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Mean of grouped data

Statistics · 20 problems, easy to harder

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How we do it

  1. Find the midpoints
  2. Multiply m × f
  3. Add the columns
  4. Share equally
  5. Check

The steps for Mean of grouped data 0-10: 3, 10-20: 7, 20-30: 6, 30-40: 4. Every solution names its steps, so you always know what comes next.

Worked examples

Mean of grouped data 0-10: 3, 10-20: 7, 20-30: 6, 30-40: 4 Easy

The question

Question: find the mean of the grouped data with classes 0 to 10, 10 to 20, 20 to 30 and 30 to 40, and frequencies 3, 7, 6 and 4.

Mean of the grouped data

Each class is a range of numbers.

f is how many numbers are in it.

So 3 numbers are from 0 to 10.

f = how many, m = the middle of the class

1 Find the midpoints

We don't know the numbers themselves.

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

m = (0 + 10) ÷ 2 = 5

The middle of 0 and 10: add them, divide by 2.

5.

Next: 15.

Each midpoint is 10 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.

5 times 3 is 15.

15 times 7 is 105.

The same for every row.

3 Add the columns

The f column: 20 numbers in all.

3 + 7 + 6 + 4 = 20

The m × f column: 410.

That is about the total of all the numbers.

15 + 105 + 150 + 140 = 410

4 Share equally

The mean shares the total equally: 410 divided by 20.

mean ≈ 41020

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 20 numbers.

Careful: not 4104 ✗

410 divided by 20 is 20.5.

mean ≈ 20.5

5 Check

Let's check: 20.5 times 20 is 410, the total.

20.5 × 20 = 410

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

So it makes sense!

5 < 20.5 < 35

Let's look back.

First: find the midpoints.

Then: multiply m × f.

Then: add the columns.

Then: share equally.

Last: check.

Answer
mean ≈ 20.5

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Mean of grouped data 10-19: 2, 20-29: 5, 30-39: 8, 40-49: 4, 50-59: 1 Medium

The question

Question: find the mean of the grouped data with classes 10 to 19, 20 to 29, 30 to 39, 40 to 49 and 50 to 59, and frequencies 2, 5, 8, 4 and 1.

Mean of the grouped data

Each class is a range of numbers.

f is how many numbers are in it.

So 2 numbers are from 10 to 19.

f = how many, m = the middle of the class

1 Find the midpoints

We don't know the numbers themselves.

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

m = (10 + 19) ÷ 2 = 14.5

The middle of 10 and 19: add them, divide by 2.

14.5.

Next: 24.5.

Each midpoint is 10 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.

14.5 times 2 is 29.

24.5 times 5 is 122.5.

The same for every row.

3 Add the columns

The f column: 20 numbers in all.

2 + 5 + 8 + 4 + 1 = 20

The m × f column: 660.

That is about the total of all the numbers.

29 + 122.5 + 276 + 178 + 54.5 = 660

4 Share equally

The mean shares the total equally: 660 divided by 20.

mean ≈ 66020

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

≈ : an estimate, from the midpoints

Careful!

Not divided by the 5 classes.

There are 20 numbers.

Careful: not 6605 ✗

660 divided by 20 is 33.

mean ≈ 33

5 Check

Let's check: 33 times 20 is 660, the total.

33 × 20 = 660

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

So it makes sense!

14.5 < 33 < 54.5

Let's look back.

First: find the midpoints.

Then: multiply m × f.

Then: add the columns.

Then: share equally.

Last: check.

Answer
mean ≈ 33

▶ Watch it solved

Mean of grouped data 0-10: 2, 10-20: 7, 20-30: 9, 30-40: 5, 40-50: 3 Harder

The question

Question: find the mean of the grouped data with classes 0 to 10, 10 to 20, 20 to 30, 30 to 40 and 40 to 50, and frequencies 2, 7, 9, 5 and 3.

Mean of the grouped data

Each class is a range of numbers.

f is how many numbers are in it.

So 2 numbers are from 0 to 10.

f = how many, m = the middle of the class

1 Find the midpoints

We don't know the numbers themselves.

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

m = (0 + 10) ÷ 2 = 5

The middle of 0 and 10: add them, divide by 2.

5.

Next: 15.

Each midpoint is 10 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.

5 times 2 is 10.

15 times 7 is 105.

The same for every row.

3 Add the columns

The f column: 26 numbers in all.

2 + 7 + 9 + 5 + 3 = 26

The m × f column: 650.

That is about the total of all the numbers.

10 + 105 + 225 + 175 + 135 = 650

4 Share equally

The mean shares the total equally: 650 divided by 26.

mean ≈ 65026

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

≈ : an estimate, from the midpoints

Careful!

Not divided by the 5 classes.

There are 26 numbers.

Careful: not 6505 ✗

650 divided by 26 is 25.

mean ≈ 25

5 Check

Let's check: 25 times 26 is 650, the total.

25 × 26 = 650

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

So it makes sense!

5 < 25 < 45

Let's look back.

First: find the midpoints.

Then: multiply m × f.

Then: add the columns.

Then: share equally.

Last: check.

Answer
mean ≈ 25

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. mean of grouped data 0-10: 3, 10-20: 7, 20-30: 6, 30-40: 4
    Answer
    mean ≈ 20.5
  2. mean of grouped data 0-4: 2, 4-8: 5, 8-12: 3
    Answer
    mean ≈ 6.4
  3. mean of grouped data 0-20: 4, 20-40: 10, 40-60: 6
    Answer
    mean ≈ 32
  4. mean of grouped data 10-20: 5, 20-30: 8, 30-40: 7
    Answer
    mean ≈ 26
  5. mean of grouped data 0-6: 1, 6-12: 4, 12-18: 3, 18-24: 2
    Answer
    mean ≈ 12.6
  6. mean of grouped data 50-60: 2, 60-70: 6, 70-80: 9, 80-90: 3
    Answer
    mean ≈ 71.5
  7. mean of grouped data 0-2: 6, 2-4: 9, 4-6: 5
    Answer
    mean ≈ 2.9

Medium

  1. mean of grouped data 10-19: 2, 20-29: 5, 30-39: 8, 40-49: 4, 50-59: 1
    Answer
    mean ≈ 33
  2. mean of grouped data 0-9: 4, 10-19: 7, 20-29: 6, 30-39: 3
    Answer
    mean ≈ 18.5
  3. mean of grouped data 0-10: 5, 10-20: 9, 20-30: 4, 30-40: 2
    Answer
    mean ≈ 16.5
  4. mean of grouped data 40-50: 3, 50-60: 7, 60-70: 12, 70-80: 6, 80-90: 2
    Answer
    mean ≈ 64
  5. mean of grouped data 100-120: 4, 120-140: 9, 140-160: 5, 160-180: 2
    Answer
    mean ≈ 135
  6. mean of grouped data 1-5: 3, 6-10: 8, 11-15: 6, 16-20: 3
    Answer
    1 decimal place: 1 more than the data
  7. mean of grouped data 0-5: 7, 5-10: 4, 10-15: 2, 15-20: 1
    Answer
    1 decimal place: 1 more than the data

Harder

  1. mean of grouped data 0-10: 2, 10-20: 7, 20-30: 9, 30-40: 5, 40-50: 3
    Answer
    mean ≈ 25
  2. mean of grouped data 15-25: 3, 25-35: 5, 35-45: 8, 45-55: 6, 55-65: 2
    Answer
    1 decimal place: 1 more than the data
  3. mean of grouped data 60-69: 2, 70-79: 6, 80-89: 9, 90-99: 4
    Answer
    1 decimal place: 1 more than the data
  4. mean of grouped data 0-30: 4, 30-60: 11, 60-90: 8, 90-120: 3
    Answer
    1 decimal place: 1 more than the data
  5. mean of grouped data 0-1: 5, 1-2: 8, 2-3: 4, 3-4: 3
    Answer
    1 decimal place: 1 more than the data
  6. mean of grouped data 20-24: 6, 25-29: 9, 30-34: 7, 35-39: 4, 40-44: 1
    Answer
    1 decimal place: 1 more than the data