Cumulative Frequency
Cumulative Frequency
Lesson duration is: 60 minItems in this lesson
Cumulative Frequency
Open with the idea that cumulative frequency is about running totals, not separate class frequencies. This helps students connect the table and the graph from the start.
Learning goal
I can interpret and use cumulative frequency to describe distributions and estimate the median, quartiles, percentiles and interquartile range.
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What words do you link with cumulative frequency?
Use this to surface prior ideas such as running total, graph, median, quartiles, estimate or total frequency. Pick out misconceptions like 'just frequency' for discussion.
What cumulative frequency represents
Frequency tells us how many values are in one class interval.
Cumulative frequency tells us how many values are at or below a given boundary.
It is a running total.
As you move through the classes in order, cumulative frequency stays the same or increases.
The final cumulative frequency is the total number of values.
Emphasise 'at or below' because students often confuse cumulative frequency with the frequency in a class alone.
From frequency to cumulative frequency
Up to 10: 3
Up to 20: 8
Up to 30: 15
Up to 40: 19
Up to 50: 20
Mass (kg)
0–10: 3
10–20: 5
20–30: 7
30–40: 4
40–50: 1
Grouped frequency table
Cumulative frequency
Model the running total clearly: 3, then 3+5=8, then 8+7=15, and so on. Keep the wording consistent with upper class boundaries.
How to calculate it
Work through the classes in order.
Start with the first frequency.
Add the next frequency to get the next total.
Keep adding each new class frequency.
Write each total against the upper class boundary.
This creates a cumulative frequency table ready for plotting.
If students have seen class boundaries before, remind them that the graph uses boundaries rather than interval labels.
Why the graph rises
Each point shows a total so far.
Totals cannot decrease.
Steeper sections mean frequencies are larger in that part of the distribution.
Flatter sections mean fewer values are being added.
Students often notice the shape before they understand it. Use the steepness to preview interpretation of distributions later.
Constructing a cumulative frequency graph
Put the class boundaries on the horizontal axis.
Put the cumulative frequency on the vertical axis.
Plot each upper boundary against its cumulative frequency.
Include the starting point at the lower boundary with cumulative frequency 0.
Join the points with a smooth curve.
For the example table, the plotted points would be (0,0), (10,3), (20,8), (30,15), (40,19), (50,20).
What does cumulative frequency tell you?
Values in one class
A running total
Class width only
Greatest value only
Checks the key distinction from ordinary frequency. 'Values in one class' is the most likely misconception.
Building the table and graph
To calculate cumulative frequency, work through the classes in order and keep a running total. On the graph, plot each upper class boundary against its cumulative frequency, and include the starting point with cumulative frequency 0.
All four blanks come directly from the method slides: in order, running total, upper class boundary, starting point at 0.
Why can a cumulative frequency graph not go down?
Expected elements: cumulative frequency cannot go down because each point is a total so far; new frequencies are added, so the total stays the same or increases.
Reading the graph
To estimate a value from the graph:
move across from a cumulative frequency,
meet the curve,
then move down to the horizontal axis.
To estimate how many values are below a threshold, start at the threshold on the horizontal axis and read up to the curve.
Stress that students should read in the correct direction depending on the question: from vertical axis to find a value, or from horizontal axis to find a number of values.
Finding the median
The median is the middle value in the data set.
First find the total frequency, n.
The median is at the 50th percentile.
On a cumulative frequency graph, use the position n ÷ 2 on the vertical axis.
Read across to the curve, then down to estimate the median value.
If n = 20, the median is read at cumulative frequency 10.
Link this directly to the final cumulative frequency being the total number of values.
Finding the quartiles
The lower quartile is the value one quarter of the way through the data.
The upper quartile is the value three quarters of the way through.
Use positions n ÷ 4 and 3n ÷ 4 on the cumulative frequency axis.
Read across to the curve and then down to the horizontal axis.
If n = 20:
lower quartile at 5
upper quartile at 15
Keep the method visual and procedural: locate the position on the vertical axis first, then trace to the graph.
Interquartile range
The interquartile range (IQR) measures the spread of the middle half of the data.
IQR = upper quartile − lower quartile
A larger IQR means the middle 50% is more spread out.
A smaller IQR means it is more tightly grouped.
Mention that the IQR is useful because it is less affected by extreme values than the full range.
Finding percentiles
A percentile gives the value below which a certain percentage of the data lies.
10th percentile → position 0.1n
90th percentile → position 0.9n
Find that position on the cumulative frequency axis.
Read across to the curve and then down to estimate the value.
With n = 20, these positions are 2 and 18.
Students may already know median as the 50th percentile, so use that link to make percentiles feel familiar.
Values below or above a threshold
A cumulative frequency graph can answer threshold questions.
Below a value: read the cumulative frequency at that value.
Above a value: subtract that cumulative frequency from the total frequency.
Example with total frequency 20:
below 25 kg → read the graph at 25
above 25 kg → 20 − (number below 25 kg)
Students often forget that 'above' needs subtraction from the total. Make that contrast explicit.
If the total frequency is 20, where is the median read?
At 5
At 10
At 15
At 20
Uses n ÷ 2. The distractors match quartile positions or the total frequency.
Reading from the graph
The median is the 50th percentile, so on a cumulative frequency graph we use position n ÷ 2 on the vertical axis. The lower and upper quartiles are at n ÷ 4 and 3n ÷ 4.
Reinforces the positional method before students move to percentiles and threshold questions.
How do you find the number above 25 kg?
Read at 25 kg
Add to the total
Use 25 ÷ 20
Total minus below 25
Targets the common error of forgetting to subtract from the total for an 'above' question.
Estimating from grouped data
What the graph gives
Why estimates are needed
Estimates, not exact values
Values are spread across each class interval
The curve helps locate positions between boundaries
Grouped data does not show every individual value
Median, quartiles and percentiles may fall inside a class
We read an approximate value from the curve
This is the key idea behind grouped-data cumulative frequency: the exact raw values are unknown, so graph answers are estimates.
Describing a distribution
Cumulative frequency helps describe how data is distributed.
The median shows a typical central value.
The quartiles show where the middle half lies.
The IQR shows spread.
A steeper part of the curve shows many values packed into a small interval.
A flatter part shows fewer values in that interval.
Keep this tied to interpretation rather than formal shape language. The lesson goal is describing distribution using cumulative frequency information.
Summary of the method
Frequency table → cumulative frequency table → graph → estimates
Add frequencies in order.
Plot upper class boundaries against cumulative frequencies.
Use positions on the vertical axis for median, quartiles and percentiles.
Read values from the horizontal axis.
Use the total frequency to answer 'above' questions.
This slide ties the whole lesson arc together and reinforces the sequence students should follow in exam questions.
Why are values from a cumulative frequency graph often estimates?
Axes are uneven
The curve is straight
Data is grouped
Totals decrease
Links directly to the grouped-data explanation: individual raw values are not known exactly.
Describing the distribution
The interquartile range, or IQR, is found by subtracting the lower quartile from the upper quartile. A steeper part of a cumulative frequency curve shows many values packed into a small interval, while a flatter part shows fewer values there.
Combines IQR and curve-shape interpretation from the final teaching block.
How can this graph help describe a distribution?
Expected elements: mention centre and spread, for example median for a typical value and quartiles or IQR for how spread out the middle half is; curve steepness/flatter sections may also be referenced.
What is one idea from today that you will remember?
Reflection prompt with no right answer. Encourage students to name one concrete idea such as median at n ÷ 2, quartiles, IQR, or above/below thresholds.
What helped you understand cumulative frequency best?
Reflection prompt with no right answer. Invite students to mention a helpful representation, for example the running-total table, the graph-reading steps, or the summary method.
Glossary
frequency - the number of values in a class interval
cumulative frequency - the running total of frequencies up to a given value
class interval - a group of values within a range
class boundary - the value used to separate class intervals on a graph
median - the middle value, or 50th percentile, of a 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
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Glossary
interquartile range - the difference between the upper quartile and lower quartile
percentile - the value below which a given percentage of the data lies
distribution - the way data is spread across different values
estimate - a value read approximately rather than exactly
threshold - a chosen value used to compare how many data points are below or above it
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