Mode & Mean from Grouped Data (Edexcel GCSE Statistics)

Revision Note

Mode & Mean from Grouped Data

How do I estimate the mean for grouped data?

  • It is impossible to find the mean for grouped data, because we don't have access to the original data values

    • i.e. there is no way to find the exact sum of all the data values

    • so we can't use the formula mean space equals space fraction numerator sum space of space values over denominator number space of space values end fraction

  • However we can estimate the mean for grouped data

    • To do this we use the class midpoints as our data values

      • e.g. if a class interval is 150 ≤ x < 160

      • we assume that all the data values are equal to the midpoint, 155

  • STEP 1
    Draw an extra two columns on the end of a table of the grouped data

    • In the first new column write down the midpoint of each class interval

    • If the midpoint isn't obvious, add the endpoints and divide by 2

      • e.g. if a class interval is 150 ≤ x < 160

      • the midpoint is fraction numerator 150 plus 160 over denominator 2 end fraction equals 310 over 2 equals 155

  • STEP 2
    Calculate "frequency" × "midpoint" (this is often called fx)

    • Write these values in the second column you added to the table

  • STEP 3
    Total the fx column and (if necessary) the frequency column

    • The question may already tell you the total number of data values (i.e. the total frequency)

      • In that case there's no need to find the total of the frequency column

  • STEP 4
    Estimate the mean by using the formula

    • estimated space mean equals fraction numerator total space of space open parentheses midpoints cross times frequencies close parentheses space over denominator total space frequency end fraction

      • i.e. divide the total of the fx column by the total number of data values

How do I find the modal class?

  • It is impossible to find the modal value (i.e. mode) for grouped data, because we don't have access to the original data values

    • i.e. it is impossible to say which value occurs the most times

  • For grouped data we talk about the modal class instead

    • This is the class with the highest frequency

      • Just find the highest frequency in the table

      • The corresponding class interval tells you the modal class

  • Be careful not to confuse the modal class with its frequency

    • e.g. if the highest frequency in the table is 34, corresponding to the class interval 40 less or equal than x less than 50

      • then the modal class is 40 less or equal than x less than 50

      • The modal class is not '34'!

What if the grouped data is in a histogram?

  • Remember that a histogram is a way of presenting grouped data as a diagram

  • The modal class will be the one with the highest bar on the histogram

  • To estimate the mean it will usually be easiest to rewrite the data as a table first

    • The class intervals can be read off the horizontal axis

    • The frequencies can be read off the vertical axis

Examiner Tips and Tricks

  • When presented with data in a table it may not be obvious whether the data is grouped or not

    • When you see the phrase “estimate the mean” you know that you are in the world of grouped data!

    • So use the midpoint technique to answer the question

Worked Example

The weights of 20 three-week-old Labrador puppies were recorded at a vet's clinic. The results are shown in the table below.

Weight, w kg

Frequency

3 ≤ w < 3.5

2

3.5 ≤ w < 4

4

4 ≤ w < 4.5

6

4.5 ≤ w < 5

5

5 ≤ w < 5.5

2

5.5 ≤ w < 6

1


(a) Estimate the mean weight of these puppies.

First add two columns to the table
Complete the first new column with the midpoints of the class intervals
Complete the second extra column by calculating "fx"
A total row is also useful

Weight, w kg

Frequency

Midpoint

"fx"

3 ≤ w < 3.5

2

3.25

2 × 3.25 = 6.5

3.5 ≤ w < 4

4

3.75

4 × 3.75 = 15

4 ≤ w < 4.5

6

4.25

6 × 4.25 = 25.5

4.5 ≤ w < 5

5

4.75

5 × 4.75 = 23.75

5 ≤ w < 5.5

2

5.25

2 × 5.5 = 10.5

5.5 ≤ w < 6

1

5.75

1 × 5.75 = 5.75

Total

20

 

87


Now we can find the mean using estimated space mean equals fraction numerator total space of space open parentheses midpoints cross times frequencies close parentheses over denominator total space frequency end fraction

estimated space mean equals 87 over 20 equals 4.35

4.35 kg

(b) Write down the modal class.

The highest frequency in the table is 6
This corresponds to the interval 4 ≤ w < 4.5

4 ≤ w < 4.5

A common error here would be to write down 6 (the frequency) as the modal class

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Roger B

Author: Roger B

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Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

Dan Finlay

Author: Dan Finlay

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Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.