Mean (AQA GCSE Statistics): Revision Notes
Mean
What is the mean from grouped data?
When data is presented in a grouped frequency table, we cannot calculate the exact mean because we don't know the individual values within each group. Instead, we can calculate an estimate of the mean using the midpoint of each interval to represent all the values in that group.
The mean from grouped data gives us a reasonable approximation of what the true mean would be if we had access to all the individual data values. This is a fundamental concept in statistics when working with grouped data.
Formula for calculating the mean
The formula for calculating the mean from grouped data is:
Where:
- = the sum of (frequency × midpoint) for each interval
- = the sum of all frequencies
- = the midpoint of each interval
- = the frequency of each interval
The symbol (pronounced "x bar") is often used to represent the mean.
Finding midpoints
To find the midpoint of each interval, you add the two boundary values together and divide by 2.
Finding a Midpoint:
For an interval 20 < T ≤ 30:
- Lower boundary = 20
- Upper boundary = 30
- Midpoint =
Step-by-step method
- Create your table with columns for intervals, frequency, midpoint, and f × x
- Calculate the midpoint for each interval
- Multiply frequency by midpoint (f × x) for each row
- Add up all frequencies to get
- Add up all f × x values to get
- Apply the formula: Mean =
Worked example 1: Homework time
Worked Example: Calculating Mean Homework Time
A teacher recorded the time students spent on homework. Here's how to calculate the estimated mean:
The table shows homework times with the following data:
- 0 < T ≤ 10 minutes: 12 students
- 10 < T ≤ 20 minutes: 8 students
- 20 < T ≤ 30 minutes: 3 students
- 30 < T ≤ 40 minutes: 2 students
Step-by-step solution:
- Calculate midpoints: 5, 15, 25, 35
- Calculate f × x: 60, 120, 75, 70
- Sum of frequencies () = 12 + 8 + 3 + 2 = 25
- Sum of f × x () = 60 + 120 + 75 + 70 = 325
- Mean =
Worked example 2: Driving experience
Worked Example: Mean Years of Driving Experience
Here's another example showing how to find the mean number of years people have been driving:
The calculation process is clearly shown with:
- Intervals from 0-10 years up to 40-50 years
- Frequencies: 8, 12, 14, 11, 5 (total = 50)
- Midpoints: 5, 15, 25, 35, 45
- f × x calculations giving a total of 1180
- Final answer:
Practice question
Test Your Understanding
You can test your understanding with this data about wasteland areas. The table shows area ranges from 0-200 m² up to 800-1000 m² with corresponding frequencies. Remember to find the midpoint of each interval first, then follow the step-by-step method.
Why is this an estimate?
Understanding Why This is an Estimate
The result we calculate is an estimate rather than the exact mean because:
- We don't know the actual individual values within each interval
- We assume all values in an interval are represented by the midpoint
- The true values could be spread differently throughout each interval
- If we had access to all the raw data, the exact mean might be slightly different
This is a crucial concept to understand for exam success!
Common exam tips
Exam Success Tips
- Always show your working by creating a clear table with all columns
- Check your arithmetic - addition errors are common in these questions
- Remember it's an estimate - exam questions often ask you to explain why
- Use appropriate units in your final answer
- Round sensibly - usually to 1 decimal place unless specified otherwise
Key Points to Remember:
- Mean from grouped data =
- Midpoint =
- The answer is always an estimate, not an exact value
- Show all your working in a clear table format
- Don't forget to include appropriate units in your final answer