What is cumulative frequency?
Think adding up.
Cumulative frequency is a running total.
Cumulative Frequency
Cumulative frequency shows the total number of values up to a certain point. Graphs based on this help estimate medians and quartiles.
Overview
Cumulative frequency means a running total of frequencies.
Instead of showing how many values are in each class only, it shows how many values are up to that point.
Cumulative frequency = running total of frequency
A cumulative frequency graph helps estimate important values such as the median, lower quartile, upper quartile and interquartile range.
What you should understand after this topic
- Calculate cumulative frequency
- Complete a cumulative frequency table
- Draw a cumulative frequency graph
- Find the median and quartiles from the graph
- Estimate the interquartile range
Key Definitions
Frequency
The number of values in a class interval.
Cumulative Frequency
The running total of frequencies.
Median
The middle value in the data set.
Lower Quartile
The value one quarter of the way through the data.
Upper Quartile
The value three quarters of the way through the data.
Interquartile Range
The difference between the upper quartile and lower quartile.
Key Rules
Add as you go
Each cumulative frequency includes all the earlier classes.
Use upper class boundaries
When drawing the graph, plot cumulative frequency against the upper boundary.
Draw a smooth curve
Do not join the points with bar-chart style blocks.
Use total frequency
The median and quartiles depend on the total number of values.
How to Solve
What is cumulative frequency?
Cumulative frequency means a running total of frequencies. Each value tells you how many data values are up to that point.
Cumulative frequency is often used with frequency tables.
Step 1: Work out cumulative frequency
Start with the first frequency, then keep adding the next frequency to the running total.
First class: \(4\)
Second class: \(4 + 7 = 11\)
Third class: \(11 + 9 = 20\)
Fourth class: \(20 + 6 = 26\)
Fifth class: \(26 + 4 = 30\)
Exam tip: The final cumulative frequency should equal the total frequency.
Step 2: Plot the graph
Plot cumulative frequency against the upper class boundary.
Plot the points: \((10,4), (20,11), (30,20), (40,26), (50,30)\).
Then draw a smooth increasing curve through the points.
Why this matters: The graph must increase because cumulative frequency is a running total.
Cumulative frequency graph placeholder
Step 3: Find the median
The total frequency is \(30\), so the median is found halfway through the data.
Median position = \( \dfrac{30}{2} = 15 \)
Go to 15 on the cumulative frequency axis.
Draw across to the curve, then down to the horizontal axis to estimate the median.
Step 4: Find the quartiles
Quartiles split the data into four equal parts. Use the cumulative frequency axis to locate them.
Lower quartile position = \( \dfrac{30}{4} = 7.5 \)
Upper quartile position = \( \dfrac{3 \times 30}{4} = 22.5 \)
Read the x-values from the graph at cumulative frequencies 7.5 and 22.5.
Step 5: Find the interquartile range
\( \text{IQR} = Q_3 - Q_1 \)
Subtract the lower quartile from the upper quartile.
This links closely to box plots, which also use quartiles and interquartile range.
What the graph shows
Exam tip: Always use the graph to estimate values, so answers may not be exact.
Median
The middle estimated value of the data.
Quartiles
They split the data into four equal parts.
Interquartile Range
A measure of spread for the middle half of the data.
Shape of the curve
A steeper section means more data is packed into that interval.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics, focusing on cumulative frequency tables.
Edexcel
The table shows the number of minutes spent on homework.
| Time (minutes) | Frequency | Cumulative frequency |
|---|---|---|
| 0 < t ≤ 10 | 3 | |
| 10 < t ≤ 20 | 5 | |
| 20 < t ≤ 30 | 4 | |
| 30 < t ≤ 40 | 6 |
Complete the cumulative frequency column.
Edexcel
The total frequency is 40.
State the cumulative frequency position used to find the median.
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on quartiles and interquartile range.
AQA
The total frequency is 36.
Find the position of the lower quartile.
AQA
The total frequency is 36.
Find the position of the upper quartile.
AQA
A cumulative frequency graph shows that \(Q_1 = 18\) and \(Q_3 = 31\).
Find the interquartile range.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising reading from cumulative frequency graphs.
OCR
The cumulative frequency graph shows the distribution of test scores.
Estimate the median score from the graph.
OCR
Explain why a cumulative frequency graph is always increasing.
Exam Checklist
Step 1
Find the running total for each class.
Step 2
Plot against the upper class boundaries.
Step 3
Draw a smooth cumulative frequency curve.
Step 4
Use \( \dfrac{n}{2}, \dfrac{n}{4}, \dfrac{3n}{4} \) to find the median and quartiles.
Most common exam mistakes
Wrong totals
Not adding the frequencies cumulatively.
Wrong plotting
Using the wrong x-values when drawing the graph.
Wrong positions
Using the wrong median or quartile position.
Reading errors
Misreading the graph when estimating values.
Common Mistakes
These are common mistakes students make when working with cumulative frequency tables and graphs in GCSE Maths.
Not using a running total
Incorrect
A student writes the frequencies without adding them cumulatively.
Correct
Cumulative frequency must be a running total. Each value should include all previous frequencies added together.
Plotting against the midpoint instead of the upper boundary
Incorrect
A student plots points using class midpoints.
Correct
Cumulative frequency graphs must be plotted against the upper class boundary, not the midpoint.
Joining the graph incorrectly
Incorrect
A student draws straight lines between points or leaves gaps.
Correct
Cumulative frequency graphs should be drawn as a smooth curve passing through the plotted points.
Using the wrong total for median and quartiles
Incorrect
A student calculates the median using an incorrect total frequency.
Correct
Use the total cumulative frequency to find positions such as median (n/2), lower quartile (n/4) and upper quartile (3n/4).
Reading values inaccurately
Incorrect
A student estimates values poorly from the graph.
Correct
Draw careful horizontal and vertical lines from the graph to read values accurately, especially when estimating medians or quartiles.
Try It Yourself
Practise constructing and interpreting cumulative frequency graphs.
Foundation
Foundation Practice
Understand cumulative frequency and calculate totals.
Frequencies are: 3, 5, 2. What is the final cumulative frequency?
Add all values.
3 + 5 + 2 = 10.
Frequencies are: 4, 6, 10. What is the cumulative frequency after the second value?
Add first two.
4 + 6 = 10.
Frequencies: 2, 3, 5, 4. Find cumulative frequency after third value.
Add first three.
2 + 3 + 5 = 10.
What does the final cumulative frequency represent?
Think total.
Final cumulative frequency = total.
Frequencies: 6, 4, 10. What is total?
Add all values.
6 + 4 + 10 = 20.
A student forgets to add previous values. What is wrong?
Think running total.
Cumulative means adding previous values.
Frequencies: 5, 5, 5. What is cumulative frequency at end?
Add all.
5 + 5 + 5 = 15.
Cumulative frequency always:
Think adding.
It never decreases.
Frequencies: 1, 2, 3, 4. Find cumulative frequency after last value.
Add all values.
1 + 2 + 3 + 4 = 10.
Higher
Higher Practice
Use cumulative frequency to find median, quartiles and interpret graphs.
A data set has 40 values. Where is the median?
Halfway point.
Median = 40 ÷ 2 = 20th value.
A data set has 60 values. Find the position of the lower quartile.
¼ of total.
60 ÷ 4 = 15.
A data set has 80 values. Where is the upper quartile?
¾ of total.
80 × 3/4 = 60.
A data set has 100 values. Find the median position.
Half of total.
100 ÷ 2 = 50.
What is the interquartile range (IQR)?
Think spread.
IQR = Q3 − Q1.
Lower quartile = 20, upper quartile = 50. Find IQR.
Subtract.
50 − 20 = 30.
A student uses total frequency instead of cumulative graph to find median. What is wrong?
Think graph reading.
Median comes from cumulative graph.
A data set has 120 values. Find position of upper quartile.
¾ of total.
120 × 3/4 = 90.
Why is cumulative frequency useful?
Think summary.
Used for medians and quartiles.
Lower quartile = 15, upper quartile = 45. Find IQR.
Subtract.
45 − 15 = 30.
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What does cumulative mean?
Running total.
What can you estimate?
Median and quartiles.
What graph is used?
Cumulative frequency curve.