Averages from Frequency Tables

What you'll learn

Read a frequency table, where frequency means “how many”.
Find the mean, median and mode: three different types of average.
Use midpoints to estimate a mean from grouped data.
Find a median class and answer percentage questions from a table.

1. Frequency tables: the basic idea

A frequency table is a compact way of showing repeated data. Instead of writing 2, 2, 2, 2, 2, you can write “2 has frequency 5”.

Definition

Frequency

Frequency means how many times a value, score, result or group occurs.

Definition

Mean, median and mode

The mean is found by adding all the data values and dividing by how many values there are.
The median is the middle value after the data has been put in order.
The mode is the value that occurs most often.

Ungrouped frequency tables

An ungrouped frequency table gives exact values, such as 0 points, 1 point, 2 points, and so on.

Key Idea

Mean from a frequency table

For an exact frequency table, multiply each value by its frequency, add these products, then divide by the total frequency.

Example

Mean number of points

A player records the points scored in 50 games.

Points	Frequency
0	8
1	12
2	17
3	8
4	4
5	1

Add the frequencies to check the total number of games:
$8 + 12 + 17 + 8 + 4 + 1 = 50$
Multiply each points score by its frequency:
$0×8=01×12=122×17=343×8=244×4=165×1=5\begin{aligned} 0 \times 8 &= 0\\ 1 \times 12 &= 12\\ 2 \times 17 &= 34\\ 3 \times 8 &= 24\\ 4 \times 4 &= 16\\ 5 \times 1 &= 5 \end{aligned}$
Add the products to find the total number of points:
$0 + 12 + 34 + 24 + 16 + 5 = 91$
Divide by the total frequency:
$mean=9150=1.82\text{mean}=\frac{91}{50}=1.82$

Common Mistake

Averaging the first column

Do not just average the values in the first column. The frequencies tell you that some values happen many more times than others.

2. Finding a missing frequency

Sometimes the frequency is unknown, often called $x$ . Use the total information to build an equation.

Example

Finding a missing frequency

A football team scored a total of 54 goals. The table shows the number of goals scored per game.

Goals	Frequency
0	9
1	13
2	x
3	5
4 or more	0

Work out the known contribution to the total goals:
$\times 9+1 \times 13+3 \times 5=28$
The 2-goal row contributes $2 x$ goals, so form an equation:
$2 x + 28 = 54$
Solve the equation:
$2x+28=542x=26x=13\begin{aligned} 2x+28&=54\\ 2x&=26\\ x&=13 \end{aligned}$

Common Mistake

Open-ended classes

A row such as “4 or more” is only harmless here because its frequency is 0. If an open-ended row has a non-zero frequency, you cannot calculate an exact mean without more information.

3. Median, mode and total from a frequency table

For the median, imagine the data written out in order. You do not need to actually write it all out; use cumulative frequency instead.

Definition

Cumulative frequency

Cumulative frequency is the running total of the frequencies as you move down a table.

Example

Median, mode and total goals

A team played 38 games.

Goals	Frequency
0	6
1	15
2	10
3	7
4 or more	0

There are 38 games, so the median is between the 19th and 20th values.
Build cumulative frequencies:
- Up to 0 goals: 6 games
- Up to 1 goal: 21 games
- Up to 2 goals: 31 games
- Up to 3 goals: 38 games
The 19th and 20th values are both in the 1-goal row, so the median is 1 goal.
The highest frequency is 15, so the mode is 1 goal.
Work out the total number of goals:
$\times 6+1 \times 15+2 \times 10+3 \times 7=56$

Tip

Median positions

For an even number of values, look for the two middle positions. If both positions are in the same row, the median is that row’s value.

4. Grouped frequency tables and midpoints

Grouped data is data shown in intervals rather than exact individual values. Each interval is called a class interval.

For example, if a table says 20 < t ≤ 30, you know the values are more than 20 and up to 30, but you do not know the exact values.

The midpoint is the halfway value in a class interval. For estimates, we pretend all the values in that interval are at the midpoint.

Number line showing a grouped class interval and its midpoint

Key Idea

Estimated mean

For grouped data, use midpoint times frequency. Then divide by the total frequency.

mean=∑(midpoint×frequency)∑frequencies\text{estimated mean}=\frac{\sum(\text{midpoint}\times\text{frequency})}{\sum \text{frequencies}}

Example

Estimating the mean height

The heights of 60 plants are grouped as follows.

Height (cm)	Frequency
140 < h ≤ 150	5
150 < h ≤ 160	11
160 < h ≤ 170	16
170 < h ≤ 180	18
180 < h ≤ 200	10

Find the midpoint of each interval:
$190145,\ 155,\ 165,\ 175,\ 190$
Multiply each midpoint by its frequency:
$145×5=725155×11=1705165×16=2640175×18=3150190×10=1900\begin{aligned} 145 \times 5 &= 725\\ 155 \times 11 &= 1705\\ 165 \times 16 &= 2640\\ 175 \times 18 &= 3150\\ 190 \times 10 &= 1900 \end{aligned}$
Add these products:
$725 + 1705 + 2640 + 3150 + 1900 = 10120$
Divide by the total frequency to get 168.7 cm to 1 decimal place:
$1012060=168.666…≈168.7\frac{10120}{60}=168.666\ldots \approx 168.7$
This is an estimate because the exact plant heights inside each interval are not known.

Common Mistake

Wrong midpoint

Do not assume every interval has the same width. For 180 < h ≤ 200, the midpoint is 190, not 185.

5. Median class and percentages

For grouped data, you usually cannot find the exact median. Instead, you find the median class, which is the class interval containing the middle value.

Example

Travel times to an event

The table shows travel times for 100 people.

Time (minutes)	Frequency
0 < t ≤ 10	12
10 < t ≤ 20	18
20 < t ≤ 30	24
30 < t ≤ 40	28
40 < t ≤ 50	13
50 < t ≤ 60	5

To find the percentage who travelled for more than 30 minutes, add the frequencies above 30 minutes:
$28 + 13 + 5 = 46$
Since the total is 100 people, the percentage is 46%.
For the median class, the middle values are the 50th and 51st values.
Use cumulative frequencies:
- Up to 10 minutes: 12 people
- Up to 20 minutes: 30 people
- Up to 30 minutes: 54 people
The 50th and 51st values are in the interval 20 < t ≤ 30, so this is the median class.
To estimate the mean, use midpoints 5, 15, 25, 35, 45 and 55:
$\times 12+15 \times 18+25 \times 24+35 \times 28+45 \times 13+55 \times 5=2770$
Divide by 100:
$2770100=27.7\frac{2770}{100}=27.7$

6. When the data is shown in a bar chart

Sometimes the frequencies are shown on a bar chart instead of already being in a table. First read the bar heights, then use the same midpoint method.

Bar chart showing grouped homework hours and number of students

Example

Estimated mean from a bar chart

A bar chart shows homework hours for 30 students. The bar heights are 5, 8, 9, 5 and 3 for the groups 0–2, 3–5, 6–8, 9–11 and 12–14 hours.

Find the midpoint of each group:
$131,\ 4,\ 7,\ 10,\ 13$
Multiply each midpoint by the frequency:
$\times 5+4 \times 8+7 \times 9+10 \times 5+13 \times 3=189$
Divide by the total number of students:
$18930=6.3\frac{189}{30}=6.3$

Tip

Bar chart first step

If the question gives a bar chart, write down the frequencies before calculating. This helps you avoid misreading a bar halfway through your working.

Exam technique

In the exam

For the mean, make a “value or midpoint times frequency” column in your working.
Always divide by the total frequency, not by the number of rows.
For medians, use cumulative frequencies and state which position or positions you are checking.
For grouped data, remember to say the mean is an estimate because the exact values are unknown.

Self review

Check yourself

If a value has frequency 12, what does that mean in words?
How would you find the midpoint of 30 < t ≤ 40?
Why does a grouped frequency table usually only give an estimated mean?

Averages from Frequency Tables

What you'll learn

Read a frequency table, where frequency means “how many”.
Find the mean, median and mode: three different types of average.
Use midpoints to estimate a mean from grouped data.
Find a median class and answer percentage questions from a table.

1. Frequency tables: the basic idea

A frequency table is a compact way of showing repeated data. Instead of writing 2, 2, 2, 2, 2, you can write “2 has frequency 5”.

Definition

Frequency

Frequency means how many times a value, score, result or group occurs.

Definition

Mean, median and mode

The mean is found by adding all the data values and dividing by how many values there are.
The median is the middle value after the data has been put in order.
The mode is the value that occurs most often.

Ungrouped frequency tables

An ungrouped frequency table gives exact values, such as 0 points, 1 point, 2 points, and so on.

Key Idea

Mean from a frequency table

For an exact frequency table, multiply each value by its frequency, add these products, then divide by the total frequency.

Example

Mean number of points

A player records the points scored in 50 games.

Points	Frequency
0	8
1	12
2	17
3	8
4	4
5	1

Add the frequencies to check the total number of games:
$8 + 12 + 17 + 8 + 4 + 1 = 50$
Multiply each points score by its frequency:
$0×8=01×12=122×17=343×8=244×4=165×1=5\begin{aligned} 0 \times 8 &= 0\\ 1 \times 12 &= 12\\ 2 \times 17 &= 34\\ 3 \times 8 &= 24\\ 4 \times 4 &= 16\\ 5 \times 1 &= 5 \end{aligned}$
Add the products to find the total number of points:
$0 + 12 + 34 + 24 + 16 + 5 = 91$
Divide by the total frequency:
$mean=9150=1.82\text{mean}=\frac{91}{50}=1.82$

Common Mistake

Averaging the first column

Do not just average the values in the first column. The frequencies tell you that some values happen many more times than others.

2. Finding a missing frequency

Sometimes the frequency is unknown, often called $x$ . Use the total information to build an equation.

Example

Finding a missing frequency

A football team scored a total of 54 goals. The table shows the number of goals scored per game.

Goals	Frequency
0	9
1	13
2	x
3	5
4 or more	0

Work out the known contribution to the total goals:
$\times 9+1 \times 13+3 \times 5=28$
The 2-goal row contributes $2 x$ goals, so form an equation:
$2 x + 28 = 54$
Solve the equation:
$2x+28=542x=26x=13\begin{aligned} 2x+28&=54\\ 2x&=26\\ x&=13 \end{aligned}$

Common Mistake

Open-ended classes

A row such as “4 or more” is only harmless here because its frequency is 0. If an open-ended row has a non-zero frequency, you cannot calculate an exact mean without more information.

3. Median, mode and total from a frequency table

For the median, imagine the data written out in order. You do not need to actually write it all out; use cumulative frequency instead.

Definition

Cumulative frequency

Cumulative frequency is the running total of the frequencies as you move down a table.

Example

Median, mode and total goals

A team played 38 games.

Goals	Frequency
0	6
1	15
2	10
3	7
4 or more	0

There are 38 games, so the median is between the 19th and 20th values.
Build cumulative frequencies:
- Up to 0 goals: 6 games
- Up to 1 goal: 21 games
- Up to 2 goals: 31 games
- Up to 3 goals: 38 games
The 19th and 20th values are both in the 1-goal row, so the median is 1 goal.
The highest frequency is 15, so the mode is 1 goal.
Work out the total number of goals:
$\times 6+1 \times 15+2 \times 10+3 \times 7=56$

Tip

Median positions

For an even number of values, look for the two middle positions. If both positions are in the same row, the median is that row’s value.

4. Grouped frequency tables and midpoints

Grouped data is data shown in intervals rather than exact individual values. Each interval is called a class interval.

For example, if a table says 20 < t ≤ 30, you know the values are more than 20 and up to 30, but you do not know the exact values.

The midpoint is the halfway value in a class interval. For estimates, we pretend all the values in that interval are at the midpoint.

Number line showing a grouped class interval and its midpoint

Key Idea

Estimated mean

For grouped data, use midpoint times frequency. Then divide by the total frequency.

mean=∑(midpoint×frequency)∑frequencies\text{estimated mean}=\frac{\sum(\text{midpoint}\times\text{frequency})}{\sum \text{frequencies}}

Example

Estimating the mean height

The heights of 60 plants are grouped as follows.

Height (cm)	Frequency
140 < h ≤ 150	5
150 < h ≤ 160	11
160 < h ≤ 170	16
170 < h ≤ 180	18
180 < h ≤ 200	10

Find the midpoint of each interval:
$190145,\ 155,\ 165,\ 175,\ 190$
Multiply each midpoint by its frequency:
$145×5=725155×11=1705165×16=2640175×18=3150190×10=1900\begin{aligned} 145 \times 5 &= 725\\ 155 \times 11 &= 1705\\ 165 \times 16 &= 2640\\ 175 \times 18 &= 3150\\ 190 \times 10 &= 1900 \end{aligned}$
Add these products:
$725 + 1705 + 2640 + 3150 + 1900 = 10120$
Divide by the total frequency to get 168.7 cm to 1 decimal place:
$1012060=168.666…≈168.7\frac{10120}{60}=168.666\ldots \approx 168.7$
This is an estimate because the exact plant heights inside each interval are not known.

Common Mistake

Wrong midpoint

Do not assume every interval has the same width. For 180 < h ≤ 200, the midpoint is 190, not 185.

5. Median class and percentages

For grouped data, you usually cannot find the exact median. Instead, you find the median class, which is the class interval containing the middle value.

Example

Travel times to an event

The table shows travel times for 100 people.

Time (minutes)	Frequency
0 < t ≤ 10	12
10 < t ≤ 20	18
20 < t ≤ 30	24
30 < t ≤ 40	28
40 < t ≤ 50	13
50 < t ≤ 60	5

To find the percentage who travelled for more than 30 minutes, add the frequencies above 30 minutes:
$28 + 13 + 5 = 46$
Since the total is 100 people, the percentage is 46%.
For the median class, the middle values are the 50th and 51st values.
Use cumulative frequencies:
- Up to 10 minutes: 12 people
- Up to 20 minutes: 30 people
- Up to 30 minutes: 54 people
The 50th and 51st values are in the interval 20 < t ≤ 30, so this is the median class.
To estimate the mean, use midpoints 5, 15, 25, 35, 45 and 55:
$\times 12+15 \times 18+25 \times 24+35 \times 28+45 \times 13+55 \times 5=2770$
Divide by 100:
$2770100=27.7\frac{2770}{100}=27.7$

6. When the data is shown in a bar chart

Sometimes the frequencies are shown on a bar chart instead of already being in a table. First read the bar heights, then use the same midpoint method.

Bar chart showing grouped homework hours and number of students

Example

Estimated mean from a bar chart

A bar chart shows homework hours for 30 students. The bar heights are 5, 8, 9, 5 and 3 for the groups 0–2, 3–5, 6–8, 9–11 and 12–14 hours.

Find the midpoint of each group:
$131,\ 4,\ 7,\ 10,\ 13$
Multiply each midpoint by the frequency:
$\times 5+4 \times 8+7 \times 9+10 \times 5+13 \times 3=189$
Divide by the total number of students:
$18930=6.3\frac{189}{30}=6.3$

Tip

Bar chart first step

If the question gives a bar chart, write down the frequencies before calculating. This helps you avoid misreading a bar halfway through your working.

Exam technique

In the exam

For the mean, make a “value or midpoint times frequency” column in your working.
Always divide by the total frequency, not by the number of rows.
For medians, use cumulative frequencies and state which position or positions you are checking.
For grouped data, remember to say the mean is an estimate because the exact values are unknown.

Self review

Check yourself

If a value has frequency 12, what does that mean in words?
How would you find the midpoint of 30 < t ≤ 40?
Why does a grouped frequency table usually only give an estimated mean?

Averages from Frequency Tables

What you'll learn

1. Frequency tables: the basic idea

Ungrouped frequency tables

2. Finding a missing frequency

3. Median, mode and total from a frequency table

4. Grouped frequency tables and midpoints

5. Median class and percentages

6. When the data is shown in a bar chart

Recap questions

You've reached the end

Averages from Frequency Tables

What you'll learn

1. Frequency tables: the basic idea

Ungrouped frequency tables

2. Finding a missing frequency

3. Median, mode and total from a frequency table

4. Grouped frequency tables and midpoints

5. Median class and percentages

6. When the data is shown in a bar chart

Recap questions

You've reached the end

Averages from Frequency Tables

What you'll learn

1. Frequency tables: the basic idea

Ungrouped frequency tables

2. Finding a missing frequency

3. Median, mode and total from a frequency table

4. Grouped frequency tables and midpoints

5. Median class and percentages

6. When the data is shown in a bar chart

Recap questions

You've reached the end

Averages from Frequency Tables Revision Guide

Averages from Frequency Tables

What you'll learn

1. Frequency tables: the basic idea

Ungrouped frequency tables

2. Finding a missing frequency

3. Median, mode and total from a frequency table

4. Grouped frequency tables and midpoints

5. Median class and percentages

6. When the data is shown in a bar chart

Recap questions

You've reached the end

Averages from Frequency Tables Revision Guide