Averages
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Revision notes for Edexcel GCSE Maths Averages. Open the guide for explanations and worked examples. Written against the Edexcel GCSE Maths (1MA1) specification, so the content matches what's examinable rather than general Maths background.

Averages

What you'll learn

  • How to find the range, mode, median and mean.
  • Why putting numbers in order is so important.
  • How to work backwards when a number is hidden.
  • How to update a mean when more data is added.

What are averages?

A set of numbers is often called data. Averages help us describe what is “typical” in the data.

At GCSE, the main averages are mode, median and mean. You will also often be asked for the range, which tells you how spread out the numbers are.

Definition

Key words for averages

  • Data means a collection of values, usually numbers.
  • The range is the largest value minus the smallest value.
  • The mode is the value that appears most often.
  • The median is the middle value after the numbers are in order.
  • The mean is found by adding all the values, then dividing by how many values there are.

First: put the numbers in order

Putting numbers in order means writing them from smallest to largest.

This is especially important for the median, because the middle number only works after the list has been ordered.

Example

Ordering before finding the middle

  1. Suppose the numbers are 9, 3, 6, 4, 8.

The unordered numbers are rearranged from smallest to largest so the middle value can be spotted correctly.

  1. Write them from smallest to largest: 3, 4, 6, 8, 9.

  2. There are 5 numbers, so the middle one is the 3rd number.

  3. The median is 6.

Common Mistake

Middle of the original list

The median is not just the number in the middle of the list as it is written. You must put the numbers in order first.

Range

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

Key Idea

Range

To find the range, do largest value minus smallest value.

Example

Finding the range of some heights

  1. The heights are 152 cm, 168 cm, 160 cm, 174 cm, 152 cm and 181 cm.

The range is the distance from the smallest value to the largest value.

  1. The smallest height is 152 cm and the largest height is 181 cm.

  2. Subtract the smallest from the largest:

    181−152=29181 - 152 = 29181−152=29

  3. The range is 29 cm.

Common Mistake

Range is not the biggest number

The range is not the largest value. It is the difference between the largest and smallest values.

Mode

The mode is the value that appears most often.

A list can have one mode, more than one mode, or no mode if no value repeats.

Example

Finding the mode

  1. The numbers are 5, 8, 8, 9, 10, 8 and 11.

Counting repeats shows that 8 occurs more often than any other number.

  1. Count how often each number appears.

  2. The number 8 appears three times.

  3. No other number appears as often.

  4. The mode is 8.

Tip

Quick mode check

Look for repeats. The mode is the number you see most often, not the biggest number.

Median

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

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

If there is an even number of values, there are two middle values. Add them together and divide by 2.

Example

Finding the median with an even number of values

  1. The weights are 118 g, 140 g, 126 g, 132 g, 150 g and 124 g.

With an even number of ordered values, the median lies halfway between the two middle values.

  1. Put them in order: 118 g, 124 g, 126 g, 132 g, 140 g, 150 g.

  2. There are 6 values, so there is no single middle value.

  3. The two middle values are 126 g and 132 g.

  4. Find the mean of these two middle values:

    126+1322=129\frac{126+132}{2}=1292126+132​=129

  5. The median weight is 129 g.

Mean

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

The total means all the values added together.

mean=total of valuesnumber of values\text{mean}=\frac{\text{total of values}}{\text{number of values}}mean=number of valuestotal of values​
Example

Finding the mean of numbers on cards

  1. The card numbers are 14, 6, 10, 6, 12 and 18.

The mean balances the total equally across all six cards.

  1. Add the numbers:

    14+6+10+6+12+18=6614+6+10+6+12+18=6614+6+10+6+12+18=66

  2. Count how many numbers there are. There are 6 numbers.

  3. Divide the total by 6:

    66÷6=1166 \div 6=1166÷6=11

  4. The mean is 11.

Tip

Mean checklist

For the mean, you need two things: the total and the number of values.

Hidden numbers using the mean

Sometimes a question tells you the mean and asks you to find a missing number.

Work backwards by finding the total first.

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

Finding one hidden number

  1. Seven cards have the numbers 12, 5, 9, 10, 8, 11 and one hidden number. The mean is 9.

The missing card is found by comparing the required total with the total of the visible cards.

  1. Find the total needed for 7 numbers with mean 9:

    9×7=639 \times 7=639×7=63

  2. Add the known numbers:

    12+5+9+10+8+11=5512+5+9+10+8+11=5512+5+9+10+8+11=55

  3. Subtract to find the hidden number:

    63−55=863-55=863−55=8

  4. The hidden number is 8.

Hidden numbers using mean and mode

Sometimes you are given both the mean and the mode. Use the mean to find the total, then use the mode clue to decide which number must repeat.

Example

Finding two hidden numbers

  1. Six cards show 2, 4, a hidden number, 7, 5 and another hidden number. The mean is 5 and the mode is 4.

The mean gives the total of the two hidden numbers, and the mode clue means one hidden card must be 4.

  1. Find the total needed for 6 numbers with mean 5:

    5×6=305 \times 6=305×6=30

  2. Add the numbers you can see:

    2+4+7+5=182+4+7+5=182+4+7+5=18

  3. The two hidden numbers must add to:

    30−18=1230-18=1230−18=12

  4. The mode is 4, so one hidden number must be 4.

  5. Find the other hidden number:

    12−4=812-4=812−4=8

  6. The two hidden numbers are 4 and 8.

Updating a mean when another value is added

If a new value is added, do not just average the old mean and the new value. First turn the old mean back into a total.

Example

Adding one more day to a mean

  1. A runner has a mean distance of 12 km for 4 days. On the next day, she runs 17 km.

To update a mean, first turn the old mean back into a total, then add the new value.

  1. Find the total for the first 4 days:

    12×4=4812 \times 4=4812×4=48

  2. Add the new distance:

    48+17=6548+17=6548+17=65

  3. There are now 5 days, so divide by 5:

    65÷5=1365 \div 5=1365÷5=13

  4. The new mean distance is 13 km.

Mean of the remaining group

If you know the mean of all the numbers and the mean of some of them, use totals.

Example

Finding the mean of the other numbers

  1. The mean of 6 numbers is 20. The mean of 2 of the numbers is 14.

  2. Find the total of all 6 numbers:

    20×6=12020 \times 6=12020×6=120

  3. Find the total of the 2 numbers:

    14×2=2814 \times 2=2814×2=28

  4. Find the total of the other 4 numbers:

    120−28=92120-28=92120−28=92

  5. Divide by 4 to find their mean:

    92÷4=2392 \div 4=2392÷4=23

  6. The mean of the other 4 numbers is 23.

Exam technique

In the exam

  1. Put the data in order before finding the median.

  2. For the mean, write down the total and how many values there are.

  3. If a number is hidden, work out the total the list should have first.

  4. Include units in your final answer if the data has units, such as cm, g or km.

Self review

Check yourself

  • Can you explain the difference between mode and median?

  • If 6 numbers have mean 10, what total must they have?

  • Why is “the middle number” unsafe unless the list is in order?

Recap questions

Test yourself with 5 quick questions on this guide. Answer them all correctly to complete it.

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