Estimated Mean from Grouped Data

Q: What Is the Midpoint of a Class Interval?

When data is grouped into class intervals (such as $10 ≤ x < 20$), you don't know the individual values within each group — only how many values fall into it. The midpoint, the value halfway between the interval's lower and upper boundary, is used as an estimate for every value in that class.

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What Is the Midpoint of a Class Interval?

When data is grouped into class intervals (such as $10 \leq x < 20$), you don't know the individual values within each group — only how many values fall into it. The midpoint, the value halfway between the interval's lower and upper boundary, is used as an estimate for every value in that class.

Midpoint = (lower boundary + upper boundary) ÷ 2

Key point: We cannot find the exact mean from grouped data because we do not know the individual values. We use the midpoint as our best single representative for each class.

✔ Quick Check emgd1 — Midpoints and why we estimate

Question 1 of 10

The Estimated Mean Formula

Once we have the midpoint for each class, we multiply each midpoint by its frequency to get $fm$, then use the formula:

$$\text{Estimated mean} = \frac{\Sigma fm}{\Sigma f}$$

Where $m$ = midpoint and $f$ = frequency. The four steps are: find each midpoint, multiply midpoint × frequency, sum all $fm$ values, then divide by the total frequency.

Full Worked Example — Heights of 30 students

Step 1 — Find midpoints

140–150: m=145, 150–160: m=155, 160–170: m=165, 170–180: m=175

Step 2 — Calculate fm

5×145=725, 12×155=1860, 9×165=1485, 4×175=700

Step 3 — Sum fm

Σfm = 725 + 1860 + 1485 + 700 = 4770

Step 4 — Divide by Σf

Estimated mean = 4770 ÷ 30 = 159 cm

✔ Quick Check emgd2 — Applying the estimated mean formula

Question 1 of 10

Interpreting the Estimated Mean

The result is an estimate — always state this in your answer. Compare the estimated mean to the modal class (the class with the highest frequency) to check your answer is reasonable. If they are in very different parts of the data, re-check your working.

Cinema ages example — Worked calculation

Ages (years): 0–9 (f=8, m=4.5), 10–19 (f=22, m=14.5), 20–29 (f=30, m=24.5), 30–39 (f=18, m=34.5), 40–49 (f=7, m=44.5)

fm values

36 + 319 + 735 + 621 + 311.5 = 2022.5

Σf

8 + 22 + 30 + 18 + 7 = 85

Estimated mean

2022.5 ÷ 85 ≈ 23.8 years (1 d.p.)

Interpretation

The estimated mean age is approximately 23.8 years. This falls within the modal class (20–29), which suggests a reasonable answer.

⚠ Always write "estimated mean" — never just "mean" when working from grouped data. Calling it the exact mean would be inaccurate.

✔ Quick Check emgd3 — Interpretation and full problems

Question 1 of 10

Practice: Estimated Mean from Grouped Data

Use each table to calculate the estimated mean, then click Check. Give answers to 1 decimal place.

1. Homework time (mins), 30 pupils

Time (mins)	f	m	f×m
0 < t ≤ 10	4	5	20
10 < t ≤ 20	8	15	120
20 < t ≤ 30	11	25	275
30 < t ≤ 40	7	35	245
Total	30	—	660

2. Age (years) of 20 people surveyed

Age	f	m	f×m
0 < a ≤ 10	3	5	15
10 < a ≤ 20	6	15	90
20 < a ≤ 30	11	25	275
Total	20	—	380

3. Distance run (km) by 25 athletes

Distance	f	m	f×m
0 < d ≤ 5	5	2.5	12.5
5 < d ≤ 10	12	7.5	90
10 < d ≤ 15	8	12.5	100
Total	25	—	202.5

4. Weight of parcels (kg), 15 parcels

Weight	f	m	f×m
0 < w ≤ 2	4	1	4
2 < w ≤ 4	7	3	21
4 < w ≤ 6	4	5	20
Total	15	—	45

5. Texts sent per day, 40 pupils

Texts	f	m	f×m
0 < t ≤ 20	10	10	100
20 < t ≤ 40	18	30	540
40 < t ≤ 60	12	50	600
Total	40	—	1240

6. Waiting time (mins) at a bus stop, 20 people

Time	f	m	f×m
0 < t ≤ 5	4	2.5	10
5 < t ≤ 10	10	7.5	75
10 < t ≤ 15	6	12.5	75
Total	20	—	160

7. Age of trees in a park (years), 25 trees

Age	f	m	f×m
0 < a ≤ 20	5	10	50
20 < a ≤ 40	10	30	300
40 < a ≤ 60	10	50	500
Total	25	—	850

8. Customers per hour, 30 hours recorded

Customers	f	m	f×m
0 < c ≤ 10	6	5	30
10 < c ≤ 20	15	15	225
20 < c ≤ 30	9	25	225
Total	30	—	480

9. Length of phone calls (mins), 20 calls

Length	f	m	f×m
0 < l ≤ 5	10	2.5	25
5 < l ≤ 10	6	7.5	45
10 < l ≤ 15	4	12.5	50
Total	20	—	120

10. Weekly grocery spend (£), 25 households

Spend	f	m	f×m
0 < s ≤ 40	5	20	100
40 < s ≤ 80	12	60	720
80 < s ≤ 120	8	100	800
Total	25	—	1620

⚠ Things to Watch Out For

Using class boundaries instead of midpoints: The estimated mean uses the MIDPOINT of each class, not the boundary values. For the class 10–20, use 15, not 10 or 20.
Dividing by number of rows instead of total frequency: Divide Σfm by Σf (sum of all frequencies), not by the number of class intervals.
Calling it the exact mean: Always say "estimated mean" — you cannot calculate the exact mean from grouped data because individual values are unknown.
Open-ended classes: A class like "40 and over" has no upper boundary. You must make a reasonable assumption for the midpoint (e.g. treat it as 40–50, midpoint 45), and state your assumption.
Midpoint calculation error: For 20 ≤ x < 30: midpoint = (20 + 30) ÷ 2 = 25. Students sometimes write 20 + 10 = 30, or forget to divide by 2.

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Learning Objectives

Real-World Applications

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What Is the Midpoint of a Class Interval?

✔ Quick Check emgd1 — Midpoints and why we estimate

The Estimated Mean Formula

Full Worked Example — Heights of 30 students

✔ Quick Check emgd2 — Applying the estimated mean formula

Interpreting the Estimated Mean

Cinema ages example — Worked calculation

✔ Quick Check emgd3 — Interpretation and full problems

Practice: Estimated Mean from Grouped Data

⚠ Things to Watch Out For