GCSE Maths • Statistics • Primarily higher tier
GCSE Cumulative Frequency Revision Guide
Turn grouped frequencies into running totals and read a whole distribution from one graph.
Foundation
Exam practice
Worked examples
What You'll Learn
Build Cumulative Frequency Tables
Form accurate running totals and check the final value against N.
Draw Cumulative Frequency Graphs
Plot upper class boundaries against cumulative totals with suitable scales.
Estimate the Median and Quartiles
Use N/2, N/4 and 3N/4 with horizontal and vertical construction lines.
Calculate and Interpret IQR
Use Q3−Q1 to describe the spread of the middle half.
Read Percentiles and Frequencies
Use kN/100 and differences between cumulative totals.
Compare Distributions
Compare typical value using the median and spread using the IQR.
Key Rules
Lesson Modules
Cumulative Frequency Graphs
Calculate running totals, plot upper class boundaries and interpret the curve.
Median And Quartiles From Graphs
Estimate Q1, the median, Q3, percentiles and IQR using construction lines.
Core Subtopics
Worked Examples
From Grouped Frequencies to Quartiles
QUESTION
The groups 0<x≤10, 10<x≤20, 20<x≤30, 30<x≤40 and 40<x≤50 have frequencies 4, 6, 10, 10 and 10. Construct cumulative totals and use the cumulative frequency graph to estimate Q1, the median, Q3 and the IQR.
- Given information: There are N=4+6+10+10+10=40 observations.
- Method: Create running totals, plot each total at its upper class boundary, then use N/4, N/2 and 3N/4.
- Running totals: 4, 10, 20, 30, 40.
- Plot (0,0), (10,4), (20,10), (30,20), (40,30) and (50,40).
- Q1 position: 40/4=10; read across from CF 10 to obtain approximately 20.
- Median position: 40/2=20; read across from CF 20 to obtain approximately 30.
- Q3 position: 3×40/4=30; read across from CF 30 to obtain approximately 40.
- IQR≈40−20=20.
- Final answer: Q1≈20, median≈30, Q3≈40 and IQR≈20.
- Independent answer check: The last cumulative total is 40=N, and 10, 20 and 30 are the 25%, 50% and 75% positions.
- One common mistake: Graph readings from grouped data should be described as estimates, even when a plotted point makes a reading appear exact.
- Common GCSE Mistakes
- Plotting Frequencies
- Using Class Midpoints
- Forgetting the Zero Start
- Drawing a Decreasing Graph
- Using N for the Median
- Swapping Quartiles
- Reading the Wrong Axis
- Giving Exact Claims
- Adding for the IQR
- Comparing Only Medians
- Exam Tips
- Write the cumulative-frequency column before plotting.
- Check that the final running total equals N.
- Use upper class boundaries and label both axes with quantities and units.
- Choose a scale that uses most of the graph and permits accurate readings.
- Show horizontal and vertical construction lines for quartiles and percentiles.
- Use a ruler for construction lines and read to sensible precision.
- Write “estimate” when values come from grouped data or a graph.
- When comparing groups, state which has the greater median and which has the smaller or larger IQR.
- Use N−CF(x) for the number above x and CF(b)−CF(a) for the number between two values.
Ready to Practice?
Printable Worksheet
Build tables, plot curves and estimate quartiles.
Interactive Quiz
Check positions, boundaries, percentiles and IQR.
Exam-Style Questions
Interpret curves and compare two distributions.
Watch the Cumulative Frequency Tutorial
Clear, step-by-step explanation of expanding brackets, simplifying complex expressions, and solving for x. Perfect for visual learners who need a breakdown of the rules in action.
- 12 minute comprehensive guide
- High-definition whiteboard walk-through
Frequently Asked Questions
It is a running total of frequencies up to each class boundary.
It shows the total number of observations, N.
For grouped data, use upper class boundaries, not midpoints.
No. A running total can stay level or rise, but it cannot decrease.
Q1 is at N/4 and Q3 is at 3N/4.
Subtract the lower quartile from the upper quartile: Q3−Q1.
Grouped observations are not known individually, and drawing or reading the curve introduces approximation.
Compare medians for typical values and IQRs for spread; a smaller IQR means the middle 50% is less spread out.
