Statistics

Cumulative frequency

Q: Where do you plot points on a cumulative frequency graph?

At the upper boundary of each class, because the running total counts everything up to the end of that class.

Q: How do you find the median from a cumulative frequency graph?

Go to half the total frequency on the vertical axis, read across to the curve, then down to the horizontal axis.

11 past-paper questions, easiest first. Try each one, then reveal the answer.

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Cumulative frequency: what you need to know

A cumulative frequency graph shows a running total of grouped data. From it you can estimate the median, the quartiles and how many values lie above or below any point. It is Higher only.

Key facts

Cumulative frequency is a running total: add each frequency to the ones before it.
Plot each running total at the upper end of its class, not the midpoint.
Join the points with a smooth curve or straight lines.
For $n$ values, read across at $\frac{n}{2}$ for the median, $\frac{n}{4}$ for the lower quartile and $\frac{3n}{4}$ for the upper quartile.
Interquartile range = upper quartile − lower quartile.

Drawing and using the graph

Add a running total column to the table.
Plot (upper class boundary, running total) for each class. Also plot the lower boundary of the first class at 0, e.g. $(0, 0)$ if the first class starts at 0.
Draw the curve.
Read across from the cumulative frequency you need, then down to the axis.

How many are more than a value? What percentage?

Read up from the value to the curve and across to find how many are below it.
Subtract from the total.
For a percentage or probability, divide by the total (and multiply by 100 for a percentage).

Worked examples

Example 1 Higher · grade 6

The masses, $m$ grams, of 80 apples are grouped: $0 < m \le 20$ : 6, $20 < m \le 40$ : 14, $40 < m \le 60$ : 28, $60 < m \le 80$ : 22, $80 < m \le 100$ : 10. Find the cumulative frequencies and the points to plot.

Show the solution

Running totals: 6, 20, 48, 70, 80
Points: (0, 0), (20, 6), (40, 20), (60, 48), (80, 70), (100, 80)
To estimate the median, read across from 40; for the quartiles, from 20 and 60.

Example 2 Higher · grade 7

A cumulative frequency graph shows the times of 120 runners. Reading across from 30, 60 and 90 to the curve and down to the time axis gives 28, 36 and 45 minutes. Reading up from 50 minutes gives a cumulative frequency of 102. (a) Find the interquartile range. (b) Estimate how many runners took more than 50 minutes. (c) What percentage of the runners is this?

Show the solution

$120 \div 4 = 30$ and $\frac{3}{4} \times 120 = 90$ , so the lower quartile is 28 minutes and the upper quartile is 45 minutes.
(a) IQR $= 45 - 28 = 17$ minutes
(b) 102 runners took 50 minutes or less.
$120 - 102 = 18$ runners
(c) $\frac{18}{120} \times 100 = 15\%$

Common mistakes

Plotting points at the midpoints of the classes instead of the upper ends.
Reading quartiles at 25 and 75 when the total is not 100. Use a quarter and three quarters of the total.
Forgetting to subtract from the total for 'more than' questions.
Not starting the graph at the lowest class boundary with a cumulative frequency of 0, or forgetting to label the axes.

Exam tips

Draw your reading lines on the graph: they can earn method marks even if your reading is a little out.
Answers are accepted within a small range, so read carefully but do not worry about tiny differences.
If asked what is wrong with someone's graph or table, check where the points are plotted, the running totals and the axis labels. Joining points with straight lines is not a mistake.

Questions people ask

Where do you plot points on a cumulative frequency graph?

At the upper boundary of each class, because the running total counts everything up to the end of that class.

How do you find the median from a cumulative frequency graph?

Go to half the total frequency on the vertical axis, read across to the curve, then down to the horizontal axis.

Exam questions: Cumulative frequency (11, easiest first)

Cumulative frequency6 marksGrade 5No calculatorHigher

November 2024 · Paper 1 Higher · Q9

Read quartiles and the median from a cumulative frequency graph to draw a box plot, then test a claim about the percentage who passed.

Show answer

(a)	Box plot: minimum 8, lower quartile 25, median 34, upper quartile 42, maximum 74	3 marks
(b)	Yes: about 42 students scored under 40, so 18 passed, and 18 is 30% of 60	3 marks

Worked solution Checked against the mark scheme

(a) Read across at cf 15, 30 and 45:

LQ $\approx 25$ , median $\approx 34$ , UQ $\approx 42$ . Plot these with the lowest mark 8 and highest 74.

(b) At mark 40 the graph reads a cumulative frequency of about 42.

Passed: $60 - 42 = 18$ , and $\frac{18}{60} \times 100 = 30\%$ . So Sian is correct.

Watch out: Reading the quartiles at cf 20 and 40 instead of 15 and 45.

Cumulative frequency6 marksGrade 5CalculatorHigher

November 2023 · Paper 3 Higher · Q13

Read a cumulative frequency graph to estimate a probability, the median and the interquartile range, then compare spreads.

Show answer

(a)	$\frac{6}{80}$ (accept $\frac{5}{80}$ )	2 marks
(b)	$60$ cm	1 mark
(c)	$35$ cm	2 marks
(d)	The heights of the plants grown outside are more spread out (IQR 35 cm compared with 30 cm)	1 mark

Worked solution Checked against the mark scheme

(a) At 90 cm the cumulative frequency is about 74, so $80 - 74 = 6$ plants are taller. Probability $\approx \frac{6}{80}$ .

(b) Median at $80 \div 2 = 40$ th plant: about 60 cm.

(d) $35 > 30$ , so the heights of the outside plants are more varied than those grown inside.

Watch out: In (a), giving 74/80, the probability of being 90 cm or shorter.

Cumulative frequency7 marksGrade 6CalculatorHigher

June 2024 · Paper 2 Higher · Q11

Complete a cumulative frequency table, draw the graph, estimate the interquartile range and count values above a given weight.

Show answer

(a)	$7, 28, 43, 54, 60$	1 mark
(b)	Cumulative frequency curve through $(2,7),(4,28),(6,43),(8,54),(10,60)$	2 marks
(c)	About $3.7$ kg (accept 3.4 to 4)	2 marks
(d)	About $8$ (accept 7 to 10)	2 marks

Worked solution Checked against the mark scheme

(a) Running totals: $7, 7+21=28, 43, 54, 60$ .

(b) Plot at the upper ends of each class, $(2,7),(4,28),(6,43),(8,54),(10,60)$ , and join with a smooth curve.

(d) At 7.4 kg the cf is about 51, so $60 - 51 \approx 9$ parcels.

Watch out: Plotting cumulative frequencies at the midpoints of the classes instead of the upper boundaries.

Cumulative frequency3 marksGrade 6CalculatorHigher

June 2023 · Paper 2 Higher · Q9

Plot a cumulative frequency graph from a table of ages and read off an estimate of the median.

Show answer

(a)	Curve through $(20, 0)$ , $(30, 20)$ , $(40, 48)$ , $(50, 64)$ , $(60, 75)$ , $(70, 80)$	2 marks
(b)	About $37$ years (36 to 38)	1 mark

Worked solution Checked against the mark scheme

(a) Plot each cumulative frequency at the upper end of its class: $(30, 20)$ , $(40, 48)$ , $(50, 64)$ , $(60, 75)$ , $(70, 80)$ , starting from $(20, 0)$ , and join with a smooth curve.

(b) Median is at $80 \div 2 = 40$ people. Read across from 40 to the curve and down: about $37$ years.

Watch out: Plotting the points at the middle of each class instead of the upper end.

Cumulative frequency6 marksGrade 6No calculatorHigher

June 2022 · Paper 1 Higher · Q10

Build a cumulative frequency table and graph for daily profits, then estimate a count and the interquartile range.

Show answer

(a)	10, 25, 50, 80, 85, 100	1 mark
(b)	Points at $(50,10), (100,25), (150,50), (200,80), (250,85), (300,100)$ joined	2 marks
(c)	About 37 days (35 to 39)	1 mark
(d)	About £89 (85 to 93)	2 marks

Worked solution Checked against the mark scheme

(a) Running totals: 10, 25, 50, 80, 85, 100

(b) Plot each total at the top of its class (50, 100, …, 300) and join.

(d) Lower quartile (25th value) $= £100$ ; upper quartile (75th value) $\approx £190$

IQR $\approx 190 - 100 = £90$

Watch out: Plotting points at class midpoints instead of upper class boundaries.

Cumulative frequency2 marksGrade 6CalculatorHigher

November 2022 · Paper 3 Higher · Q13

Identify two faults in a drawn cumulative frequency graph.

Show answer

Answer	1. The horizontal axis has no label. 2. The points are plotted at the middle of each class instead of at the upper end	2 marks

Worked solution Checked against the mark scheme

Check the axes: the time axis has no label (it should say time in seconds).

Check the points: cumulative frequency is plotted at the upper end of each class (10, 20, 30, …), but these are plotted at 5, 15, 25, …

Watch out: Saying the curve should be joined with straight lines, which is not wrong.

Cumulative frequency3 marksGrade 6No calculatorHigher

November 2021 · Paper 1 Higher · Q12

Plot a cumulative frequency curve for plant heights and read off the median.

Show answer

(a)	Points (5, 4), (10, 11), (15, 24), (20, 34), (25, 38), (30, 40) joined with a smooth curve or straight lines	2 marks
(b)	Between $13$ and $14$ cm	1 mark

Worked solution Checked against the mark scheme

(a) Plot each cumulative frequency at the upper end of its interval: (5, 4), (10, 11), (15, 24), (20, 34), (25, 38), (30, 40). Start at (0, 0) and join with a smooth curve.

(b) The median is at half of 40, which is 20. Read across from 20 to the curve and down: about $13.5$ cm.

Watch out: Plotting points at the midpoints of the intervals instead of the upper ends.

Cumulative frequency1 markGrade 6CalculatorHigher

November 2021 · Paper 2 Higher · Q8

Spot what is wrong with a cumulative frequency table built from grouped data.

Show answer

Answer	The class intervals are wrong: each should start at 0, e.g. $0 < t \le 20$ not $10 < t \le 20$	1 mark

Worked solution Checked against the mark scheme

Cumulative frequency counts everyone up to the end of each class.

So 13 is the number with $t \le 20$ , which covers $0 < t \le 20$ , not just $10 < t \le 20$ .

The running totals are right, but every interval after the first should start at 0.

Watch out: Saying the totals have been added wrongly, when the totals are correct.

Cumulative frequency6 marksGrade 6No calculatorHigher

November 2020 · Paper 1 Higher · Q12

Complete a cumulative frequency table for wages, draw the graph, and use it to test a claim about 60% of people.

Show answer

(a)	5, 15, 35, 55, 70, 80	1 mark
(b)	Points plotted at (250, 5), (300, 15), (350, 35), (400, 55), (450, 70), (500, 80), joined with a smooth curve or straight segments	2 marks
(c)	No: 60% is 48 people, but only about 40 people earn £360 or less (about 50%)	3 marks

Worked solution Checked against the mark scheme

(a) Add the frequencies as you go: 5, 15, 35, 55, 70, 80

(b) Plot each total at the top of its class (250, 300, ..., 500) and join the points.

Reading the graph at £360 gives about 40 people, which is $\frac{40}{80} = 50\%$ .

So Juan is wrong: only about 50% earn £360 or less.

Watch out: Plotting the points at the midpoints of the classes instead of the upper boundaries.

Cumulative frequency6 marksGrade 6No calculatorHigher

November 2019 · Paper 1 Higher · Q10

Draw a cumulative frequency graph, then use it to estimate the interquartile range and a probability.

Show answer

(a)	Graph through (40, 5), (60, 25), (80, 35), (100, 38), (120, 40)	2 marks
(b)	About 25 minutes (21 to 28)	2 marks
(c)	About $\frac{21.5}{40}$ (between $\frac{19}{40}$ and $\frac{24}{40}$ )	2 marks

Worked solution Checked against the mark scheme

(a) Plot each cumulative frequency at the end of its interval and join with a smooth curve or straight lines, starting at (20, 0).

(b) LQ at $cf = 10$ : about 45; UQ at $cf = 30$ : about 70

IQR $\approx 70 - 45 = 25$ minutes

$36.5 - 15 = 21.5$ , so probability $\approx \frac{21.5}{40} \approx 0.54$

Watch out: Plotting points at the middle of each interval instead of the upper end.

Cumulative frequency6 marksGrade 6CalculatorHigher

June 2019 · Paper 2 Higher · Q11

Build a cumulative frequency table for journey times, plot the curve, and use it to estimate the percentage taking over 90 minutes.

Show answer

(a)	$5, 35, 55, 70, 78, 80$	1 mark
(b)	Cumulative frequency graph through $(20,5), (40,35), (60,55), (80,70), (100,78), (120,80)$	2 marks
(c)	$7.5\%$ (5% to 10% from the reading)	3 marks

Worked solution Checked against the mark scheme

(a) Add the frequencies as you go: $5, 5+30=35, 55, 70, 78, 80$ .

(b) Plot each total at the top of its interval, e.g. $(40, 35)$ , and join with a smooth curve.

$\frac{6}{80} \times 100 = 7.5\%$

Watch out: Plotting points at the middle of each interval instead of the upper end.

Also tested in these questions

Their main topic is different, but part of each one tests this topic.

June 2023 · Paper 1 Foundation · Q23: Draw a frequency polygon from a grouped frequency table.
November 2022 · Paper 2 Higher · Q11: Use a cumulative frequency graph to find the quartiles and median, then draw a box plot.
November 2018 · Paper 1 Higher · Q9: Read quartiles from a cumulative frequency graph to draw a box plot, compare it with another box plot, and judge a claim about the data.