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 10.
Its f is 4: 4 numbers are from 0 to 10.
Next: 10 to 20, with 9 numbers.
Next: 20 to 30, with 5 numbers.
Last: 30 to 40, 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 10, we add them and divide by 2.
That is 5.
Next: the middle of 10 and 20 is 15.
Each class starts 10 later, so each midpoint is 10 more.
Then 25.
Then 35.
2 Multiply m × f
Each number in a class counts as its midpoint.
So a class adds up to m times f.
First row: 5 times 4 is 20.
15 times 9 is 135.
25 times 5 is 125.
35 times 2 is 70.
3 Add the columns
We add the f column: 4 + 9 + 5 + 2 is 20.
So there are 20 numbers in all.
Then the m × f column: 20 + 135 + 125 + 70 is 350.
350 is about the total of all the numbers.
4 Share equally
The mean shares the total equally: 350 divided by 20.
It is only an estimate: the midpoints stand in for the real numbers.
Careful!
Not divided by the 4 classes.
There are 20 numbers.
350 divided by 20 is 17.5.
5 Check
Let's check: 17.5 times 20 is 350, 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.