GCSE specification fit
Cumulative Frequency is part of GCSE Maths Statistics.
Build cumulative frequency tables, plot upper class boundaries, and estimate the median, quartiles and interquartile range from a curve. Questions often ask you to compare two distributions using both typical value and spread.
What you will learn
Why this matters
Cumulative frequency is used for large grouped data where individual values are not listed.
Prior knowledge
You should already be comfortable with:
Clear explanation
Main idea
Cumulative frequency means running total. For grouped data, each plotted point shows how many values are less than or equal to the upper class boundary.
Method
Add the frequencies down the table, plot each upper boundary against its cumulative frequency, then draw a smooth increasing curve. Use half the total for the median, one quarter for Q1 and three quarters for Q3.
For grouped intervals such as 20 < x ≤ 30, the plotted x-value is 30 because the cumulative total counts everything up to that upper boundary. If the first class begins at 0, it is often sensible to begin the curve at (0, 0) before plotting the running totals.
Exam tip
Graph readings are estimates. Show construction lines and compare distributions with two ideas: median for typical value and IQR for spread.
Worked examples
Finding quartile positions
A grouped data set has total frequency 80.
Building the running totals
A grouped table has intervals 0 < x ≤ 10, 10 < x ≤ 20, 20 < x ≤ 30 with frequencies 7, 11 and 12.
Estimating spread from graph readings
A cumulative frequency curve represents 120 plants. From the graph, Q1 is about 18 cm, the median is about 25 cm and Q3 is about 34 cm.
Quick checks
Choose an answer, then check your thinking.
1. Which class value do you plot on a cumulative frequency graph?
2. For 120 values, where is the median read?
Practice questions
Question 1
A grouped table records 6, 9, 15 and 10 pupils in four time intervals for a homework task. Write the cumulative frequencies ready for plotting.
Reveal answer and marking guidance
Answer: 6, 15, 30, 40.
Marking: Add each new frequency to the running total.
Question 2
A cumulative frequency graph represents 64 students. At which cumulative frequency should you draw the horizontal line to read the median from the curve?
Reveal answer and marking guidance
Answer: 32.
Marking: The median is at half the total frequency: 64 ÷ 2 = 32.
Question 3
For the same 64 students, at which cumulative frequencies should you read Q1 and Q3 before using construction lines on the graph?
Reveal answer and marking guidance
Answer: Q1 at 16 and Q3 at 48.
Marking: Use one quarter and three quarters of 64.
Question 4
A cumulative frequency graph for journey times gives Q1 = 22 minutes and Q3 = 37 minutes. Find the interquartile range and include the unit.
Reveal answer and marking guidance
Answer: 15 minutes.
Marking: IQR = Q3 − Q1 = 37 − 22 = 15 minutes.
Question 5
A grouped table for waiting times uses intervals 0 < x ≤ 10, 10 < x ≤ 20, 20 < x ≤ 40 and 40 < x ≤ 60 with frequencies 8, 12, 25 and 15. Write the points you would plot for a cumulative frequency curve.
Reveal answer and marking guidance
Answer: (10, 8), (20, 20), (40, 45), (60, 60).
Marking: Use upper class boundaries for the x-coordinates and running totals for the y-coordinates.
Question 6
Two cumulative frequency curves each represent 80 values. Group A has median 34 and IQR 12. Group B has median 31 and IQR 20. Compare the two groups.
Reveal answer and marking guidance
Answer: Group A has the higher typical value and is more consistent, because its median is higher and its IQR is smaller.
Marking: Mention both median and IQR, and use comparative language rather than only listing the numbers.
Question 7
Frequencies in four classes are 4, 13, 18 and 5. Write the final cumulative frequency and explain what it represents.
Reveal answer and marking guidance
Answer: 40; it represents the total number of values.
Marking: Add all frequencies: 4 + 13 + 18 + 5 = 40.
Question 8
A grouped table uses classes 0 < x ≤ 5, 5 < x ≤ 15 and 15 < x ≤ 25 with frequencies 3, 9 and 8. Write the cumulative frequency plotting points.
Reveal answer and marking guidance
Answer: (5, 3), (15, 12), (25, 20).
Marking: Use upper class boundaries and cumulative totals, not class widths or midpoints.
Question 9
A cumulative frequency curve for 100 runners gives Q1 = 18 minutes and Q3 = 31 minutes. Find the interquartile range and interpret it.
Reveal answer and marking guidance
Answer: IQR = 13 minutes; the middle 50% of runners' times are spread over about 13 minutes.
Marking: Subtract Q1 from Q3 and describe the spread in the context.
Question 10
A cumulative frequency curve represents 96 delivery times. State the cumulative frequencies for Q1, the median and Q3. The graph readings are Q1 = 14 minutes, median = 21 minutes and Q3 = 33 minutes. Find the IQR.
Reveal answer and marking guidance
Answer: Q1 is read at 24, the median at 48 and Q3 at 72. The IQR is 19 minutes.
Marking: Use one quarter, one half and three quarters of 96 for the graph positions. Then subtract the graph readings: 33 − 14 = 19 minutes.
Answers and marking guidance
The exact practice answers are hidden under each question so you can try first. For cumulative frequency, marks usually come from correct running totals, plotting upper class boundaries, drawing a smooth increasing curve and reading estimates with construction lines. When comparing two distributions, mention the median for the typical value and the interquartile range for consistency or spread.
Common mistakes
- Plotting midpoints: cumulative frequency graphs use upper class boundaries, not class midpoints.
- Losing the running total: each cumulative frequency must include all previous groups.
- Using quartile positions incorrectly: Q1, median and Q3 are read at one quarter, one half and three quarters of the total frequency.
- Only comparing medians: distribution comparisons usually need a comment about spread as well.
Extension challenge
Create a GCSE-style question on cumulative frequency, solve it, then write one sentence explaining why your method works.
Reveal answer
Example answer: A good answer includes a correct method, a checked final answer and a short reason using the key vocabulary from this lesson.
Exam-board guidance
Cumulative Frequency appears within GCSE Maths statistics, especially for grouped data and distribution comparison. The shared skill is to build running totals, plot the curve accurately, estimate median and quartiles, and explain what those estimates mean in context.
AQA GCSE Maths
Plot upper class boundaries against cumulative frequency, show construction lines, then read the median, quartiles and IQR as estimates from the curve.
OCR GCSE Maths
Keep the running total accurate, use the total frequency to locate quartiles, and compare data sets using both median and IQR in context.
Pearson Edexcel GCSE Maths
Plot at upper class boundaries, start from the lower boundary when appropriate, and remember graph readings are estimates rather than exact data values.
Eduqas GCSE Maths
Show the cumulative totals and construction lines so the examiner can see how you estimated the median, quartiles and interquartile range.
WJEC Wales
Expect real-data contexts; state what a higher median or larger IQR means for the situation rather than just quoting two numbers.
CCEA GCSE Maths
Check the unit and scale on each axis, and write comparison sentences using median for typical value and IQR for consistency or spread.
Next lesson
Next, continue with Box Plots.