Cumulative Frequency and Box Plots (Higher)

This lesson is Higher tier. When data is grouped, you cannot read exact quartiles from a table — but you can draw a cumulative frequency curve and read estimates of the median, quartiles and interquartile range straight off it. A box plot (box-and-whisker diagram) then summarises those five key values in a single, comparison-friendly picture. Both are core Higher-tier tools in the OCR GCSE Mathematics (J560) Statistics strand, and questions almost always end by asking you to compare two data sets using a median and a measure of spread.

The topic blends AO1 (plotting the curve and drawing the box plot accurately) with AO2 (reading off and interpreting the median, quartiles and IQR) and AO3 (comparing distributions and judging which is more consistent). OCR command words include "Draw", "Estimate", "Find", "Compare" and "Give a reason for your answer".

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Key Vocabulary

Term	Meaning
Cumulative frequency	A running total of frequencies up to the top of each class
Cumulative frequency curve	A smooth S-shaped graph of cumulative frequency against the upper class boundary
Median ($Q_2$Q2​)	The middle value; at cumulative frequency $\frac{n}{2}$2n​
Lower quartile ($Q_1$Q1​)	The value a quarter of the way through; at $\frac{n}{4}$4n​
Upper quartile ($Q_3$Q3​)	The value three-quarters through; at $\frac{3n}{4}$43n​
Interquartile range (IQR)	$Q_3 - Q_1$Q3​−Q1​; the spread of the middle 50%
Box plot	A diagram showing minimum, $Q_1$Q1​, median, $Q_3$Q3​ and maximum

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Building Cumulative Frequency

Cumulative frequency is a running total: for each class you add its frequency to all those before it. You always plot the cumulative frequency against the upper boundary of the class, because by that point all of those values have been counted.

Worked Example 1

Complete the cumulative frequency column for the times, $t$t minutes, taken by 60 students to finish a quiz.

Time $t$t (min)	Frequency	Cumulative frequency
$0 < t \le 10$0<t≤10	4	4
$10 < t \le 20$10<t≤20	8	12
$20 < t \le 30$20<t≤30	18	30
$30 < t \le 40$30<t≤40	18	48
$40 < t \le 50$40<t≤50	9	57
$50 < t \le 60$50<t≤60	3	60

Solution: Add each frequency to the running total: $4$4; $4+8 = 12$4+8=12; $12+18 = 30$12+18=30; $30+18 = 48$30+18=48; $48+9 = 57$48+9=57; $57+3 = 60$57+3=60. The final cumulative frequency equals $n = 60$n=60, a useful check.

Common error: plotting cumulative frequency against the class midpoint. It must be plotted against the upper boundary (10, 20, 30, …).

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Drawing the Cumulative Frequency Curve

Plot each cumulative frequency against the upper class boundary, then join the points with a smooth curve (not straight segments). The result is a characteristic S-shape. The curve for the quiz data above is shown below.

The dashed line shows how to read the median: go across from cumulative frequency $\frac{60}{2} = 30$260​=30 to the curve, then down to the time axis.

Worked Example 2

Use the curve (or the table) to estimate the median time.

Solution: The median is at cumulative frequency $\frac{60}{2} = 30$260​=30. Reading across to the curve and down gives a time of about 30 minutes (the curve passes through the point (30, 30)).

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Quartiles and the Interquartile Range

On a cumulative frequency curve of $n$n values:

• Lower quartile $Q_1$Q1​ is read at cumulative frequency $\dfrac{n}{4}$4n​,
• Median $Q_2$Q2​ at $\dfrac{n}{2}$2n​,
• Upper quartile $Q_3$Q3​ at $\dfrac{3n}{4}$43n​.

Then $\text{IQR} = Q_3 - Q_1$IQR=Q3​−Q1​.

Worked Example 3

For the quiz data ($n = 60$n=60), estimate $Q_1$Q1​, $Q_3$Q3​ and the IQR.

Solution:

• $Q_1$Q1​ at cumulative frequency $\dfrac{60}{4} = 15$460​=15 → reading across and down gives about 22 minutes.
• $Q_3$Q3​ at cumulative frequency $\dfrac{3 \times 60}{4} = 45$43×60​=45 → about 38 minutes.
• IQR $= Q_3 - Q_1 = 38 - 22 = 16$=Q3​−Q1​=38−22=16 minutes.

Common error: using $\frac{n+1}{4}$4n+1​ on a cumulative frequency curve. For grouped data read from a curve, use $\frac{n}{4}$4n​, $\frac{n}{2}$2n​ and $\frac{3n}{4}$43n​.

Worked Example 4

Use the curve to estimate how many students took more than 35 minutes.

Solution: Read up from 35 minutes to the curve and across: the cumulative frequency is about 39. That is the number who took 35 minutes or less, so the number taking more than 35 minutes is $60 - 39 =$60−39= about 21 students.

Worked Example 4b

A cumulative frequency curve for the test marks of 160 students gives $Q_1 = 42$Q1​=42, median $= 55$=55 and $Q_3 = 68$Q3​=68. (a) Work out the interquartile range. (b) The pass mark is 50. Roughly what can you say about the proportion who passed?

Solution: (a) IQR $= Q_3 - Q_1 = 68 - 42 = 26$=Q3​−Q1​=68−42=26 marks. (b) The median is 55, which is above the pass mark of 50, so more than half the students passed. (To be precise you would read the cumulative frequency at mark 50 from the curve and compare with 160.)

Worked Example 4c

Explain why $\frac{n}{4}$4n​, $\frac{n}{2}$2n​ and $\frac{3n}{4}$43n​ are used on a cumulative frequency curve rather than $\frac{n+1}{4}$4n+1​ etc.

Solution: The $\frac{n+1}{4}$4n+1​ formulae locate the position of a value in a small, listed data set. A cumulative frequency curve treats the data as a smooth, continuous quantity over the whole range, so we simply split the total frequency $n$n into quarters and read the corresponding values. With large grouped data the difference between $\frac{n}{4}$4n​ and $\frac{n+1}{4}$4n+1​ is negligible, and the curve method uses $\frac{n}{4}$4n​, $\frac{n}{2}$2n​, $\frac{3n}{4}$43n​ by convention.

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Box Plots

A box plot displays five numbers — the minimum, $Q_1$Q1​, the median, $Q_3$Q3​ and the maximum — on a number line. The box spans $Q_1$Q1​ to $Q_3$Q3​ (so its width is the IQR), a line inside marks the median, and "whiskers" reach out to the smallest and largest values.

The box plot below summarises the quiz times: minimum 8, $Q_1$Q1​ 22, median 30, $Q_3$Q3​ 38, maximum 56 minutes.

Worked Example 5

From a box plot you read: minimum 12, $Q_1$Q1​ 19, median 24, $Q_3$Q3​ 31, maximum 45. Find (a) the range and (b) the interquartile range.

Solution: (a) Range $= 45 - 12 = 33$=45−12=33. (b) IQR $= Q_3 - Q_1 = 31 - 19 = 12$=Q3​−Q1​=31−19=12.

Worked Example 6

Explain what the width of the box tells you in a box plot.

Solution: The box spans $Q_1$Q1​ to $Q_3$Q3​, so its width is the interquartile range — the spread of the middle 50% of the data. A narrow box means the central half of the data is tightly clustered (consistent); a wide box means it is more spread out.

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Comparing Distributions with Box Plots

Box plots are designed for comparison. Compare two data sets by their medians (which is higher on average) and their IQRs (which is more consistent), always in context.

Worked Example 7

Two classes sat a test. Class P: median 24, IQR 12. Class Q: median 28, IQR 5. Compare the two classes.