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Find the Mean from a Grouped Frequency Table

In this worksheet, students find the mean from a grouped frequency table.

'Find the Mean from a Grouped Frequency Table' worksheet

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR

Curriculum topic:   Statistics

Curriculum subtopic:   Statistics, Analysing Data

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

Finding an average is incredibly useful when you want to find one piece of data that describe a whole set of data.

You will already know that if I wanted to find the average from a list of numbers, I would just add the numbers and then divide by how many there were.

 

What if the list is really, really long?

Imagine that you collected the shoe size of 500 people. Would you want to add them all together?

There has to be a quicker way.

 

Using a grouped frequency table.

If we have loads of data and it was all discrete, we would have to use a grouped frequency tablle rather than a frequency table. Let's look at an example.

Example 1: I collect the amount of pocket money from some children and put it in a grouped frequency table. Estimate the mean of the pocket money.

Pocket Money (p) 0<p<10 10<p<20 20<p<30 30<p<40 40<p<50
Frequency 4 6 11 17 9

When we found the mean from a frequency table, we just multiplied the numbers together. We can't do that here becuase we don't know exactly what the pupils each got. We have to make an assumption that all the students got the amount at the midpoint of the group.

 

Pocket Money (p) 0≤p≤10 10<p≤20 20<p≤30 30<p≤40 40<p≤50
Frequency 4 6 11 17 9
Midpoint 5 15 25 35 45

Now we can multiply together the midpoints and the frequency

Pocket Money (p) 0≤p≤10 10<p≤20 20<p≤30 30<p≤40 40<p≤50
Frequency 4 6 11 17 9
Midpoint 5 15 25 35 45
mp x f 20 90 275 595 405

 

Once we have this, we can work out an estimate of the total pocket money would be added together would be by adding the numbers in red. (1385 )

 

We now have one of the bits of information I need to find the mean (what they all add up to)

All I need now is to find out how many people I asked. I can get this by adding up all the numbers in the frequency row (47)

 

To find the mean, all I now have to do is to divide one of these numbers by the other (1385 ÷ 47 ) to get an average of £29.47

 

A key point.

You will probably see the word estimate in the question. We are not finding the mean, we are estimating it.

When you have grouped data, we made the assumption earlier that everone got the value in the middle, this means we cannot s for certain what the mean is, we can only estimate it.

 

 

Can you summarise for me?

Step 1: Find the midpoints

Step 2: Multiply the midpoints and the frequencies together

Step 3: Add all these new numbers up

Step 4: Add up all the frequencies

Step 5: Divide to find the mean.

 

To find the the mean from a frequency table we ...

I measure the heights of 100 trees and put the data into a grouped frequency table

Height (cm) 40 ≤ h ≤ 60 60 ≤ h ≤ 80 80 ≤ h ≤ 100 100 ≤ h ≤ 120 120 ≤ h ≤ 140 140 ≤ h ≤ 160
Frequency 5 9 22 27 26 11
mp            
mp x f A B C D E F

I want to find an estimate for the mean height of the trees.

 

What numbers need to go in boxes A-F

 Number
A
B
C
D
E
F

I measure the heights of 100 trees and put the data into a grouped frequency table

Height (cm) 40 ≤ h ≤ 60 60 ≤ h ≤ 80 80 ≤ h ≤ 100 100 ≤ h ≤ 120 120 ≤ h ≤ 140 140 ≤ h ≤ 160
Frequency 5 9 22 27 26 11
mp 50 70 90 110 130 150
mp x f 250 630 1980 2970 3380 1650

I want to find an estimate for the mean height of the trees.

 

What is the total height for the trees?

I survey a group of students about how many pets they have.

I display the data in a frequency table.

Pets 0 1 2 3 4 5
Frequency 3 7 9 6 2 1
  0 7 18 18 8 5

 

What is the mean number of pets

108.6

0.009

I record the amount of hours a bike tyre is ridden before it gets a puncture.

Time (hrs) 150 < t < 200 200 < t < 250 250 < t < 300 300 < t < 350 350 < t < 400
Frequency 24 45 18 10 3
mp          
mp x f A B C D E

I want to find an estimate for the average time before it get's a puncture

 

What numbers need to go in boxes A - E

 Number
A
B
C
D
E

I record the amount of hours a bike tyre is ridden before it gets a puncture.

Time (hrs) 150 < t < 200 200 < t < 250 250 < t < 300 300 < t < 350 350 < t < 400
Frequency 24 45 18 10 3
mp 175 225 275 325 375
mp x f 4200 10125 4950 3250 1050

I want to find an estimate for the average time before it get's a puncture

 

What is the total time the bike tyres are ridden?

I record the amount of hours a bike tyre is ridden before it gets a puncture.

Time (hrs) 150 < t < 200 200 < t < 250 250 < t < 300 300 < t < 350 350 < t < 400
Frequency 24 45 18 10 3
mp          
mp x f A B C D E

I want to find an estimate for the average time before it get's a puncture

235.75

0004

I record the ages of the people who come into a coffee shop in the course of a day.

Age (Years) 14 - 18 19 - 20 21 - 26 27 - 35 36-50
Frequency 13 12 9 8 2
mp          
mp x f A B C D E

I want to find an estimate for the average age of the customer

 

What numbers need to go in boxes A - E

 Number
A
B
C
D
E

I record the ages of the people who come into a coffee shop in the course of a day.

Age (Years) 14 - 18 19 - 20 21 - 26 27 - 35 36-50
Frequency 13 12 9 8 2
mp 17 19.5 23.5 31 43
mp x f 221 234 211.5 248 86

I want to find an estimate for the average age of the customer

What is the total age of all the customers?

I record the ages of the people who come into a coffee shop in the course of a day.

Age (Years) 14 - 18 19 - 20 21 - 26 27 - 35 36-50
Frequency 13 12 9 8 2
mp 17 19.5 23.5 31 43
mp x f 221 234 211.5 248 8

Find an estimate for the average age of the customer

22.7

0.043

  • Question 1

To find the the mean from a frequency table we ...

CORRECT ANSWER
EDDIE SAYS
I sometimes remember the steps a bit easier than the words. Midpoints, Multiply, Add, Add, Divide
  • Question 2

I measure the heights of 100 trees and put the data into a grouped frequency table

Height (cm) 40 ≤ h ≤ 60 60 ≤ h ≤ 80 80 ≤ h ≤ 100 100 ≤ h ≤ 120 120 ≤ h ≤ 140 140 ≤ h ≤ 160
Frequency 5 9 22 27 26 11
mp            
mp x f A B C D E F

I want to find an estimate for the mean height of the trees.

 

What numbers need to go in boxes A-F

CORRECT ANSWER
 Number
A
B
C
D
E
F
EDDIE SAYS
Remember the first two rules? Midpoints then Multiply...
  • Question 3

I measure the heights of 100 trees and put the data into a grouped frequency table

Height (cm) 40 ≤ h ≤ 60 60 ≤ h ≤ 80 80 ≤ h ≤ 100 100 ≤ h ≤ 120 120 ≤ h ≤ 140 140 ≤ h ≤ 160
Frequency 5 9 22 27 26 11
mp 50 70 90 110 130 150
mp x f 250 630 1980 2970 3380 1650

I want to find an estimate for the mean height of the trees.

 

What is the total height for the trees?

CORRECT ANSWER
10860
EDDIE SAYS
All we have to do here is to add up the mp x f row
  • Question 4

I survey a group of students about how many pets they have.

I display the data in a frequency table.

Pets 0 1 2 3 4 5
Frequency 3 7 9 6 2 1
  0 7 18 18 8 5

 

What is the mean number of pets

CORRECT ANSWER
108.6
EDDIE SAYS
If we find the number of trees, there are 100. We need to divide so we either do 10860 ÷ 100 or 100 ÷ 10860. Which one gives the sensible answer?
  • Question 5

I record the amount of hours a bike tyre is ridden before it gets a puncture.

Time (hrs) 150 < t < 200 200 < t < 250 250 < t < 300 300 < t < 350 350 < t < 400
Frequency 24 45 18 10 3
mp          
mp x f A B C D E

I want to find an estimate for the average time before it get's a puncture

 

What numbers need to go in boxes A - E

CORRECT ANSWER
 Number
A
B
C
D
E
EDDIE SAYS
Remember the first two rules? Midpoints then Multiply...
  • Question 6

I record the amount of hours a bike tyre is ridden before it gets a puncture.

Time (hrs) 150 < t < 200 200 < t < 250 250 < t < 300 300 < t < 350 350 < t < 400
Frequency 24 45 18 10 3
mp 175 225 275 325 375
mp x f 4200 10125 4950 3250 1050

I want to find an estimate for the average time before it get's a puncture

 

What is the total time the bike tyres are ridden?

CORRECT ANSWER
23575
EDDIE SAYS
All we need to do is to add up these mp x f row.
  • Question 7

I record the amount of hours a bike tyre is ridden before it gets a puncture.

Time (hrs) 150 < t < 200 200 < t < 250 250 < t < 300 300 < t < 350 350 < t < 400
Frequency 24 45 18 10 3
mp          
mp x f A B C D E

I want to find an estimate for the average time before it get's a puncture

CORRECT ANSWER
235.75
EDDIE SAYS
If we add up the number of bikes, we get 100 We now know we have 23575 miles split between 100 bikes. Which way round do I need to divide them? If you can't remember, just try them both and decide which of your answers is more sensible.
  • Question 8

I record the ages of the people who come into a coffee shop in the course of a day.

Age (Years) 14 - 18 19 - 20 21 - 26 27 - 35 36-50
Frequency 13 12 9 8 2
mp          
mp x f A B C D E

I want to find an estimate for the average age of the customer

 

What numbers need to go in boxes A - E

CORRECT ANSWER
 Number
A
B
C
D
E
EDDIE SAYS
Remember the first two rules? Midpoints then Multiply...
  • Question 9

I record the ages of the people who come into a coffee shop in the course of a day.

Age (Years) 14 - 18 19 - 20 21 - 26 27 - 35 36-50
Frequency 13 12 9 8 2
mp 17 19.5 23.5 31 43
mp x f 221 234 211.5 248 86

I want to find an estimate for the average age of the customer

What is the total age of all the customers?

CORRECT ANSWER
1000.5
EDDIE SAYS
All we need to do is to add up these new numbers.
  • Question 10

I record the ages of the people who come into a coffee shop in the course of a day.

Age (Years) 14 - 18 19 - 20 21 - 26 27 - 35 36-50
Frequency 13 12 9 8 2
mp 17 19.5 23.5 31 43
mp x f 221 234 211.5 248 8

Find an estimate for the average age of the customer

CORRECT ANSWER
22.7
EDDIE SAYS
If we add up the number of people we get 44 We now know we have 1000.5 years hours split between 44 people Which way round do I need to divide them? If you can't remember, just try them both and decide which of your answers is more sensible. Does it make sense that the average age is 0.043 years old for a coffee shop?
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