Frequency Tables & Averages Lesson | GCSE Maths Edexcel IGCSE Higher

Frequency Tables & Averages

This lesson covers:

What a frequency table is
How to identify the mode, range, mean and median from a frequency table

Introduction to frequency tables

Frequency tables sort data into categories and tally how often each category occurs.

The table below shows the frequency table for the ages of students in a class.

Age	Frequency
14	4
15	7
16	5
	Total = 16

Using frequency tables

Frequency tables can be used to find the mode, range, median and mean of a set of data.

The frequency table below shows the test scores of students in a class.

Score	Frequency
65	2
66	4
67	6
68	5
69	3
70	1
	Total = 21

Finding the mode

The mode is the most frequently occurring value in the data.

67 is the mode of the test scores as it has the highest frequency.

Finding the range

The range gives an indication of the spread of the data.

It is calculated by subtracting the smallest value from the largest value.

The range of the test scores = 70 - 65 = 5

Finding the mean

The mean is the total sum of all the values divided by the number of values.

To find the mean from a frequency table:

Multiply each category by its frequency.
Find the total of the products in step 1.
Divide by the total frequency.

Score	Frequency	Score x Frequency
65	2	65 x 2 = 130
66	4	66 x 4 = 264
67	6	67 x 6 = 402
68	5	68 x 5 = 340
69	3	69 x 3 = 207
70	1	70 x 1 = 70
	Total = 21	Total = 1413

mean = $\frac{1413}{21}$ = 67.29

Finding the median

The median is the middle value when data are ordered from smallest to largest.

In a frequency table, the median category contains the middle value, which can be found using the formula:

$\text{median position = }\frac{n+1}{2}$

Where:

n = total frequency

The median position of the test scores is:

median position = $\frac{n + 1}{2}=\frac{21+1}{2}=11$

The 11^thvalue in the data lies in the 67 category. The median is 67.

Worked Example 1: Using frequency tables

Identify the mode, range, mean, and median from the frequency table below.

Age	Frequency
14	4
15	7
16	5
	Total = 16

Finding the mode:

The mode is the most frequently occurring age = 15

Finding the range:

range = largest value - smallest value = 16 - 14 = 2

Finding the mean:

To calculate the mean we need to add a column containing the age multiplied by the frequency.

Age	Frequency	Age x Frequency
14	4	56
15	7	105
16	5	80
	Total = 16	Total = 241

$\text{mean = }\frac{241}{16}\text{ = 15.1 (to 3 s.f.)}$

Finding the median:

median position = $\frac{\text{n + 1} }{2}=\frac{16+1}{2}=\frac{17}{2}=8.5$

Counting through the frequency values, the 8th and 9th values fall within the Age = 15 category. The median age is 15.

Worked Example 2: Using frequency tables

Find the mode, range, mean, and median from the frequency table below.

Wait time (mins)	Frequency
10	4
20	7
30	5
40	10
50	4
60	1
	Total = 31

Finding the mode:

The mode is the most frequently occurring wait time = 40 mins

Finding the range:

range = largest value - smallest value = 60 - 10 = 50 mins

Finding the mean:

To calculate the mean we need to add a column containing the age multiplied by the frequency.

Wait time (mins)	Frequency	Wait time x Frequency
10	4	40
20	7	140
30	5	150
40	10	400
50	4	200
60	1	60
	Total = 31	Total = 990

$\text{mean = }\frac{990}{31}\text{ = 31.9 (to 3s.f.)}$

Finding the median:

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

Counting through the frequency values, the 16th value falls within the wait time = 30 mins category. The median age is 30 mins.

A survey of 13 people collected data on the number of pets they owned.

What is the mode of the data?

number of pets	frequency
0	4
1	6
2	2
3	1

1 pet

3 pets

2 pets

0 pets

0

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1

Four team members compared the number of items they each sold in a day.

What is the range of the data?

number sold	frequency
50	8
51	20
52	25
53	15

51.5

53

3

50

0

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1

The frequency table below shows the wait times for 24 people at a petrol station.

Calculate the mean wait time using the frequency table below.

wait time (minutes)	frequency
5	2
6	4
7	6
8	5
9	4
10	3

10.7 minutes

7.6 minutes

6.3 minutes

5.4 minutes

0

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1

A group of friends played a game. The table below summarises the points awarded.

What is the median of the data in the table?

number of points	frequency
10	3
20	2
30	6
40	4

10

30

20

40

0

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1