Finding an average is incredibly useful when you want to find one piece of data that describe a whole set of data.
You will already know that if I wanted to find the average from a list of numbers, we would just add the numbers and then divide by how many there were.
What if the list is really, really long?
Imagine that you collected the shoe size of 500 people. Would you want to add them all together?
There has to be a quicker way!
Using a grouped frequency table.
If we had lots of data and it was all discrete, we would have to use a grouped frequency table rather than a frequency table. Let's look at an example.
Example 1: I collect the amount of pocket money from some children and put it in a grouped frequency table. Estimate the mean of the pocket money.
Pocket Money (p)  0

10

20

30

40

Frequency  4  6  11  17  9 
When we found the mean from a frequency table, we just multiplied the numbers together. We can't do that here because we don't know exactly what the pupils each got. We have to make an assumption that all the students got the amount at the midpoint of the group.
Pocket Money (p)  0≤p≤10  10

20

30

40

Frequency  4  6  11  17  9 
Midpoint  5  15  25  35  45 
Now we can multiply together the midpoints and the frequency
Pocket Money (p)  0≤p≤10  10

20

30

40

Frequency  4  6  11  17  9 
Midpoint  5  15  25  35  45 
mp x f  20  90  275  595  405 
Once we have this, we can work out an estimate of the total pocket money would be added together would be by adding the numbers in red. (1385 )
We now have one of the bits of information we need to find the mean (what they all add up to)
All we need now is to find out how many people were asked. We can get this by adding up all the numbers in the frequency row (47)
To find the mean, all we now have to do is to divide one of these numbers by the other (1385 ÷ 47 ) to get an average of £29.47
A key point.
You will probably see the word estimate in the question. We are not finding the mean, we are estimating it.
When you have grouped data, we made the assumption earlier that everyone got the value in the middle, this means we cannot for certain what the mean is, we can only estimate it.
Can you summarise?
Step 1: Find the midpoints
Step 2: Multiply the midpoints and the frequencies together
Step 3: Add all these new numbers up
Step 4: Add up all the frequencies
Step 5: Divide to find the mean.