Ch 3 · 9.6–9.7 Cumulative Frequency Diagrams and Histograms · IGCSE Mathematics 0580
Topic 3 of 3 in Statistics — IGCSE Mathematics 0580 — 18 lessons.
Introduction
For large grouped data sets, two diagrams do the heavy lifting. A cumulative frequency curve gives the median and quartiles, so you can compare spreads. A histogram shows the shape of a distribution correctly even when the class widths are unequal. (Syllabus sections 9.6 and 9.7.)
Key ideas
Add up the frequencies as you go: the cumulative frequency is the running total up to the upper boundary of each class. Plot each running total against its upper boundary (small crosses), start from the lower boundary of the first class at 0, and join with a smooth curve.
🧑🏫 Teacher's explanation
Imagine you are collecting marbles in jars. If you have 5 red, 8 blue, and 3 green marbles, the cumulative frequency isn't just the count in each jar, but the 'running total' of all marbles collected so far. It tells you how many items fall below a certain threshold.
📖 Definitions
🔍 Worked example — Easy
A set of data has n=40 values. Using a cumulative frequency diagram, estimate the median and the interquartile range if Q_1 is at 25 and Q_3 is at 55.
🔍 Worked example — Standard
A histogram has a class interval from 10 ≤ x < 30 with a frequency of 40. Calculate the frequency density for this bar.
🔍 Worked example — Exam-style
In a histogram, a bar has a frequency density of 5 and a class width of 4. Calculate the frequency of this class.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples
Journey times of 40 students: frequencies 6, 14, 11, 7, 2 for classes of width 10 from 0 to 50 minutes. Cumulative frequencies: 6, 20, 31, 38, 40 at 10, 20, 30, 40, 50 minutes. Median at 20th value ≈ 20 minutes. Q_1 (10th value) ≈ 13 minutes; Q_3 (30th) ≈ 29 minutes, so the IQR ≈ 16 minutes. (Readings from a smooth curve vary slightly.)
🔍 Worked example — Easy
A group of 50 students took a test. The cumulative frequencies for the marks are: 10 marks arrow 5 students, 20 marks arrow 15 students, 30 marks arrow 30 students, 40 marks arrow 45 students, 50 marks arrow 50 students. Estimate the median and the interquartile range.
🔍 Worked example — Standard
A histogram is drawn for the following data: Class 10 < x ≤ 20 has frequency 40; Class 20 < x ≤ 30 has frequency 30; Class 30 < x ≤ 50 has frequency 60. Calculate the frequency density for each class.
🔍 Worked example — Exam-style
In a histogram, a bar represents a class with a frequency of 45 and a class width of 15. Another bar represents a class with a frequency of 30 and a class width of 5. Which class has a higher frequency density?
⚠️ Common errors
Common mistakes
Practice
Answers: (1) 20, 40 and 60. (2) 9 cm. (3) 2.4 × 15 = 36. (4) 36 ÷ 30 = 1.2.