Cumulative frequency and box plots | AQA GCSE Maths Notes

Visual model

Box plots compare spread and median

medianinterquartile rangeminmaxcompare centre and spread

Median is the middle value.

The box shows the interquartile range.

Compare spread as well as centre.

Gold-standard guide

26 mins

What you will learn

Compare distributions and estimate quartiles.

Use a clear step-by-step method for cumulative frequency and box plots.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier statistics skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Cumulative frequency graph: read off values by going across from the cumulative frequency axis to the curve, then down to the data axis

Step 1

Find median (at n/2 = 40th value)

Read across from 40 on the CF axis → score ≈ 60

Step 2

Find Q1 (at n/4 = 20th value)

Read across from 20 → score ≈ 50

Step 3

Find Q3 (at 3n/4 = 60th value)

Read across from 60 → score ≈ 70

Watch out

Students read from the data axis to the graph rather than from the CF axis

Median

median is the 50th percentile.

IQR

$interquartile range = upper quartile - lower quartile.$

Worked example

From a cumulative frequency graph for 80 students' test scores, find the median, lower quartile, upper quartile and interquartile range. The graph passes through (50, 20), (60, 40), (70, 60), (80, 80).

Find median (at n/2 = 40th value): Read across from 40 on the CF axis → score ≈ 60.

Find Q1 (at n/4 = 20th value): Read across from 20 → score ≈ 50.

Find Q3 (at 3n/4 = 60th value): Read across from 60 → score ≈ 70.

Calculate the IQR: IQR = Q3 − Q1 = 70 − 50 = 20.

Final answer

Median ≈ 60; Q1 ≈ 50; Q3 ≈ 70; IQR = 20

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

From a cumulative frequency graph for 80 students' test scores, find the median, lower quartile, upper quartile and interquartile range. The graph passes through (50, 20), (60, 40), (70, 60), (80, 80).

4 marks4 minscumulative-frequency-and-box-plots-worked

Show solution

Worked solution

1.Find median (at n/2 = 40th value): Read across from 40 on the CF axis → score ≈ 60.
2.Find Q1 (at n/4 = 20th value): Read across from 20 → score ≈ 50.
3.Find Q3 (at 3n/4 = 60th value): Read across from 60 → score ≈ 70.
4.Calculate the IQR: IQR = Q3 − Q1 = 70 − 50 = 20.

Final answer

Median ≈ 60; Q1 ≈ 50; Q3 ≈ 70; IQR = 20

Mark points

M1: find median (at n/2 = 40th value)
M1: find q1 (at n/4 = 20th value)
M1: find q3 (at 3n/4 = 60th value)
M1: calculate the iqr
A1: Median ≈ 60; Q1 ≈ 50; Q3 ≈ 70; IQR = 20

Watch out

Students read from the data axis to the graph rather than from the CF axis.Always go to the CF axis first (n/2, n/4, 3n/4), draw a horizontal line to the curve, then drop vertically to the data axis.

Diagnosticrecall

n = 60. Find the CF values for median, Q1 and Q3.

1 mark2 minscumulative-frequency-and-box-plots-q1

Show solution

Worked solution

1.Spot the skill: Cumulative frequency graph: read off values by going across from the cumulative frequency axis to the curve, then down to the data axis.
2.Use the find median (at n/2 = 40th value) stage first, then find q1 (at n/4 = 20th value).
3.Keep the final answer visible: Median at 30, Q1 at 15, Q3 at 45.

Final answer

Median at 30, Q1 at 15, Q3 at 45

Mark points

M1: use the correct cumulative frequency graph: read off values by going across from the cumulative frequency axis to the curve, then down to the data axis.median at n/2, q1 at n/4, q3 at 3n/4. iqr = q3 − q1. box plots show: minimum, q1, median, q3, maximum.
A1: Median at 30, Q1 at 15, Q3 at 45

Watch out

Students read from the data axis to the graph rather than from the CF axis.Always go to the CF axis first (n/2, n/4, 3n/4), draw a horizontal line to the curve, then drop vertically to the data axis.

Easyprocedure

Q1 = 25, Q3 = 45. Find IQR and identify any outlier if a value is 80.

2 marks3 minscumulative-frequency-and-box-plots-q2

Show solution

Worked solution

1.Spot the skill: Cumulative frequency graph: read off values by going across from the cumulative frequency axis to the curve, then down to the data axis.
2.Use the find q1 (at n/4 = 20th value) stage first, then find q3 (at 3n/4 = 60th value).
3.Keep the final answer visible: IQR = 20; outlier boundary = 45 + 1.5×20 = 75; 80 > 75 so it is an outlier.

Final answer

IQR = 20; outlier boundary = 45 + 1.5×20 = 75; 80 > 75 so it is an outlier

Mark points

M1: use the correct cumulative frequency graph: read off values by going across from the cumulative frequency axis to the curve, then down to the data axis.median at n/2, q1 at n/4, q3 at 3n/4. iqr = q3 − q1. box plots show: minimum, q1, median, q3, maximum.
A1: IQR = 20; outlier boundary = 45 + 1.5×20 = 75; 80 > 75 so it is an outlier

Watch out

Students read from the data axis to the graph rather than from the CF axis.Always go to the CF axis first (n/2, n/4, 3n/4), draw a horizontal line to the curve, then drop vertically to the data axis.

Mediumreasoning

A box plot has min=10, Q1=20, median=30, Q3=45, max=60. Find the IQR and range.

3 marks4 minscumulative-frequency-and-box-plots-q3

Show solution

Worked solution

1.Spot the skill: Cumulative frequency graph: read off values by going across from the cumulative frequency axis to the curve, then down to the data axis.
2.Use the find q3 (at 3n/4 = 60th value) stage first, then calculate the iqr.
3.Keep the final answer visible: IQR = 25; Range = 50.

Final answer

IQR = 25; Range = 50

Mark points

M1: use the correct cumulative frequency graph: read off values by going across from the cumulative frequency axis to the curve, then down to the data axis.median at n/2, q1 at n/4, q3 at 3n/4. iqr = q3 − q1. box plots show: minimum, q1, median, q3, maximum.
A1: IQR = 25; Range = 50

Watch out

Students read from the data axis to the graph rather than from the CF axis.Always go to the CF axis first (n/2, n/4, 3n/4), draw a horizontal line to the curve, then drop vertically to the data axis.

Hardproblem solving

Describe the shape of the distribution when Q3−median > median−Q1.

3 marks5 minscumulative-frequency-and-box-plots-q4

Show solution

Worked solution

1.Spot the skill: Cumulative frequency graph: read off values by going across from the cumulative frequency axis to the curve, then down to the data axis.
2.Use the calculate the iqr stage first, then find median (at n/2 = 40th value).
3.Keep the final answer visible: Positively skewed (stretched on the upper end).

Final answer

Positively skewed (stretched on the upper end)

Mark points

M1: use the correct cumulative frequency graph: read off values by going across from the cumulative frequency axis to the curve, then down to the data axis.median at n/2, q1 at n/4, q3 at 3n/4. iqr = q3 − q1. box plots show: minimum, q1, median, q3, maximum.
A1: Positively skewed (stretched on the upper end)

Watch out

Students read from the data axis to the graph rather than from the CF axis.Always go to the CF axis first (n/2, n/4, 3n/4), draw a horizontal line to the curve, then drop vertically to the data axis.

Exam-stylemulti-step

Compare: Group A median 55, IQR 12; Group B median 50, IQR 20.

4 marks6 minscumulative-frequency-and-box-plots-q5

Show solution

Worked solution

1.Spot the skill: Cumulative frequency graph: read off values by going across from the cumulative frequency axis to the curve, then down to the data axis.
2.Use the find median (at n/2 = 40th value) stage first, then find q1 (at n/4 = 20th value).
3.Keep the final answer visible: A has higher typical score; B has more variation in scores.

Final answer

A has higher typical score; B has more variation in scores

Mark points

M1: use the correct cumulative frequency graph: read off values by going across from the cumulative frequency axis to the curve, then down to the data axis.median at n/2, q1 at n/4, q3 at 3n/4. iqr = q3 − q1. box plots show: minimum, q1, median, q3, maximum.
A1: A has higher typical score; B has more variation in scores

Watch out

Students read from the data axis to the graph rather than from the CF axis.Always go to the CF axis first (n/2, n/4, 3n/4), draw a horizontal line to the curve, then drop vertically to the data axis.

Grade 9 stretchproblem solving

Two classes have the same median. Class A has IQR 14 and Class B has IQR 9. Compare their consistency.

4 marks7 minsboxplot-g9

Show solution

Worked solution

1.Recall that a smaller IQR means less spread in the middle half.
2.Compare the two IQR values.

Final answer

Class B is more consistent because its IQR is smaller

Mark points

C1: compare IQR values
C1: link smaller spread to greater consistency

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint

16 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Cumulative frequency and box plots - 2 marksn = 60. Find the CF values for median, Q1 and Q3.Mark answer

Answer

Median at 30, Q1 at 15, Q3 at 45

2Collecting and sampling data - 2 marksWhy might a questionnaire question be biased?Mark answer

Answer

Leading wording, only offering responses that agree, or not including a 'no' option

3Averages and range - 2 marksThe mean of 5 numbers is 12. Four of them are 8, 14, 10, 15. Find the fifth.Mark answer

Answer

4Grouped data and estimated mean - 3 marksA survey records [10,20): 5 responses and [20,30): 15. Estimate total mean across both groups.Mark answer

Answer

Use midpoints 15 and 25: (5×15 + 15×25)/20 = $45\frac{0}{2}0$ = 22.5

Mastery check

I can explain the method for cumulative frequency and box plots.
I can show clear working without skipping key steps.
I can avoid this mistake: Students read from the data axis to the graph rather than from the CF axis.Always go to the CF axis first (n/2, n/4, 3n/4), draw a horizontal line to the curve, then drop vertically to the data axis.

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