Averages

What you'll learn

How to find the range, mode, median and mean from a list of numbers.
Why putting values in order is often the first useful step.
How to work backwards when a mean is given but one number is missing.
How to update or combine means using totals.

Starting point: data values and ordering

In averages questions, you are usually given a list of numbers. Each number is called a data value.

Averages help describe what a set of data is “typically” like. The range is slightly different: it describes how spread out the data is.

Definition

The key words

Data means the numbers or information you are working with.
An average is a number that represents a typical value in the data. The main averages here are the mean, median and mode.
The range tells you how spread out the data is.

Before finding the median, and sometimes before checking the mode, it helps to put the numbers in ascending order, which means from smallest to largest.

Example

Putting values in order

A set of cards shows these numbers:

9, 3, 12, 7, 3, 10

Put the numbers in ascending order.

Find the smallest number first. The smallest is 3.
There are two 3s, so write both of them.
Continue from smallest to largest:
$123,\ 3,\ 7,\ 9,\ 10,\ 12$
The ordered list is 3, 3, 7, 9, 10, 12.

Tip

Keep repeats

If a number appears more than once, write it more than once when ordering the list. Repeated values matter for the mode, median and mean.

The range

The range tells you the gap between the largest value and the smallest value.

Definition

Range

The range is found by subtracting the smallest value from the largest value.

value\text{range}=\text{largest value}-\text{smallest value}

Range is not an average, but it often appears in the same topic because it tells you about the spread of the data.

Example

Finding the range

The weights of six parcels, in grams, are:

114, 96, 130, 121, 118, 105

Find the range.

Identify the smallest value and the largest value. The smallest is 96 and the largest is 130.
Subtract the smallest from the largest:
$130 - 96 = 34$
The range is 34 grams.

Tip

Range sanity check

The range should never be bigger than the largest value unless the data includes negative numbers. For these Grade 2 questions, your range will usually be a positive number.

The mode

The mode is the value that appears most often.

Definition

Mode

The mode is the most frequent value in the data.

You do not calculate the mode using a formula. You just count how many times each value appears.

Example

Finding the mode

A list of numbers is:

4, 8, 6, 8, 10, 5, 8, 6

Find the mode.

Count how many times each repeated number appears.
The number 6 appears twice.
The number 8 appears three times.
No other number appears as often as 8, so the mode is 8.

Common Mistake

Choosing the biggest number

The mode is not the largest number. It is the number that appears the most often.

The median

The median is the middle value when the data is in order.

Definition

Median

The median is the middle value after the data has been arranged from smallest to largest.

This diagram shows why ordering matters, and what changes when there is an odd or even number of values.

Diagram showing how to find the median for odd and even numbers of values

Median with an odd number of values

If there is an odd number of values, there is one middle value.

Example

Median of an odd-sized list

Find the median of:

13, 6, 9, 15, 7

Put the numbers in ascending order:
$156,\ 7,\ 9,\ 13,\ 15$
There are five values, so the middle one is the third value.
The third value is 9, so the median is 9.

Median with an even number of values

If there is an even number of values, there are two middle values. The median is halfway between them, so you add the two middle values and divide by 2.

Example

Median of an even-sized list

The masses of six potatoes, in grams, are:

142, 118, 135, 126, 110, 129

Find the median mass.

Put the values in ascending order:
$142110,\ 118,\ 126,\ 129,\ 135,\ 142$
There are six values, so the two middle values are the third and fourth values: 126 and 129.
Find the mean of these two middle values:
$126+1292=127.5\frac{126+129}{2}=127.5$
The median mass is 127.5 grams.

Common Mistake

Median before ordering

You cannot just pick the number written in the middle of the original list. The median is the middle value after the numbers have been put in order.

Example

Explaining why a median answer is wrong

A student says, “The median of 18, 12, 15, 20, 12 is 15 because it is in the middle of the list.”

Explain why this is not correct.

The student has used the middle position in the original list.
For the median, the numbers must first be put in order:
$2012,\ 12,\ 15,\ 18,\ 20$
The middle value of the ordered list is 15, so in this particular case the answer happens to be 15, but the student’s method was not safe.
A correct explanation is: the list must be ordered before choosing the middle value.

The mean

The mean is what many people call “the average”. You find it by adding all the values, then dividing by how many values there are.

Definition

Mean

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

values\text{mean}=\frac{\text{total of values}}{\text{number of values}}

Key Idea

Mean as a fair share

The mean tells you what each value would be if the total were shared out equally between all the values.

Example

Calculating the mean

A list of five numbers is:

9, 4, 12, 8, 7

Calculate the mean.

Add all the values:
$9 + 4 + 12 + 8 + 7 = 40$
Count how many values there are. There are five values.
Divide the total by the number of values:
$40 \div 5 = 8$
The mean is 8.

Tip

Show the total

For mean questions, write down the total first. It earns method marks and makes hidden-number questions much easier.

Working backwards from the mean

Sometimes the mean is given, but one value is missing. In that case, use:

values\text{total}=\text{mean}\times\text{number of values}

Then subtract the values you already know.

Example

Finding a hidden value from the mean

Seven numbers have a mean of 9:

8, 12, 5, 10, 9, 7, ?

Find the hidden number.

There are seven numbers and the mean is 9, so find the total:
$\times 7 = 63$
Add the six known numbers:
$8 + 12 + 5 + 10 + 9 + 7 = 51$
Subtract the known total from the full total:
$63 - 51 = 12$
The hidden number is 12.

You can also combine mean information with the mode.

Example

Two hidden cards using mean and mode

Six cards have these numbers:

2, 5, 6, 7, ?, ?

The mode is 2. The mean is 5. Find the two hidden numbers.

Use the mean to find the total of all six numbers:
$\times 6 = 30$
Add the four visible numbers:
$2 + 5 + 6 + 7 = 20$
The two hidden numbers must add to 10:
$30 - 20 = 10$
Because the mode is 2, one hidden number must be 2.
Find the other hidden number:
$10 - 2 = 8$
The two hidden numbers are 2 and 8.

Common Mistake

Forgetting the number of values

If the mean is 5 for six values, the total is 30, not 5. Always multiply the mean by the number of values first.

Updating or combining means

A mean can change when a new value is added. The safest method is to go back to the total.

Example

Adding one more value

A runner ran a mean distance of 6.4 km over five days. On the sixth day, she ran 11 km. Find the mean distance for all six days.

Find the total distance for the first five days:
$6.4 \times 5 = 32$
Add the sixth day’s distance:
$32 + 11 = 43$
Divide by the new number of days:
$43 \div 6 = 7.166\ldots$
The mean distance is about 7.17 km.

Sometimes you are given the mean of a whole group and the mean of part of the group. Again, use totals.

Example

Finding the mean of the remaining group

The mean of eight scores is 40. The mean of two of the scores is 28. Work out the mean of the other six scores.

Find the total of all eight scores:
$40 \times 8 = 320$
Find the total of the two known scores:
$28 \times 2 = 56$
Find the total of the other six scores:
$320 - 56 = 264$
Divide by six to find their mean:
$264 \div 6 = 44$
The mean of the other six scores is 44.

Exam technique

In the exam

For the median, always put the data in order first, even if the list looks nearly ordered.
For the mean, write the total and the number of values clearly before dividing.
For hidden-number or combined-mean questions, use total = mean × number of values as your starting point.

Self review

Check yourself

Can you explain the difference between mean, median, mode and range?
If there are six ordered values, which two positions do you use to find the median?
If the mean of five numbers is 12, what total must the five numbers add to?

Averages

What you'll learn

How to find the range, mode, median and mean from a list of numbers.
Why putting values in order is often the first useful step.
How to work backwards when a mean is given but one number is missing.
How to update or combine means using totals.

Starting point: data values and ordering

In averages questions, you are usually given a list of numbers. Each number is called a data value.

Averages help describe what a set of data is “typically” like. The range is slightly different: it describes how spread out the data is.

Definition

The key words

Data means the numbers or information you are working with.
An average is a number that represents a typical value in the data. The main averages here are the mean, median and mode.
The range tells you how spread out the data is.

Before finding the median, and sometimes before checking the mode, it helps to put the numbers in ascending order, which means from smallest to largest.

Example

Putting values in order

A set of cards shows these numbers:

9, 3, 12, 7, 3, 10

Put the numbers in ascending order.

Find the smallest number first. The smallest is 3.
There are two 3s, so write both of them.
Continue from smallest to largest:
$123,\ 3,\ 7,\ 9,\ 10,\ 12$
The ordered list is 3, 3, 7, 9, 10, 12.

Tip

Keep repeats

If a number appears more than once, write it more than once when ordering the list. Repeated values matter for the mode, median and mean.

The range

The range tells you the gap between the largest value and the smallest value.

Definition

Range

The range is found by subtracting the smallest value from the largest value.

value\text{range}=\text{largest value}-\text{smallest value}

Range is not an average, but it often appears in the same topic because it tells you about the spread of the data.

Example

Finding the range

The weights of six parcels, in grams, are:

114, 96, 130, 121, 118, 105

Find the range.

Identify the smallest value and the largest value. The smallest is 96 and the largest is 130.
Subtract the smallest from the largest:
$130 - 96 = 34$
The range is 34 grams.

Tip

Range sanity check

The range should never be bigger than the largest value unless the data includes negative numbers. For these Grade 2 questions, your range will usually be a positive number.

The mode

The mode is the value that appears most often.

Definition

Mode

The mode is the most frequent value in the data.

You do not calculate the mode using a formula. You just count how many times each value appears.

Example

Finding the mode

A list of numbers is:

4, 8, 6, 8, 10, 5, 8, 6

Find the mode.

Count how many times each repeated number appears.
The number 6 appears twice.
The number 8 appears three times.
No other number appears as often as 8, so the mode is 8.

Common Mistake

Choosing the biggest number

The mode is not the largest number. It is the number that appears the most often.

The median

The median is the middle value when the data is in order.

Definition

Median

The median is the middle value after the data has been arranged from smallest to largest.

This diagram shows why ordering matters, and what changes when there is an odd or even number of values.

Diagram showing how to find the median for odd and even numbers of values

Median with an odd number of values

If there is an odd number of values, there is one middle value.

Example

Median of an odd-sized list

Find the median of:

13, 6, 9, 15, 7

Put the numbers in ascending order:
$156,\ 7,\ 9,\ 13,\ 15$
There are five values, so the middle one is the third value.
The third value is 9, so the median is 9.

Median with an even number of values

If there is an even number of values, there are two middle values. The median is halfway between them, so you add the two middle values and divide by 2.

Example

Median of an even-sized list

The masses of six potatoes, in grams, are:

142, 118, 135, 126, 110, 129

Find the median mass.

Put the values in ascending order:
$142110,\ 118,\ 126,\ 129,\ 135,\ 142$
There are six values, so the two middle values are the third and fourth values: 126 and 129.
Find the mean of these two middle values:
$126+1292=127.5\frac{126+129}{2}=127.5$
The median mass is 127.5 grams.

Common Mistake

Median before ordering

You cannot just pick the number written in the middle of the original list. The median is the middle value after the numbers have been put in order.

Example

Explaining why a median answer is wrong

A student says, “The median of 18, 12, 15, 20, 12 is 15 because it is in the middle of the list.”

Explain why this is not correct.

The student has used the middle position in the original list.
For the median, the numbers must first be put in order:
$2012,\ 12,\ 15,\ 18,\ 20$
The middle value of the ordered list is 15, so in this particular case the answer happens to be 15, but the student’s method was not safe.
A correct explanation is: the list must be ordered before choosing the middle value.

The mean

The mean is what many people call “the average”. You find it by adding all the values, then dividing by how many values there are.

Definition

Mean

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

values\text{mean}=\frac{\text{total of values}}{\text{number of values}}

Key Idea

Mean as a fair share

The mean tells you what each value would be if the total were shared out equally between all the values.

Example

Calculating the mean

A list of five numbers is:

9, 4, 12, 8, 7

Calculate the mean.

Add all the values:
$9 + 4 + 12 + 8 + 7 = 40$
Count how many values there are. There are five values.
Divide the total by the number of values:
$40 \div 5 = 8$
The mean is 8.

Tip

Show the total

For mean questions, write down the total first. It earns method marks and makes hidden-number questions much easier.

Working backwards from the mean

Sometimes the mean is given, but one value is missing. In that case, use:

values\text{total}=\text{mean}\times\text{number of values}

Then subtract the values you already know.

Example

Finding a hidden value from the mean

Seven numbers have a mean of 9:

8, 12, 5, 10, 9, 7, ?

Find the hidden number.

There are seven numbers and the mean is 9, so find the total:
$\times 7 = 63$
Add the six known numbers:
$8 + 12 + 5 + 10 + 9 + 7 = 51$
Subtract the known total from the full total:
$63 - 51 = 12$
The hidden number is 12.

You can also combine mean information with the mode.

Example

Two hidden cards using mean and mode

Six cards have these numbers:

2, 5, 6, 7, ?, ?

The mode is 2. The mean is 5. Find the two hidden numbers.

Use the mean to find the total of all six numbers:
$\times 6 = 30$
Add the four visible numbers:
$2 + 5 + 6 + 7 = 20$
The two hidden numbers must add to 10:
$30 - 20 = 10$
Because the mode is 2, one hidden number must be 2.
Find the other hidden number:
$10 - 2 = 8$
The two hidden numbers are 2 and 8.

Common Mistake

Forgetting the number of values

If the mean is 5 for six values, the total is 30, not 5. Always multiply the mean by the number of values first.

Updating or combining means

A mean can change when a new value is added. The safest method is to go back to the total.

Example

Adding one more value

A runner ran a mean distance of 6.4 km over five days. On the sixth day, she ran 11 km. Find the mean distance for all six days.

Find the total distance for the first five days:
$6.4 \times 5 = 32$
Add the sixth day’s distance:
$32 + 11 = 43$
Divide by the new number of days:
$43 \div 6 = 7.166\ldots$
The mean distance is about 7.17 km.

Sometimes you are given the mean of a whole group and the mean of part of the group. Again, use totals.

Example

Finding the mean of the remaining group

The mean of eight scores is 40. The mean of two of the scores is 28. Work out the mean of the other six scores.

Find the total of all eight scores:
$40 \times 8 = 320$
Find the total of the two known scores:
$28 \times 2 = 56$
Find the total of the other six scores:
$320 - 56 = 264$
Divide by six to find their mean:
$264 \div 6 = 44$
The mean of the other six scores is 44.

Exam technique

In the exam

For the median, always put the data in order first, even if the list looks nearly ordered.
For the mean, write the total and the number of values clearly before dividing.
For hidden-number or combined-mean questions, use total = mean × number of values as your starting point.

Self review

Check yourself

Can you explain the difference between mean, median, mode and range?
If there are six ordered values, which two positions do you use to find the median?
If the mean of five numbers is 12, what total must the five numbers add to?

Averages

What you'll learn

Starting point: data values and ordering

The range

The mode

The median

Median with an odd number of values

Median with an even number of values

The mean

Working backwards from the mean

Updating or combining means

Recap questions

You've reached the end

Averages

What you'll learn

Starting point: data values and ordering

The range

The mode

The median

Median with an odd number of values

Median with an even number of values

The mean

Working backwards from the mean

Updating or combining means

Recap questions

You've reached the end

Averages

What you'll learn

Starting point: data values and ordering

The range

The mode

The median

Median with an odd number of values

Median with an even number of values

The mean

Working backwards from the mean

Updating or combining means

Recap questions

You've reached the end

Averages Revision Guide

Averages

What you'll learn

Starting point: data values and ordering

The range

The mode

The median

Median with an odd number of values

Median with an even number of values

The mean

Working backwards from the mean

Updating or combining means

Recap questions

You've reached the end

Averages Revision Guide