GCSE MATHS • HIGHER TIER
Cumulative Frequency Graphs Made Simple
Learn how to calculate cumulative frequency, draw cumulative frequency graphs, and estimate medians, quartiles, interquartile range and percentiles.
Clear method
Exam practice
Worked proofs
Cumulative Frequency Graphs in Four Moves
1
Calculate
Calculate the running cumulative totals.
2
Plot
Plot cumulative frequency against each upper class boundary.
3
Draw
Join the points using a smooth increasing curve.
4
Interpret
Use the graph to estimate medians, quartiles, IQR and percentiles.
What You’ll Learn
Understand Cumulative Frequency
Understand cumulative frequency as a running total of frequencies.
Draw the Graph
Plot upper class boundaries against cumulative frequencies.
Find Median and Quartiles
Estimate the median, lower quartile and upper quartile from the graph.
Calculate IQR
Use Q3 − Q1 to find the interquartile range.
Find Percentiles
Locate percentile positions using cumulative frequency.
Key Rules
What Is Cumulative Frequency?
Cumulative frequency is a running total of frequencies. Instead of showing the frequency for each class interval separately, cumulative frequency shows the total frequency up to a particular value. This allows us to see how data accumulates across a dataset.
What Is a Cumulative Frequency Graph?
A cumulative frequency graph plots cumulative frequency against the upper class boundary of each class interval. The graph usually forms a smooth increasing curve because cumulative frequency can only stay the same or increase.
Worked Proof: Cumulative Frequency Graphs
How to Calculate Cumulative Frequency
- 1. Start with the first frequency.
- 2. Add the next frequency.
- 3. Continue adding frequencies as you move through the table.
- 4. Record each running total.
- These cumulative totals are then plotted on a graph.
More Worked Examples
1
Finding the Median
A cumulative frequency graph represents 80 data values.
Step 1: Find the middle value.
80 ÷ 2 = 40.
Step 2: Locate 40 on the cumulative frequency axis.
Step 3: Draw a horizontal line to the curve and then a vertical line down to the data axis.
80 ÷ 2 = 40.
Step 2: Locate 40 on the cumulative frequency axis.
Step 3: Draw a horizontal line to the curve and then a vertical line down to the data axis.
2
Finding the Lower Quartile
A dataset contains 120 values.
Step 1:Calculate one-quarter of the data.
120 ÷ 4 = 30.
Step 2:Locate cumulative frequency 30.
Step 3:Read the corresponding value from the graph.
120 ÷ 4 = 30.
Step 2:Locate cumulative frequency 30.
Step 3:Read the corresponding value from the graph.
3
Finding the Upper Quartile
Using the same dataset of 120 values:
Step 1: Calculate three-quarters of the data.
120 × 3 ÷ 4 = 90.
Step 2: Locate cumulative frequency 90.
Step 3: Read the corresponding value.
120 × 3 ÷ 4 = 90.
Step 2: Locate cumulative frequency 90.
Step 3: Read the corresponding value.
4
Finding a Percentile
A dataset contains 200 values. Find the 90th percentile.
Step 1: Calculate 90% of 200.
0.9 × 200 = 180.
Step 2: Locate cumulative frequency 180.
Step 3: Read the corresponding value from the graph.
0.9 × 200 = 180.
Step 2: Locate cumulative frequency 180.
Step 3: Read the corresponding value from the graph.
- Common GCSE Mistakes
- Wrong x-values: Use upper class boundaries, not class midpoints.
- Wrong totals: Plot cumulative frequencies, not individual frequencies.
- Wrong graph style: Draw a smooth increasing curve, not a bar chart.
- Quartile errors: Use the correct cumulative-frequency positions.
- Reading errors: Use clear guide lines when reading from the graph.
- Exam Tips
- Build totals carefully: Check each running total before plotting.
- Use upper boundaries: Plot the end of each class interval.
- Label axes clearly: Show the variable and cumulative frequency.
- Show guide lines: Draw horizontal and vertical lines for estimates.
- Compare spread: Use IQR when comparing two distributions.
Skills Used in Cumulative Frequency Graphs
Grouped Frequency Tables
Histograms
Interpreting Histograms
Box Plots
Averages
Statistics
Ready to Practise?
Printable Worksheet
Download and print practice questions with answers.
Interactive Quiz
Test your knowledge with instant feedback and hints.
Exam-Style Questions
Timed questions to build confidence for the real exam.
Watch the Cumulative Frequency Graphs Tutorial
Step-by-step walkthroughs of Cumulative Frequency Graphs with clear methods and exam tips.
Frequently Asked Questions
Cumulative frequency is a running total of frequencies.
Plot upper class boundaries on the horizontal axis and cumulative frequencies on the vertical axis.
Find half the total frequency, locate it on the cumulative-frequency axis, and read across and down from the curve.
Use one-quarter and three-quarters of the total frequency, then read the corresponding values from the graph.
Subtract the lower quartile from the upper quartile: IQR = Q3 − Q1.
Calculate the required percentage of the total frequency and read the corresponding value from the graph.
