Statistics
Grade 6-7Cumulative Frequency Graphs
A cumulative frequency graph plots the running total of frequencies against value, and its S-shaped curve lets you estimate the median, quartiles and interquartile range, or read off how many items fall below or between given values. This lesson covers reading a value from the curve, estimating the median, finding the quartiles and interquartile range, and finding the number of values between two points.
What you need to know
- Cumulative frequency is a running total of the frequencies.
- Plot each cumulative total against the upper boundary of its class.
- The curve is an S shape; read the median at n over 2 and the quartiles at n over 4 and 3n over 4.
- Reading up from a value tells you how many pieces of data are below it.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Reading a cumulative frequency value
To find how many items are below a given value, find that value on the horizontal axis, trace up to the curve, then trace across to the vertical axis to read the cumulative frequency.
A graph shows the cumulative frequency for the time taken by 80 runners to finish a race. Tracing up from 33 minutes to the curve and across gives a cumulative frequency of 33, so about 33 runners took less than 33 minutes.
Estimating the median
The median is the value at half the total frequency. Find n ÷ 2 on the vertical axis, trace across to the curve, then down to the horizontal axis to read the estimate.
With 80 runners, the median lies at cumulative frequency 80 ÷ 2 = 40. Tracing across to the curve and down gives an estimated median time of 34.5 minutes.
Finding the quartiles and interquartile range
The lower quartile is the value at one quarter of the total frequency, n ÷ 4, and the upper quartile is the value at three quarters, 3n ÷ 4. The interquartile range is the upper quartile minus the lower quartile.
With 80 runners, the lower quartile is at cumulative frequency 20, giving 30 minutes, and the upper quartile is at cumulative frequency 60, giving 39.5 minutes. The interquartile range is 39.5 − 30 = 9.5 minutes.
Finding the number of values between two points
Read the cumulative frequency at each of the two values separately, then subtract the smaller cumulative frequency from the larger to find how many values lie between them.
Reading the cumulative frequency at 33 minutes gives 33, and at 42 minutes gives 68. The number of runners who took between 33 and 42 minutes is 68 − 33 = 35.
Worked examples
Three exam-style questions, fully solved
The graph shows the cumulative frequency for the time taken by 80 runners to finish a race. Use the graph to estimate the number of runners who took less than 33 minutes.
Easy- 1.Find 33 minutes on the horizontal axis and trace up to the curve
- 2.Trace across to the vertical axis to read the cumulative frequency
Answer: 33 runners
The graph shows the cumulative frequency for the time taken by 80 runners to finish a race. Use the graph to estimate the median time.
Medium- 1.Find half the total frequency: 80 ÷ 2 = 40
- 2.Trace across from 40 on the vertical axis to the curve, then down to the horizontal axis
Answer: 34.5 minutes
The graph shows the cumulative frequency for the time taken by 80 runners to finish a race. Use the graph to estimate the interquartile range.
Hard- 1.Find the upper quartile at cumulative frequency 80 × ¾ = 60: 39.5 minutes
- 2.Find the lower quartile at cumulative frequency 80 ÷ 4 = 20: 30 minutes
- 3.Subtract to find the interquartile range: 39.5 − 30
Answer: 9.5 minutes
Practice
6 questions, marked instantly
Type an answer and check it. The worked solution appears once you have had a go. No login needed, and your progress saves in this browser.
Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
A cumulative frequency graph is drawn for 40 values. At what cumulative frequency do you read across to estimate the median?
For those 40 values, at what cumulative frequency do you read across to estimate the lower quartile?
For those 40 values, at what cumulative frequency do you read across to estimate the upper quartile?
From a cumulative frequency graph, the lower quartile is 22 and the upper quartile is 38. Work out the interquartile range.
A cumulative frequency graph for 80 people shows a cumulative frequency of 60 at a time of 25 minutes. How many people took longer than 25 minutes?
A cumulative frequency table shows: up to 10 → 5, up to 20 → 18, up to 30 → 30. How many values lie in the class from 10 to 20?
Avoid these
The mistakes students make most often
Reading the graph in the wrong direction, such as starting from the vertical axis when a value on the horizontal axis is given, or vice versa.
Using the total frequency n instead of n ÷ 2, n ÷ 4, or 3n ÷ 4 for the position of the median, lower quartile, or upper quartile.
Forgetting to subtract the lower quartile from the upper quartile when finding the interquartile range, and instead giving just the upper quartile.
When finding the number of values between two points, reading only one of the two required cumulative frequencies, or subtracting them in the wrong order.
Misplacing the trace lines on the graph, which leads to reading the wrong value from the curve.
FAQ
Questions students and parents ask
Before this topic, make sure you know
What to learn next
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