Averages (AQA GCSE Statistics): Revision Notes
Averages
An average is a single number that represents a whole set of data by showing its central tendency. Understanding averages is essential for summarising and comparing data sets effectively.
What are the three types of averages?
There are three main types of averages you need to know for your GCSE exam: the mode, the median, and the mean. Each one tells us something different about our data and is useful in different situations.
Each type of average gives you different insights into your data set. The mode shows you the most common value, the median shows you the middle value, and the mean gives you the overall average when all values are considered equally.
The mode
The mode is the value that appears most frequently in your data set. It's the easiest average to find because you simply look for the number that shows up the most times.
How to find the mode:
- Look through your data set
- Count how many times each value appears
- The value that appears most often is the mode
Worked Example: Finding the Mode
Here are the numbers of children in nine families: 6, 3, 2, 2, 3, 1, 1, 8, 1
To find the mode, count each value:
- 1 appears 3 times
- 2 appears 2 times
- 3 appears 2 times
- 6 appears 1 time
- 8 appears 1 time
Since 1 appears most frequently (3 times), the mode is 1.
The median
The median is the middle value when all your data is arranged in order from smallest to largest. Think of it as the value that sits right in the centre of your ordered list.
How to find the median:
Step 1: Always arrange your data in order from smallest to largest
Step 2: Use the formula (n + 1) ÷ 2 to find the position of the median, where n is the number of values
For an odd number of values: If you have 9 values: (9 + 1) ÷ 2 = 5, so the 5th value is the median.
For an even number of values: If you have 14 values: (14 + 1) ÷ 2 = 7.5, so the median is halfway between the 7th and 8th values.
Worked Example: Finding the Median (Odd Number of Values)
Children in families: 6, 3, 2, 2, 3, 1, 1, 8, 1
First, put in order: 1, 1, 1, 2, 2, 3, 3, 6, 8
Using (n + 1) ÷ 2 = (9 + 1) ÷ 2 = 5
The 5th value is 2, so the median is 2.
Worked Example: Finding the Median (Even Number of Values)
Number of cars in a car park over 14 days: 14, 16, 18, 15, 14, 18, 13, 20, 22, 19, 12, 12, 21, 25
First, put in order: 12, 12, 13, 14, 14, 15, 16, 18, 18, 19, 20, 21, 22, 25
Using (n + 1) ÷ 2 = (14 + 1) ÷ 2 = 7.5
This means the median is halfway between the 7th and 8th values. The 7th value is 16 and the 8th value is 18. So the median = (16 + 18) ÷ 2 = 17.
Therefore, the median is 17.
The arithmetic mean
The arithmetic mean (usually just called "the mean") is what most people think of when they hear "average". You calculate it by adding up all the values and dividing by how many values you have.
Formula for the mean:
Where:
- x̄ (x-bar) represents the mean
- Σx means "the sum of all the x-values"
- n is the number of values
How to calculate the mean:
- Add up all the values (find Σx)
- Count how many values you have (find n)
- Divide the sum by the number of values
Worked Example: Calculating the Mean
Number of pupils in a primary school each morning for eight days: 152, 165, 165, 163, 159, 160, 160, 158
Step 1: Find Σx (sum of all values) Σx = 152 + 165 + 165 + 163 + 159 + 160 + 160 + 158 = 1282
Step 2: Count the values n = 8 (there are 8 values)
Step 3: Calculate the mean Mean = x̄ = 1282 ÷ 8 = 160.25
So the mean is 160.25 pupils.
Common exam tips and traps
Always remember to:
- Put data in order before finding the median
- Check your arithmetic when calculating the mean
- Make sure you haven't missed any values when counting
- Be careful with the (n + 1) ÷ 2 rule for median - it gives you the position, not the actual value
Common traps:
- Forgetting to order the data for median calculations
- Using n ÷ 2 instead of (n + 1) ÷ 2 for finding median position
- Not taking the average of two middle values when n is even
Key Points to Remember:
- Mode = the value that appears most often (most frequent)
- Median = the middle value when data is in order (always order first!)
- Mean = add all values together and divide by how many values you have
- Use the formula (n + 1) ÷ 2 to find the position of the median
- For even numbers of values, the median is the average of the two middle values