Extended. A cumulative frequency curve turns a grouped frequency table into a graph you can read: the median, the quartiles, any percentile, and how many values lie above or below any point.
Cumulative frequency is a running total. Plot each total at the UPPER boundary of its class and join with a smooth curve. For \(n\) values, read across at \(\dfrac{n}{4}\), \(\dfrac{n}{2}\) and \(\dfrac{3n}{4}\) for the lower quartile, median and upper quartile.
Fig. 1: cumulative frequency curve for the times of 80 people, drawn to scale from the table in the first worked example. Reading across at 20, 40 and 60 gives LQ = 20, median ≈ 28 and UQ ≈ 38 minutes.
Section 01
Building the curve.
In the lab, add up the frequencies as you go to fill the cumulative frequency column. Then watch where each cross is plotted: at the upper end of its class, not the middle. Change a frequency and see the curve and every reading move.
Cumulative frequency (cf) is the running total of the frequencies. The last cf is the total, \(n\).
Plot each cf at the upper class boundary: the cf for \(20 < t \le 30\) tells you how many are 30 or less.
Start the curve at the lower boundary of the first class with cf 0, mark the points as small crosses (\(\times\)), and join them with a smooth curve (not straight segments).
Worked example From a frequency table to a curve
The times, \(t\) minutes, taken by 80 people to finish a puzzle are shown. Make a cumulative frequency table and list the points to plot.
Plotting at the midpoints (5, 15, 25, ...) shifts the whole curve 5 minutes to the left and every reading is wrong. The cf belongs to the upper boundary.
Section 02
Median, quartiles and IQR.
Every reading is the same move: start on the cumulative frequency axis, go across to the curve, then down to the horizontal axis. With a large data set, use \(\dfrac{n}{2}\) for the median, not \(\dfrac{n+1}{2}\).
Measure
Read across at
For \(n = 80\)
Lower quartile
\(\dfrac{n}{4}\)
20
Median
\(\dfrac{n}{2}\)
40
Upper quartile
\(\dfrac{3n}{4}\)
60
Worked example Read the curve
Use the curve in Fig. 1 to estimate the median, the quartiles and the interquartile range of the times.
Median: across from 40 to the curve, then down: about 28 minutes.
Lower quartile: across from 20: 20 minutes (the point (20, 20) is on the curve).
Upper quartile: across from 60: about 38 minutes.
IQR \(= 38 - 20 = 18\) minutes.
Answer: Median \(\approx 28\) min, LQ = 20 min, UQ \(\approx 38\) min, IQR \(\approx 18\) min
Examiner trap
The IQR is the difference of two TIMES read from the horizontal axis (\(38 - 20\)), not the difference of the cumulative frequencies (\(60 - 20 = 40\)).
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
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Frequencies 3, 7, 10, 5. What is the third cumulative frequency?
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Where is the cumulative frequency for the class \(20 < t \le 30\) plotted?
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A cumulative frequency curve shows 60 values. At what cumulative frequency do you read the median?
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From a curve, LQ = 34 kg and UQ = 52 kg. What is the interquartile range?
Section 03
Percentiles and how many questions.
The \(p\)th percentile is read across at \(p\%\) of \(n\). Questions that ask how many values are above a point need one extra step: the curve gives how many are BELOW, so subtract from the total.
Worked example Percentile
Use Fig. 1 to estimate the 90th percentile of the times.
\(90\%\) of 80 \(= 72\).
Across from 72 to the curve, then down: about 46 minutes.
Answer: About 46 minutes: 90% of the people finished within 46 minutes.
Worked example More than a value
Use Fig. 1 to estimate how many people took more than 45 minutes.
Up from 45 on the time axis to the curve, then across: cf \(\approx 71\). So about 71 people took 45 minutes or less.
More than 45 minutes: \(80 - 71 = 9\) people.
Answer: About 9 people
Your turn
A cumulative frequency curve for the marks of 200 students gives 64 marks at the 80th percentile. How many students scored more than 64?
80th percentile: 80% of 200 = 160 students scored 64 or less. So \(200 - 160 = 40\) students scored more.
Examiner trap
'How many took MORE than 45 minutes' is the total minus the reading. Writing 71, the number who took 45 minutes or less, is the most common lost mark on this topic.
Stage 2 · Exam
Exam-style questions
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Use the curve in Fig. 1 (times of 80 people) to estimate the interquartile range.
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Use the curve in Fig. 1 to estimate how many people took more than 45 minutes.
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A class \(20 < t \le 30\) has frequency 24 and cumulative frequency 44. Which point is plotted on the cumulative frequency diagram?
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The marks of 200 students are shown on a cumulative frequency curve. The 80th percentile is 64 marks. How many students scored more than 64 marks?
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Cumulative frequency, read.
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