Grouped Data, Percentiles & Modal Interval (Grade 10)

Estimating the mean of grouped data using midpoints, identifying the modal interval and the interval containing the median, and finding percentiles and the semi-interquartile range.

When data is grouped into intervals you no longer know the individual values, so the mean can only be estimated.

1The estimated mean

estimated mean = ∑(midpoint × frequency)∑ frequency

midpoint = lower + upper2

Interval	f	midpoint	f × mid
0 ≤ x < 10	4	5	20
10 ≤ x < 20	9	15	135
20 ≤ x < 30	14	25	350
30 ≤ x < 40	8	35	280
40 ≤ x < 50	5	45	225
Total	40		1010

Worked ExampleEstimate the mean of the grouped data above

1
Work out each midpoint: 0 + 102 = 5, then 15, 25, 35, 45.
Assume every value in an interval sits at its midpoint.
2
Multiply each midpoint by its frequency and add: 20 + 135 + 350 + 280 + 225 = 1010.
3
Estimated mean = 101040.
Divide by the total frequency.
4
= 25,25.

⚠️

Do not divide by the number of intervals (5). Divide by the total frequency (40).

2Modal interval and the median interval

Modal interval — the interval with the highest frequency. Here 20 ≤ x < 30 (frequency 14).
Median interval — the interval containing the n2th value. Here the 20th value, which falls in 20 ≤ x < 30.

💡

You cannot give an exact mode or median for grouped data — only the interval it lies in. Writing a single number instead of an interval loses the mark.

3Percentiles and quartiles

position of the p-th percentile = p100 × n

Q₁ = 25th percentile · median = 50th · Q₃ = 75th

Worked ExampleDetermine the position of the 25th percentile for n = 40

1
25100 × 40 = 10.
2
So Q₁ is at the 10th value.
Read this off an ogive or a cumulative frequency column.

4Semi-interquartile range

IQR = Q₃ − Q₁ · semi-IQR = Q₃ − Q₁2

Worked ExampleQ₁ = 24 and Q₃ = 42. Calculate the IQR and the semi-IQR.

1
IQR = 42 − 24 = 18.
2
Semi-IQR = 182 = 9.
Simply half the IQR.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Write down the midpoint of the interval 20 ≤ x < 30.
Show answer ▾Hide answer ▴
20 + 302 = 25
2
Using the table (f = 4, 9, 14, 8, 5 for intervals of width 10 from 0), estimate the mean.
Show answer ▾Hide answer ▴
101040 = 25,25
3
Write down the modal interval for that data.
Show answer ▾Hide answer ▴
20 ≤ x < 30 (highest frequency, 14)
4
In which interval does the median lie for that data?
Show answer ▾Hide answer ▴
The 20th value → 20 ≤ x < 30
5
Determine the position of the 25th percentile when n = 40.
Show answer ▾Hide answer ▴
25100 × 40 = the 10th value
6
Q₁ = 24 and Q₃ = 42. Calculate the interquartile range.
Show answer ▾Hide answer ▴
42 − 24 = 18
7
Q₁ = 24 and Q₃ = 42. Calculate the semi-interquartile range.
Show answer ▾Hide answer ▴
182 = 9
8
Why can the mean of grouped data only be estimated?
Show answer ▾Hide answer ▴
Because the individual values are unknown — we assume each one sits at its interval's midpoint.