🎯 Learning Objectives
- find class midpoints;
- estimate the mean of grouped data;
- identify the modal class;
- understand why grouped means are estimates;
- use an assumed mean efficiently.
Introduction
This lesson develops Grouped Data and Estimated Mean from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
Estimated Mean
Estimated mean = Σ(f×midpoint) ÷ Σf
| Class | f | Midpoint x | fx |
|---|---|---|---|
| 0≤t<10 | 3 | 5 | 15 |
| 10≤t<20 | 5 | 15 | 75 |
| 20≤t<30 | 2 | 25 | 50 |
Estimated mean=(15+75+50)÷10=14.
The exact values inside each class are unknown, so the midpoint calculation gives an estimate.
Assumed Mean Idea
Choose a convenient midpoint A and calculate deviations d=x−A. Then use A+Σfd÷Σf.
🧮 Estimated Mean from Grouped Data
Edit the class frequencies. Midpoints and products are calculated automatically.
| Class | Midpoint x | Frequency f | fx |
|---|
Σf
Σfx
Estimated mean
Worked Examples
Reason through the method
- Identify the known information and what must be found.
- Choose the definition, property or formula that connects them.
- Substitute carefully and show each step.
- Check the notation, units and whether the result is sensible.
Practice Questions
- Find the midpoint of 30≤x<40.
- Identify the modal class in the table above.
- Explain why the grouped mean is estimated.
- Use the table above to verify the estimated mean.
✅ Check Your Work
Show answers and working
- (30+40)÷2=35.
- 10≤t<20, because it has the highest frequency 5.
- The individual values are not known; every value in a class is represented by the midpoint.
- Σfx=140 and Σf=10, so mean=140÷10=14.
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Midpoints stand in for unknown values.
- Always divide Σfx by Σf, not by the number of classes.