To estimate the mean from a grouped table, multiply each midpoint by its frequency, add the products, and divide by the total frequency. The answer is an estimate, because the exact values are not in the table.
This lesson is part of dispersion of grouped data. It uses the midpoints from reading class intervals and cumulative frequency.
What is the method?
- Find the midpoint x of each class.
- Multiply the midpoint by the frequency to get fx.
- Add the fx column to get Σfx.
- Divide Σfx by the total frequency Σf.
Worked example: marks of 40 students
| Marks | Midpoint x | Frequency f | fx |
|---|---|---|---|
| 10 to 19 | 14.5 | 4 | 58 |
| 20 to 29 | 24.5 | 9 | 220.5 |
| 30 to 39 | 34.5 | 14 | 483 |
| 40 to 49 | 44.5 | 8 | 356 |
| 50 to 59 | 54.5 | 5 | 272.5 |
| Total | 40 | 1 390 |
Check each product: 4 × 14.5 = 58, 9 × 24.5 = 220.5, 14 × 34.5 = 483, 8 × 44.5 = 356 and 5 × 54.5 = 272.5. The sum is 58 + 220.5 + 483 + 356 + 272.5 = 1 390.
The mean is 1 390 ÷ 40 = 34.75.
Reasonableness check. The modal class is 30 to 39, with midpoint 34.5. The mean is close to it and lies between 14.5 and 54.5. The answer is sensible.
The slip that costs the most marks
The common slip is to add the midpoints and divide by the number of classes: (14.5 + 24.5 + 34.5 + 44.5 + 54.5) ÷ 5 = 34.5. This happens to be close here because the frequencies are roughly balanced, which hides the error.
When the frequencies are lopsided, the wrong method gives a clearly wrong answer. Try frequencies 1, 1, 1, 1 and 36. Nearly everyone is in the last class, yet the wrong method still gives 34.5. Always multiply by frequency.
Why the answer is only an estimate
Suppose the 14 students in 30 to 39 all scored 30. Then the true mean would be smaller than your estimate. If they all scored 39, it would be larger.
You cannot know which, so the paper asks for an estimate. Write the word “estimate” or “approximately” in your answer when the question uses it.
Check yourself
Waiting times at a clinic have frequencies 6, 11, 8 and 3 in the classes 1 to 5, 6 to 10, 11 to 15 and 16 to 20 minutes. Estimate the mean waiting time, to two decimal places.
Answer
Midpoints: 3, 8, 13, 18. Products: 6 × 3 = 18, 11 × 8 = 88, 8 × 13 = 104, 3 × 18 = 54. Σfx = 18 + 88 + 104 + 54 = 264. Σf = 28.
Mean = 264 ÷ 28 = 9.43 minutes (estimate). It sits inside the modal class 6 to 10, which is sensible.
What to study next
Continue with constructing and interpreting an ogive, which uses the cumulative column instead of the midpoints. Check your arithmetic on invented data using the descriptive statistics explorer.
For a teacher to watch your fx column, see online one-to-one Mathematics tuition.