Averages from Tables

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 Tables (CIE IGCSE Maths: Core)

Revision Note