IGCSE Mathematics: Cumulative Frequency and Histograms Practice Questions
On a cumulative frequency curve, the median is read at half the total frequency and the quartiles at one quarter and three quarters. On a histogram the vertical axis is frequency density, which is frequency divided by class width.
Topic 9.5 of Cambridge IGCSE Mathematics 0580 is the highest scoring statistics topic on the Extended papers, and the single most common error is treating a histogram like a bar chart. Frequency is the area of the bar, not its height. The questions below make that explicit.
What you need to know for Cumulative Frequency and Histograms
- Cumulative frequencyA running total of the frequencies. Plot each cumulative total against the upper boundary of its class, then join with a smooth curve.
- Reading the medianRead across from half the total frequency on the vertical axis to the curve, then down to the horizontal axis.
- QuartilesThe lower quartile is read at one quarter of the total and the upper quartile at three quarters.
- Interquartile rangeUpper quartile minus lower quartile. It measures the spread of the middle half of the data and is not affected by extreme values.
- Frequency densityFrequency divided by class width. It is what the vertical axis of a histogram shows, which is why unequal class widths can be compared fairly.
- Frequency from a histogramFrequency = frequency density x class width, which is the area of the bar.
IGCSE Mathematics Cumulative Frequency and Histograms questions and answers
4 exam-style questions written to the 0580 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
A cumulative frequency curve is drawn for 80 students. Explain how to find the median, the lower quartile and the upper quartile from the graph.
Show the worked answer
- The median is at half the total frequency: 80 divided by 2 = 40. Read across from 40 to the curve, then down to the horizontal axis.
- The lower quartile is at one quarter of the total: 80 divided by 4 = 20. Read across from 20 in the same way.
- The upper quartile is at three quarters: 3 x 80 divided by 4 = 60. Read across from 60.
- Draw the reading lines on the graph with a ruler, because Cambridge awards marks for showing the construction as well as the values.
For a set of data, the lower quartile is 24 and the upper quartile is 41. Calculate the interquartile range and explain what advantage it has over the range.
Show the worked answer
- Interquartile range = upper quartile minus lower quartile.
- 41 minus 24 = 17.
- The range uses the largest and smallest values, so a single unusually high or low value distorts it completely.
- The interquartile range covers only the middle half of the data, so extreme values at either end have no effect on it. It is therefore a more reliable measure of spread.
A histogram class covers 0 to 20 with a frequency of 30, and another covers 20 to 50 with a frequency of 45. Calculate the frequency density of each class.
Show the worked answer
- Frequency density = frequency divided by class width.
- First class: width = 20 minus 0 = 20, so frequency density = 30 divided by 20 = 1.5.
- Second class: width = 50 minus 20 = 30, so frequency density = 45 divided by 30 = 1.5.
- Both bars have the same height on the histogram, even though their frequencies differ, because the second class is wider. This is exactly why frequency density is used rather than frequency.
On a histogram, a bar has a frequency density of 2.5 and covers the class 40 to 60. Calculate the frequency of that class.
Show the worked answer
- Rearrange the frequency density formula: frequency = frequency density x class width.
- The class width is 60 minus 40 = 20.
- Frequency = 2.5 x 20.
- Frequency = 50. This is the area of the bar, which is why frequency is represented by area rather than height on a histogram.
Common mistakes in this topic
- Reading quartiles from the horizontal axis rather than the vertical one.
- Treating the height of a histogram bar as the frequency.
- Using an upper boundary as the class width.
- Plotting cumulative frequency against the midpoint rather than the upper boundary.
- Joining cumulative frequency points with straight lines when a smooth curve is expected.
Exam tips
- Always plot cumulative frequency against the upper class boundary, and start the curve at zero.
- Mark your reading lines on the graph in pencil. Marks are given for showing the method.
- On a histogram, remember area equals frequency. Height alone tells you nothing without the width.
- Add a frequency density column to any histogram table before drawing.
- Use the interquartile range rather than the range when comparing spread, and say why.
Practise 20 more questions like this, free
vStudyWise marks every answer instantly, tracks the topics you keep dropping marks on and turns them into a weekly study plan.
Cumulative Frequency and Histograms FAQs
How do I find the median from a cumulative frequency curve?
Halve the total frequency, find that value on the vertical axis, read horizontally across to the curve, then read vertically down to the horizontal axis. Draw the lines on the graph, since marks are awarded for showing the construction.
What is frequency density?
Frequency divided by class width. It is what a histogram plots on the vertical axis, which allows classes of different widths to be compared fairly. The frequency of a class is then the area of its bar, not its height.
How do I find a frequency from a histogram?
Multiply the frequency density by the class width, which gives the area of the bar. A bar with a frequency density of 2.5 covering the class 40 to 60 has a width of 20, so the frequency is 50.
Why use the interquartile range instead of the range?
The range uses only the highest and lowest values, so a single extreme value distorts it. The interquartile range measures the spread of the middle half of the data, so it is unaffected by extreme values and gives a more reliable picture of how the data is spread.
Continue through the IGCSE Mathematics syllabus
Related IGCSE Mathematics topics
See all 41 IGCSE Mathematics practice topics ›
Written to the published Cambridge IGCSE Mathematics (0580) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.