Masses, m kg, of 30 parcels: 0 < m ≤ 4: 5, 4 < m ≤ 8: 12, 8

Step by step

The question

Question: find the mean of this grouped data.

Mean of the grouped data

Each class is a range of numbers.

f is how many numbers are in it.

The first class is 0 to 4.

Its f is 5: 5 numbers are from 0 to 4.

Next: 4 to 8, with 12 numbers.

Next: 8 to 12, with 9 numbers.

Last: 12 to 16, with 4 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 4, we add them and divide by 2.

m = the middle of the class

m = (0 + 4) ÷ 2

That is 2.

Next: the middle of 4 and 8 is 6.

Each class starts 4 later, so each midpoint is 4 more.

Then 10.

Then 14.

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 times 5 is 10.

6 times 12 is 72.

10 times 9 is 90.

14 times 4 is 56.

3 Add the columns

We add the f column: 5 + 12 + 9 + 4 is 30.

5 + 12 + 9 + 4 = 30

So there are 30 numbers in all.

Then the m × f column: 10 + 72 + 90 + 56 is 228.

10 + 72 + 90 + 56 = 228

228 is about the total of all the numbers.

4 Share equally

The mean shares the total equally: 228 divided by 30.

mean ≈ 22830

It is only an estimate: the midpoints stand in for the real numbers.

≈ : an estimate, from the midpoints

Careful!

Not divided by the 4 classes.

There are 30 numbers.

Careful: not 2284 ✗

228 divided by 30 is 7.6.

mean ≈ 7.6

5 Check

Let's check: 7.6 times 30 is 228, the total.

7.6 × 30 = 228

And the mean is between the first and the last midpoints.

So it makes sense!

2 < 7.6 < 14

Let's look back.

First: find the midpoints.

Then: multiply m × f.

Then: add the columns.

Then: share equally.

Last: check.

Answer