Averages from Tables (Edexcel GCSE Maths: Foundation)

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Averages from Tables & Charts

What are frequency tables?

Frequency tables are used to summarise data in a neat format
- They also put the data in order
For example, the table below shows the number of pets in different houses along a street
- The number of pets is the data value, x
- The number of houses is the frequency, f
  - The frequency is how many times a data value is recorded (or seen)
The total frequency, n, can be calculated by adding together all the values in the frequency column

Number of pets (data value, x)	Number of houses (frequency, f)
0	2
1	7
2	6
3	4
4	1
Total frequency (n) = 20

How do I find the mode from a frequency table?

The mode is the data value with the highest frequency
- The mode for the example above is 1 pet per house
  - The mode is not the frequency, 7, this is the number of houses that have exactly 1 pet

How do I find the median from a frequency table?

The median is the data value in the middle of the frequency
- It is the ${(\frac{n + 1}{2})}^{t h}$ value, where $n$ is the total frequency
From above, so the median is the = 10.5^th value in the table
- The first two rows have a combined (cumulative) frequency of 2 + 7 = 9
- The first three rows have a combined frequency of 2 + 7 + 6 = 15
- Therefore the 10th and 11th values are in the third row (x = 2)
  - The median is 2 pets per house

How do I find the mean from a frequency table?

The mean from a frequency table has the following formula:
- - It helps to create a new column of 'data value × frequency'
  - Add up the values in this column
  - Divide by the total frequency
The mean is = 1.75 pets per house
- Means do not need to be whole numbers

Number of pets (data value, x)	Number of houses (frequency, f)	data value × frequency (xf)
0	2	0 × 2 = 0
1	7	1 × 7 = 7
2	6	2 × 6 = 12
3	4	3 × 4 = 12
4	1	4 × 1 = 4
	Total = 20	Total = 35

How do I find the range from frequency tables?

The range is the difference of the largest and smallest data values
- The range above is 4 - 0 = 4
  - The range is not the difference of the largest and smallest frequencies

What else should I know about frequency tables?

Tables can be converted back into a list of data values using their frequencies
- From above, 0 pets were recorded twice, 1 pet was recorded 7 times, 2 pets were recorded 6 times, etc
  - The list of pets recorded is 0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4
You could then find the mode, median and mean from this list of numbers

Worked example

The table shows data for the shoe sizes of pupils in class 11A.

Shoe size	Frequency
6	1
6.5	1
7	3
7.5	2
8	4
9	6
10	11
11	2
12	1

(a)

Find the mean shoe size for the class, giving your answer to 3 significant figures.

It helps to label shoe size (x) and frequency (f)
Add an extra column and calculate the values of 'shoe size × frequency', (xf)
Find the total frequency and total xf value

Shoe size (x)	Frequency (f)	xf
6	1	6 × 1 = 6
6.5	1	6.5 × 1 = 6.5
7	3	7 × 3 = 21
7.5	2	7.5 × 2 = 15
8	4	8 × 4 = 32
9	6	9 × 6 = 54
10	11	10 × 11 = 110
11	2	11 × 2 = 22
12	1	12 × 1 = 12
	Total = 31	Total = 278.5

Use the formula that the mean is the total of the xf column divided by the total frequency

Mean $= \frac{278.5}{31} = 8.983 870 . . .$

Give your final answer to 3 significant figures

The mean shoe size is 8.98 (to 3 s.f.)

Note that the mean does not have to be an actual shoe size

(b)

Find the median shoe size.

The median is the ${(\frac{n + 1}{2})}^{t h}$ value where $n$ is the total frequency

$\frac{n + 1}{2} = \frac{31 + 1}{2} = \frac{32}{2} = 16$

The median is the 16^th value
There are 1 + 1 + 3 + 2 + 4 = 11 values in the first five rows of the table
There are 11 + 6 = 17 values in the first six rows of the table
Therefore the 16^th value must be in the sixth row

The median shoe size is 9

(c)

Find the range of the shoe sizes.

The range is the highest shoe size subtract the lowest show size

12 - 6

The range of the shoe sizes is 6

Averages from Grouped Data

What is grouped data and why use it?

In some scenarios, the data can vary a lot
- e.g. heights of people
  - It is unlikely several people will be exactly the same height
Data like height is also continuous (data that can be measured)
- It would be difficult to list every value in a table
- There is also little difference between someone who is, say, 176 cm tall and someone who is 177cm tall
The data is grouped into classes
- However, when data is grouped, we lose the raw data
- With height, we know that 10 people have a height between 150 cm and 160 cm
  - but we won't know exactly what those 10 heights are
This means we cannot find the actual mean, median and mode from grouped data by their original definitions
- but we can estimate the mean
- we can talk about which class interval that the median lies in
- and we can talk about the modal class (the class interval the mode lies in)

How do I estimate the mean from grouped data?

There is one extra stage to this method compared to finding the mean from ungrouped frequency tables
- We use the class midpoints as our data values
- For example, if heights are split into class intervals 150 ≤ x < 160, 160 ≤ x < 170, etc
  - the midpoints would be 155, 165, etc
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
- Be careful with midpoints; the class intervals may be of different sizes
STEP 2
Work out "frequency" × "midpoint" (This is often called fx )
STEP 3
Total the fx column, and if not mentioned in the question, total the frequency column to find the number of data values involved
STEP 4
Estimate the mean by using: "total of fx" ÷ "no. of data values"

How do I find which class interval the median lies in?

Find the position of the median using $\frac{n + 1}{2}$ , where $n$ is the number of data values (total of the frequency column)
Use the frequencies in the table to deduce the class interval containing the ${(\frac{n + 1}{2})}^{th}$ value
- e.g. if the median is the 7^th value and the frequency of the first two class intervals are 4 and 10
  - the median will lie in the second class interval of the table
You will only be asked to find class interval containing the median
- You will not be asked to estimate the median from a grouped frequency table

How do I find the modal class interval?

The modal class interval is the interval containing the most data values
- This is the class with the highest frequency
Take care not to confuse the class interval with the frequency
- Questions will usually ask for the class interval, rather than the frequency

Exam Tip

Spotting the word "estimate" in a question can be a prompt to use the above methods for grouped data

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	3
3.5 ≤ w < 4	4
4 ≤ w < 4.5	6
4.5 ≤ w < 5	5
5 ≤ w < 6	2

(a)

Estimate the mean weight of these puppies.

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

Weight, w kg	Frequency	Midpoint	"fx"
3 ≤ w < 3.5	3	3.25	3 × 3.25 = 9.75
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 < 6	2	5.5	2 × 5.5 = 11
Total	20		85

Now we can find the mean

Mean $= \frac{85}{20} = 4.25$

An estimate of the mean weight of the puppies is 4.25 kg

(b)

Write down the modal class.

Looking for the highest frequency in the table we can see it is 6
This corresponds to the interval 4 ≤ w < 4.5

The modal class is 4 ≤ w < 4.5

A common error is to write down 6 as the mode (6 is the frequency)

(c)

Find the interval that contains the median.

There are 20 dogs
The median interval will be the interval containing the 10.5^th dog
Keep a running total

Weight, w kg	Frequency	Running Total
3 ≤ w < 3.5	3	3
3.5 ≤ w < 4	4	3 + 4 = 7
4 ≤ w < 4.5	6	7 + 6 = 13
4.5 ≤ w < 5	5	13 + 5 = 18
5 ≤ w < 6	2	18 + 2 = 20

The 10^th dog is in the third interval

The median is in the interval 4 ≤ w < 4.5

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