IGCSE Maths · Statistics · Extended only

Cumulative Frequency Diagrams and Histograms Extended only

This entire note is Extended-only, and it’s the final note of the whole course. Cumulative frequency diagrams reveal the spread of grouped data — the median, quartiles, and interquartile range — while histograms show the true shape of a distribution, even when class widths aren’t equal.

1. Building a Cumulative Frequency Table

A cumulative frequency table runs a “so far” total alongside each class’s upper boundary.

Worked example
ScoreFrequencyCumulative Frequency
0–2044
20–401014
40–602034
60–801246
80–100450
Each cumulative frequency adds the new class’s frequency to the running total — e.g. 14 + 20 = 34, and 34 + 12 = 46.

2. Drawing a Cumulative Frequency Diagram

Plot each point at the upper boundary of its class, against the cumulative frequency — then join with a smooth curve (an “ogive”), always starting from (lowest boundary, 0).

The diagram, using the table above Score Cumulative frequency 25 (median line)

3. Estimating the Median, Quartiles and IQR

Median position = n/2    Lower quartile (Q1) = n/4    Upper quartile (Q3) = 3n/4
Worked example — using the diagram above (n=50) Estimate the median, lower quartile, upper quartile, and interquartile range.
Median position: 50/2 = 25th value. Reading across from 25 to the curve, then down: median ≈ 51
Lower quartile position: 50/4 = 12.5th value → Q1 ≈ 37
Upper quartile position: 3(50)/4 = 37.5th value → Q3 ≈ 65.8
Interquartile range = Q3 − Q1 = 65.8 − 37 = 28.8
Try it — Median and Quartiles
  1. A cumulative frequency diagram has n=80. Find the positions (not values) used to estimate the median, Q1, and Q3.
  2. Using a diagram, the median position gives a reading of 42, Q1 gives 28, and Q3 gives 58. Find the interquartile range.
Show answers
  1. Median: 80/2=40th value. Q1: 80/4=20th value. Q3: 3(80)/4=60th value
  2. 58−28 = 30

4. Percentiles

A percentile works the same way as a quartile, but divides the data into 100 parts instead of 4. The kth percentile is found at position (k/100) × n.

Worked example Using the diagram from Section 2 (n=50), estimate the 90th percentile.
Position: (90/100) × 50 = 45th value. Reading across from 45 on the cumulative frequency axis to the curve, then down to the score axis gives an estimate of approximately 78.
Try it — Percentiles
  1. Find the position used to estimate the 20th percentile, for n=200.
  2. Find the position used to estimate the 60th percentile, for n=150.
Show answers
  1. (20/100)×200 = 40th value
  2. (60/100)×150 = 90th value

5. Histograms with Equal Class Widths

When all classes have the same width, a histogram looks just like a bar chart — except the bars touch, since the horizontal axis represents a continuous scale with no gaps between classes.

6. Histograms with Unequal Class Widths — Frequency Density

When class widths differ, using raw frequency for the bar height would be misleading (a wide class could look artificially tall just because it covers more ground). Instead, histograms always plot frequency density on the vertical axis.

Frequency density = frequency ÷ class width
Worked example Age (years) Frequency density An age distribution has classes 0–10 (freq 8), 10–20 (freq 12), 20–40 (freq 18), 40–70 (freq 15), 70–90 (freq 4). Find the frequency density for each class, and draw the histogram.
0–10: 8÷10 = 0.8    10–20: 12÷10 = 1.2    20–40: 18÷20 = 0.9    40–70: 15÷30 = 0.5    70–90: 4÷20 = 0.2
Each bar’s width matches its class width, and its height is the frequency density — so the bar’s area represents the frequency, not the height alone.
Worked example — working backwards from a histogram A histogram bar has frequency density 0.6 and covers a class width of 15. Find the frequency it represents.
Frequency = frequency density × class width = 0.6 × 15 = 9
Try it — Frequency Density
  1. A class has frequency 24 and width 8. Find the frequency density.
  2. A histogram bar has frequency density 1.5 and width 12. Find the frequency.
  3. A class 30–50 has frequency 16. Find its frequency density.
Show answers
  1. 24÷8 = 3
  2. 1.5×12 = 18
  3. Width = 50−30 = 20. Density = 16÷20 = 0.8

7. Estimating the Mean, Median and Mode from Grouped Data

These use the same estimation methods from the earlier Statistics note — midpoints for the mean, the modal class for the mode — but now you may need to first recover frequencies from a histogram’s frequency densities before calculating.

Worked example — using the histogram data above Find the modal class, and estimate the mean age.
Modal class: the tallest bar (highest frequency density, not frequency) is 10–20 (density 1.2) — this is the modal class, since it represents the most concentrated group of ages.
Estimated mean: using midpoints 5, 15, 30, 55, 80 with frequencies 8, 12, 18, 15, 4 (total 57):
Mean = (8×5 + 12×15 + 18×30 + 15×55 + 4×80) ÷ 57 = (40+180+540+825+320)/57 = 1905/57 = 33.4 (3 s.f.)
Try it — Estimating Averages
  1. Explain why the modal class is identified using frequency density, not raw frequency, when class widths are unequal.
  2. A histogram has classes 0–20 (density 0.4), 20–30 (density 1.1), 30–50 (density 0.3). Find the frequency for each class, then find the total frequency.
Show answers
  1. A wider class can have a high raw frequency just because it covers more ground, without actually being densely packed — frequency density corrects for this by showing concentration per unit width, which is what “most common” really means.
  2. 0–20: 0.4×20=8. 20–30: 1.1×10=11. 30–50: 0.3×20=6. Total=8+11+6=25
Common mistakes to watch for
  • Plotting cumulative frequency against the class midpoint instead of the upper boundary.
  • Forgetting the cumulative frequency curve must start at (lowest boundary, 0), not at the first plotted data point.
  • Using n (not n/2, n/4, or 3n/4) directly as the position when reading off the median or quartiles.
  • Plotting raw frequency instead of frequency density on a histogram with unequal class widths — this badly distorts the shape of the distribution.
  • Identifying the modal class using bar height on a bar chart-style read, without checking whether the histogram actually has equal class widths first.

8. Quick-Fire Challenge Round

True or False?
  1. Cumulative frequency points are plotted at the upper boundary of each class.
  2. The median position on a cumulative frequency diagram is found at n/2.
  3. A histogram’s bars always have the same width.
  4. Frequency density = frequency × class width.
Show answers
  1. True.
  2. True.
  3. False — histogram bar widths can differ, representing different class widths; that’s exactly why frequency density is needed.
  4. False — frequency density = frequency ÷ class width, not multiplied.
Spot the Error
  1. “For n=60, the median position is found at 60.”
  2. “A class 20–50 has frequency 12, so its frequency density is 12×30=360.”
  3. “A histogram’s tallest bar always has the highest frequency.”
Show answers
  1. Median position is n/2, not n itself: 60/2 = 30th value.
  2. Frequency density = frequency ÷ class width, not multiplied: 12÷30 = 0.4, not 360.
  3. Not necessarily — the tallest bar has the highest frequency density; if class widths differ, a shorter, wider bar could represent a larger actual frequency.

9. Mixed Practice — Bringing It All Together

  1. A cumulative frequency table gives n=120. Find the positions for the median, Q1, and Q3.
  2. A class has frequency 21 and width 7. Find the frequency density.
  3. A histogram bar has frequency density 2.4 and width 5. Find its frequency.
  4. Using the age histogram from Section 6, identify the modal class.
Show answers
  1. Median: 60th value. Q1: 30th value. Q3: 90th value
  2. 21÷7 = 3
  3. 2.4×5 = 12
  4. 10–20 (highest frequency density, 1.2)

Practice at Home

A final large set to close out the entire course — take it steadily.

Part A — Cumulative frequency tables and positions
  1. A table has class frequencies 6, 14, 22, 10, 8. Find the cumulative frequency for each class, and state n.
  2. For the n found in question 1, find the positions for the median, Q1, and Q3.
Part B — Reading estimates from a diagram
  1. A diagram gives a median reading of 46, Q1 of 32, and Q3 of 61. Find the interquartile range.
  2. Find the position used to estimate the 75th percentile, for n=160.
  3. Find the position used to estimate the 10th percentile, for n=90.
Part C — Frequency density
  1. A class 0–15 has frequency 18. Find its frequency density.
  2. A class 40–100 has frequency 24. Find its frequency density.
  3. A histogram bar has frequency density 0.9 and width 20. Find its frequency.
  4. A histogram bar has frequency density 3.2 and width 5. Find its frequency.
Part D — Estimating averages
  1. A grouped table has classes 0–20 (f=10, midpoint 10), 20–30 (f=15, midpoint 25), 30–60 (f=12, midpoint 45). Find the estimated mean.
  2. Using the same data, identify the modal class by frequency density (class widths are 20, 10, 30 respectively).
Part E — Applied / challenge
  1. A school’s cumulative frequency diagram for exam scores (n=200) gives Q1=45, median=58, Q3=72. Find the interquartile range, and explain what this value tells you about the spread of the middle 50% of scores.
  2. A histogram has classes 0–10 (density 0.5), 10–25 (density 1.4), 25–30 (density 0.8). Find the total number of data values represented across all three classes.
  3. Explain why a histogram with unequal class widths would give a misleading picture of the data if frequency (instead of frequency density) were plotted directly on the vertical axis — use a specific example with numbers to illustrate your point.

Key Vocabulary Recap

Cumulative frequency — a running total of frequencies up to each class boundary
Ogive — the S-shaped cumulative frequency curve
Quartile — one of three values dividing data into four equal parts
Interquartile range (IQR) — Q3 − Q1, the spread of the middle 50% of data
Frequency density — frequency ÷ class width, used for histogram bar heights

Syllabus Reference

This note covers Cambridge IGCSE Mathematics (0580) sections E9.6 and E9.7 (Extended subject content) — Cumulative frequency diagrams, and Histograms. There is no Core equivalent for either section. This is the final note of the entire Cambridge IGCSE Mathematics (0580) syllabus — covering all nine topics across both Core and Extended tiers.

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