Grouped Data, Percentiles & Modal Interval — Grade 10 (CAPS)

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 + upper)/2

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		1 010

Worked ExampleEstimate the mean of the grouped data above

1
Work out each midpoint: (0 + 10)/2 = 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 = 1 010.
3
Estimated mean = (1 010)/40.
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 n/2th 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 = p/100 × n

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

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

1
25/100 × 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 − Q₁)/2

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

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

5Histograms

Once data is grouped into intervals, the natural picture is a histogram: one bar per interval, its height the frequency.

A histogram of 40 test marks. The bars touch because the intervals are continuous — there is no gap between 20–40 and 40–60.

6Histogram or bar graph?

A histogram shows continuous grouped data — marks, heights, times. The bars touch, because one interval ends exactly where the next begins.
A bar graph shows categorical data — favourite sport, home language. The bars have gaps, because there is nothing between the categories.
The horizontal axis of a histogram is a number line; on a bar graph it is a list of labels, and the bars could be put in any order.

Reading a histogram is the same as reading a frequency table: the tallest bar marks the modal interval, and adding the heights gives the total number of values.

Worked ExampleUsing the histogram above: how many learners wrote, and in which interval does the median lie?

1
Total = 4 + 9 + 14 + 8 + 5 = 40 learners
Add the frequencies — the heights of all five bars.
2
The median is the average of the 20th and 21st values.
n = 40 is even, so the median sits between two values.
3
Cumulative: 4, then 13, then 27.
Running totals to find which interval the 20th value falls in.
4
The 20th and 21st values both fall in 40–60, so the median lies in the interval 40 ≤ x < 60.

⚠️

Leaving gaps between the bars of a histogram. The gaps are the visual signal that data is categorical — drawing them on continuous data says the wrong thing, and papers award a mark for touching bars.

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 + 30)/2 = 25
2
Using the table (f = 4, 9, 14, 8, 5 for intervals of width 10 from 0), estimate the mean.
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(1 010)/40 = 25,25
3
Write down the modal interval for that data.
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20 ≤ x < 30 (highest frequency, 14)
4
In which interval does the median lie for that data?
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The 20th value → 20 ≤ x < 30
5
Determine the position of the 25th percentile when n = 40.
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25/100 × 40 = the 10th value
6
Q₁ = 24 and Q₃ = 42. Calculate the interquartile range.
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42 − 24 = 18
7
Q₁ = 24 and Q₃ = 42. Calculate the semi-interquartile range.
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18/2 = 9
8
Why can the mean of grouped data only be estimated?
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Because the individual values are unknown — we assume each one sits at its interval's midpoint.
9
State TWO differences between a histogram and a bar graph.
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A histogram shows continuous grouped data and a bar graph shows categorical data ✓; the bars of a histogram touch while those of a bar graph have gaps ✓
10
The bars of a histogram have frequencies 6, 10, 14, 9 and 5. How many values are in the data set, and which is the modal interval?
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6 + 10 + 14 + 9 + 5 = 44 values ✓; the modal interval is the one with frequency 14 ✓
11
Explain why the bars of a histogram must touch.
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The intervals are continuous ✓ — one interval ends exactly where the next begins, so there is no value that could fall in a gap ✓