the list of numbers, 80 and 110 are both at the high end of the scale, and
20 is on its own as a much lower number. So the average is balanced towards
where most of the numbers are – and that is at the higher end.
Averages can be very useful to us if they are used in appropriate situations.
They can also be very misleading if they are used in the wrong situations. We
will spend most of the time looking at useful averages, but at the end of the
section we will look at how averages can be unhelpful.
The idea of an average is to ‘level out’ a series of numbers. We will start with
small groups of numbers in order to make the idea of an average make sense.
Later, we will look at much larger groups of numbers.
,The Average of 2 Numbers
Q: Jill and Bill each have some money. Jill has £80 and Bill has £50.
What is the average of the two amounts of money?
Basically, what the question is asking is ‘if they both had the same amount of
money, how much would they each have?’
Between them they have £130 (80 + 50)
If they shared that £130 equally, then 130 ÷ 2 = 65
They would each have £65 if they shared the money equally.
So the average of £80 and £50 is £65.
And that’s what an average is. It levels out the differences between all the
amounts and makes it so that every amount is the same.
The total amount of money is the same (£130) but the average tells us what
each person would have if their shares were levelled out.
,Let’s look at another example.
Q: Sue and Lou have measured their heights.
Sue is 145cm tall and Lou is 153cm tall.
What is their average height?
The two people are different sizes. If we add up their total height (145
+ 153) and then share it between both of them, that will tell us the
average.
145 + 153 = 298cm
If they stood on top of each other, they would reach a height of 298cm!
If we now split the 198cm between the two people (298 ÷ 2 = 149) their
average height is 149cm.
149cm is a bit taller than Sue and a bit shorter than Lou – somewhere
in-between.
The average will always be somewhere in between. It can never be smaller
than the smallest number in the list, and it can never be bigger than the
biggest number in the list. But it isn’t always exactly in the middle in big lists
of numbers, as we will see later.
So let’s recap. When we find the average of 2 numbers, we add them up and
divide the answer by 2 to share the total equally.
, What happens if we need to find the average of 3 numbers?
Q: Wayne, Shane and Jane have some money.
Wayne has £20, Shane has £80 and Jane has £110
What is the average amount of money?
As we did before, we start by adding up all the money to see how much
there is altogether.
20 + 80 + 110 = £210
They have a total of £210 between them. The average is the amount each
person would have if the £210 was shared equally.
This time we have to share it by 3, because there are 3 people.
210 ÷ 3 = £70
If the money was shared equally between the three people, they would
each have £70 (and it would still add up to £210)
So the average of £20, £80 and £110 is £70.
Have you noticed that £70 is not midway between the smallest and the
biggest number? 70 is more than half way between 20 and 110 but it is the
average of the three numbers.
Why is that?
In the list of numbers, 80 and 110 are both at the high end of the scale, and
20 is on its own as a much lower number. So the average is balanced towards
where most of the numbers are – and that is at the higher end.
