Averages from Frequency Tables

What you'll learn

How to read a frequency table without writing out every value.
How to calculate the mean from exact data in a table.
How to find the median and mode using frequencies.
How to estimate the mean from grouped data using midpoints.

Frequency Tables: The Starting Point

A frequency table is a neat way to show repeated data. Instead of listing every result separately, it tells you how often each result happened.

Definition

Frequency

The frequency is how many times a value, score, or group occurs. The total frequency is the total number of items, found by adding all the frequencies.

Example

Reading a frequency table

A small game was played several times.

Points	Frequency
0	3
1	5
2	4
3	2

Find the total number of games played.

Each frequency shows how many games had that points score, so the total is found by counting all the game icons.

Add the frequencies, not the point values.
$3 + 5 + 4 + 2 = 14$
The total frequency is 14, so 14 games were played.

The Mean from an Exact Frequency Table

The mean is the average found by adding all the values and dividing by how many values there are.

In a frequency table, each value may happen lots of times, so you multiply each value by its frequency first.

Key Idea

Mean from a frequency table

For an exact frequency table, use:

frequency\text{mean}=\frac{\text{total of value-frequency products}}{\text{total frequency}}

Example

Mean points scored

A player recorded these scores.

Score	Frequency	Product
0	4	0
1	6	6
2	10	20
3	5	15
4	2	8

Find the mean score.

The product column represents each score repeated by its frequency before dividing by the total frequency.

Multiply each score by its frequency. The products are shown in the final column.
Add the products.
$0 + 6 + 20 + 15 + 8 = 49$
Add the frequencies.
$4 + 6 + 10 + 5 + 2 = 27$
Divide the product total by the frequency total.
$mean=4927=1.814…\text{mean}=\frac{49}{27}=1.814\ldots$
The mean score is about 1.8 points.

Common Mistake

Forgetting the frequencies

Do not just add the different scores and divide by how many rows there are. A frequency table tells you that some scores happened more often than others.

Finding an Unknown Frequency

Sometimes one frequency is given as a letter, often $x$ . Use the total score or total amount to form an equation.

Example

Finding a missing frequency

A football team scored:

0 goals in 6 matches
1 goal in 8 matches
2 goals in $x$ matches
3 goals in 5 matches
4 or more goals in 0 matches

The team scored 39 goals altogether. Find $x$ .

The unknown frequency row contributes 2x goals, while the known rows contribute 23 goals.

Work out the known goal total.
$0×6+1×8+3×5=230\times 6+1\times 8+3\times 5=23$
The 2-goal row contributes $2 x$ goals, so form an equation.
$23 + 2 x = 39$
Solve the equation.
$2 x = 16$
Therefore $x = 8$ .

Tip

Rows with frequency 0

If the frequency is 0, that row contributes nothing to the total, even if the category says something like “4 or more”.

Median and Mode from Frequency Tables

Definition

Median and mode

The median is the middle value when the data is in order.
The mode is the value with the highest frequency.

A cumulative frequency is a running total of the frequencies. It helps you locate the middle value without writing out the whole list.

Example

Median, mode and total from goals

A team played 32 matches.

Goals	Frequency
0	5
1	12
2	9
3	6

Find the median, the mode, and the total number of goals.

Cumulative frequencies locate the 16th and 17th values, both inside the 1-goal group.

Add the frequencies.
$5 + 12 + 9 + 6 = 32$
There are 32 values, so the middle positions are the 16th and 17th values.
Build a running total.
- Up to 0 goals: 5
- Up to 1 goal: 17
- Up to 2 goals: 26
- Up to 3 goals: 32
The 16th and 17th values are both in the 1-goal row, so the median is 1 goal.
The highest frequency is 12, so the mode is 1 goal.
Work out the total number of goals.
$0×5+1×12+2×9+3×6=480\times 5+1\times 12+2\times 9+3\times 6=48$
The team scored 48 goals altogether.

Common Mistake

Choosing the middle row

The median is not always in the middle row of the table. Use the middle position and cumulative frequencies.

Grouped Frequency Tables

Grouped data gives ranges instead of exact values. For example, you might know that someone took over 20 minutes and up to 25 minutes, but not their exact time.

Definition

Class interval and midpoint

A class interval is a group or range of values. The midpoint is halfway between the lower and upper ends of the interval.

Because the exact values are unknown, the mean from grouped data is only an estimate.

Key Idea

Estimated mean

For grouped data, use the midpoint of each class interval:

frequency\text{estimated mean}=\frac{\text{total of midpoint-frequency products}}{\text{total frequency}}

Example

Estimating the mean time

A group of pupils completed a fitness run.

Time interval	Frequency	Midpoint	Product
Over 10, up to 15 minutes	2	12.5	25
Over 15, up to 20 minutes	6	17.5	105
Over 20, up to 25 minutes	8	22.5	180
Over 25, up to 35 minutes	4	30	120

Estimate the mean time.

For grouped data, each interval is represented by its midpoint before multiplying by frequency.

Find each midpoint. For the first class:
$10+152=12.5\frac{10+15}{2}=12.5$
Multiply each midpoint by its frequency. These products are shown in the final column.
Add the products.
$25 + 105 + 180 + 120 = 430$
Add the frequencies.
$2 + 6 + 8 + 4 = 20$
Divide to estimate the mean.
$mean=43020=21.5\text{estimated mean}=\frac{430}{20}=21.5$
The estimated mean time is 21.5 minutes.

Tip

Unequal class widths

Do not assume every midpoint goes up by the same amount. Always find the midpoint from the two ends of the interval.

Median Class from Grouped Data

For grouped data, you can usually find the median class: the interval containing the middle value.

Example

Finding the median class

The table shows travel times for 40 people.

Time interval	Frequency
Over 0, up to 10 minutes	5
Over 10, up to 20 minutes	9
Over 20, up to 30 minutes	11
Over 30, up to 40 minutes	10
Over 40, up to 50 minutes	5

Find the median class.

The cumulative frequency reaches the 20th and 21st values in the over 20, up to 30 minutes interval.

There are 40 people, so the middle positions are the 20th and 21st values.
Use cumulative frequencies.
- Up to 10 minutes: 5
- Up to 20 minutes: 14
- Up to 30 minutes: 25
The 20th and 21st values are in the “over 20, up to 30 minutes” class.
The median class is over 20 minutes, up to 30 minutes.

Exam technique

In the exam

Check whether the table uses exact values or grouped intervals.
For the mean, make a product column: value or midpoint times frequency.
For the median, use positions and cumulative frequencies, not just the middle row.

Self review

Check yourself

Why is the mean from grouped data only an estimate?
In a table with 30 values, which position or positions help you find the median?
What two totals do you need before calculating the mean from a frequency table?

Averages from Frequency Tables

What you'll learn

Frequency Tables: The Starting Point

The Mean from an Exact Frequency Table

Finding an Unknown Frequency

Median and Mode from Frequency Tables

Grouped Frequency Tables

Median Class from Grouped Data

Recap questions

Averages from Frequency Tables Revision Guide