Step by step
The question
Question: find the mean of this grouped data.
Each class is a range of numbers.
f is how many numbers are in it.
The first class is 0 to 5.
Its f is 6: 6 numbers are from 0 to 5.
Next: 5 to 10, with 10 numbers.
Next: 10 to 15, with 7 numbers.
Last: 15 to 20, with 2 numbers.
1 Find the midpoints
We don't know the numbers themselves.
So each class uses its middle: m.
m is the midpoint: the middle of the class.
To find the middle of 0 and 5, we add them and divide by 2.
That is 2.5.
Next: the middle of 5 and 10 is 7.5.
Each class starts 5 later, so each midpoint is 5 more.
Then 12.5.
Then 17.5.
2 Multiply m × f
Each number in a class counts as its midpoint.
So a class adds up to m times f.
First row: 2.5 times 6 is 15.
7.5 times 10 is 75.
12.5 times 7 is 87.5.
17.5 times 2 is 35.
3 Add the columns
We add the f column: 6 + 10 + 7 + 2 is 25.
So there are 25 numbers in all.
Then the m × f column: 15 + 75 + 87.5 + 35 is 212.5.
212.5 is about the total of all the numbers.
4 Share equally
The mean shares the total equally: 212.5 divided by 25.
It is only an estimate: the midpoints stand in for the real numbers.
Careful!
Not divided by the 4 classes.
There are 25 numbers.
212.5 divided by 25 is 8.5.
5 Check
Let's check: 8.5 times 25 is 212.5, the total.
And the mean is between the first and the last midpoints.
So it makes sense!
Let's look back.
First: find the midpoints.
Then: multiply m × f.
Then: add the columns.
Then: share equally.
Last: check.