Averages & Frequency Tables

 

What Are Frequency Tables?

A frequency table is a way of organising data to show how often each value appears. Each value is called a data value (x), and the number of times it appears is its frequency (f).

 

Data Value (x)	Frequency (f)
0	3
1	5
2	7
3	4
4	1

 

Total frequency $n = 3 + 5 + 7 + 4 + 1 = 20$

 

How to Find Averages from a Frequency Table

Mode

The mode is the data value with the highest frequency.

In the example above, the mode is 2 (it occurs 7 times).

Median

To find the median, use this formula: $\text{Median position} = \frac{n + 1}{2}$

• Here, $n = 20 \Rightarrow \frac{20 + 1}{2} = 10.5$
• Add cumulative frequencies:
  • 0 $\to$ 3
  • 0 + 5 = 8
  • 8 + 7 = 15 $\to$ The 10.5th value lies in the "2" row.
• Median = 2

Mean

To calculate the mean:

1. Create a new column: $x \times f$
2. Add up all the values of $xf$ and divide by the total frequency $n$

 

Data Value (x)	Frequency (f)	$x \times f$
0	3	0
1	5	5
2	7	14
3	4	12
4	1	4

 

Total: $\sum f = 20, \sum xf = 35$

$\text{Mean} = \frac{35}{20} = 1.75$

Range

$\text{Range} = \text{Highest data value} - \text{Lowest data value} = 4 - 0 = 4$

 

Example

The table below shows the number of goals scored by a football team over 15 matches:

 

Goals Scored (x)	Frequency (f)
0	1
1	2
2	5
3	3
4	3
5	1

 

(a) Find the mean number of goals.

x	f	$xf$
0	1	0
1	2	2
2	5	10
3	3	9
4	3	12
5	1	5

Total $f = 15, xf = 38$

$\text{Mean} = \frac{38}{15} = 2.53 \text{ goals (2 d.p.)}$

(b) Find the median number of goals

$\frac{15 + 1}{2} = 8 \text{th value}$ Cumulative frequencies:

• 1 (0 goals)
• 1 + 2 = 3 (1 goal)
• 3 + 5 = 8 $\to$ The 8th value is in the “2 goals” row.

$\text{Median} = 2 \text{ goals}$

(c) Find the mode

• The highest frequency is 5 (for 2 goals).
• $\text{Mode} = 2 \text{ goals}$

(d) Find the range

$\text{Range} = 5 - 0 = 5 \text{ goals}$