Mean (Edexcel GCSE Statistics): Revision Notes
Mean
What is the mean from grouped data?
When working with grouped data in frequency tables, we cannot calculate the exact mean because we don't know the individual values within each group or interval. Instead, we can estimate the mean using a special formula and technique.
The key to finding the mean from grouped data is using the midpoint of each interval to represent all the values in that group. This gives us a reasonable approximation of where the data points might be located within each interval.
Since we don't have access to individual data values in grouped data, we must make assumptions about how the data is distributed within each interval. The midpoint method assumes values are evenly distributed, which provides a good approximation in most cases.
The formula for mean from grouped data
The formula for estimating the mean from a grouped frequency table is:
Where:
- f = the frequency of each interval
- x = the midpoint of each interval
- Σfx = the sum of (frequency × midpoint) for all intervals
- Σf = the total frequency (sum of all frequencies)
You may also see the mean written using the symbol (pronounced "x bar").
Remember this formula: Mean = Σfx ÷ Σf. This is the cornerstone of calculating mean from grouped data and will appear in most exam questions on this topic.
How to find midpoints
The midpoint of each interval represents the middle value of that group. To calculate the midpoint:
Finding a Midpoint
If you have the interval 20 < T ≤ 30:
- Lower boundary = 20
- Upper boundary = 30
- Midpoint = (20 + 30) ÷ 2 = 25
Step-by-step method
Here's the systematic approach to calculating the mean from grouped data:
Step 1: Calculate the midpoints
Work out the midpoint (x) for each interval using the formula above.
Step 2: Calculate fx for each interval
Multiply the frequency (f) by the midpoint (x) for each interval.
Step 3: Find the totals
- Add up all the frequencies to get Σf
- Add up all the fx values to get Σfx
Step 4: Apply the formula
Divide Σfx by Σf to get your estimated mean.
Worked example
Let's look at data showing the number of years people have been driving:
Worked Example: Calculating Mean Years Driving
| Number of years driving (N) | Frequency | Midpoint (x) | f × x |
|---|---|---|---|
| 0 < N ≤ 10 | 8 | 5 | 40 |
| 10 < N ≤ 20 | 12 | 15 | 180 |
| 20 < N ≤ 30 | 14 | 25 | 350 |
| 30 < N ≤ 40 | 11 | 35 | 385 |
| 40 < N ≤ 50 | 5 | 45 | 225 |
| Totals | Σf = 50 | Σfx = 1180 |
Step-by-step calculation:
- Midpoints calculated: 5, 15, 25, 35, 45
- fx values: 40, 180, 350, 385, 225
- Total frequency (Σf): 50
- Total of fx (Σfx): 1180
- Mean = Σfx ÷ Σf = 1180 ÷ 50 = 23.6 years
Why is this an estimate?
The result we get is called an estimate because we're making several assumptions about the data:
Limitations of the Grouped Data Method:
- We don't know the exact values within each interval
- We assume all values in an interval are located at the midpoint
- In reality, the data might be spread differently within each interval
- To find the exact mean, we would need to know every individual value
Common exam tips
Exam Success Tips:
- Always show your working clearly in a table format
- Remember to calculate midpoints first
- Double-check your arithmetic, especially when adding up the fx column
- State that your answer is an estimate when asked
- Make sure your final answer makes sense in the context of the question
Remember!
Key Points to Remember:
- The mean from grouped data is always an estimate, not an exact value
- Use midpoints to represent each interval: (lower + upper) ÷ 2
- The formula is
- Work systematically: midpoints first, then fx values, then totals
- Always check your arithmetic and ensure your answer is reasonable