Mean of grouped data 1-5: 3, 6-10: 8, 11-15: 6, 16-20: 3

Step by step

The question

Question: find the mean of the grouped data with classes 1 to 5, 6 to 10, 11 to 15 and 16 to 20, and frequencies 3, 8, 6 and 3.

Each class is a range of numbers.

f is how many numbers are in it.

So 3 numbers are from 1 to 5.

1 Find the midpoints

We don't know the numbers themselves.

So each class uses its middle, m: the midpoint.

The middle of 1 and 5: add them, divide by 2.

3.

Next: 8.

Each midpoint is 5 more.

And so on.

2 Multiply m × f

Each number in a class counts as its midpoint.

So a class adds up to m times f.

3 times 3 is 9.

8 times 8 is 64.

The same for every row.

3 Add the columns

The f column: 20 numbers in all.

The m × f column: 205.

That is about the total of all the numbers.

4 Share equally

The mean shares the total equally: 205 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.

205 divided by 20 is 10.25.

The books round to one more decimal place than the data.

So the mean is about 10.3.

5 Check

Let's check: 10.3 times 20 is about 205, 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.