Mode & Mean from Grouped Data (Edexcel GCSE Statistics)

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Author

Roger

Expertise

Maths

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 $mean = \frac{sum of values}{number of values}$
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 $\frac{150 + 160}{2} = \frac{310}{2} = 155$
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
- $estimated mean = \frac{total of (midpoints \times frequencies)}{total frequency}$
  - i.e. divide the total of the fx column by the total number of data values

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 $40 \leq x < 50$
  - then the modal class is $40 \leq x < 50$
  - 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
To estimate the mean or find the modal class it will usually be easiest to rewrite the data as a table first
- The class intervals can be read off the horizontal axis
- You may need to calculate the frequencies using
  $frequency = frequency density \times class width$
If the class widths are equal then the modal class will be the one with the highest bar on the histogram
- But for unequal class widths you need to calculate the frequencies to determine the modal class

Exam Tip

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 $estimated mean = \frac{total of (midpoints \times frequencies)}{total frequency}$

$estimated mean = \frac{87}{20} = 4.35$

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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