Contents · 0% complete

Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

Learn9709Paper 5 · Probability & Statistics 1Representation of dataStem-and-leaf diagrams and box plots

Representation of data · Subtopic 1 of 4

Stem-and-Leaf Diagrams and Box Plots

Two ways to see a whole data set at a glance. A stem-and-leaf diagram keeps every value; a box plot squeezes the data into five numbers that show its centre, its spread and its shape.

Cambridge 9709 Paper 5Edexcel YMA01 WST01Edexcel 9MA0 Paper 3AQA 7357 Paper 3≈ 30 minRevision 5 · updated 2026-10-27

Practise this topicTimed test Ask Salam TutorSwitch board

Section 1 of 14

Learning objectives

Draw and read stem-and-leaf diagrams, including back-to-back ones with a key.
Find the median, lower and upper quartiles and the interquartile range.
Draw and interpret box-and-whisker plots, and identify outliers.
Describe skew and compare two data sets using an average and a measure of spread.

Pause and think

Two classes took the same test. Class A's marks were all between $50$ and $70$ ; class B's ranged from $10$ to $100$ . Both had a median of $60$ . Are the classes really "the same"?

Section 2 of 14

Core idea

Start hereThirteen scores, one glance

A teacher has $13$ test scores: $12,15,18,21,23,23,26,28,30,34,35,41,47$ . A list hides the story. Is the class bunched together or spread out? Are there stragglers? A good diagram answers in a second. Stem-and-leaf diagrams keep every value; box plots boil the data down to five numbers.

The big idea

A stem-and-leaf diagram orders the data and shows its shape. The median splits ordered data in half; the quartiles split each half again. A box plot draws min, $Q_1$ , median, $Q_3$ , max; outliers are usually beyond $1.5\times$ IQR from the quartiles.

1. Stem-and-leaf diagrams

Stem-and-leaf diagram of 13 test scores

The data:

12, 15, 18, 21, 23, 23, 26, 28, 30, 34, 35, 41, 47

. The longest row (the 20s) shows where most values are.

Stem = tens, leaf = units (here). Leaves in increasing order.
Always give a key, e.g. $2\mid3$ means $23$ .
Back-to-back diagrams compare two groups on a shared stem.

2. Median and quartiles

\text{median position}=\frac{n+1}2,\qquad\text{IQR}=Q_3-Q_1

$n$: number of values.
$Q_1$ , $Q_3$: lower and upper quartiles: medians of the lower and upper halves.

Worked exampleThe 13 scores

Find the median, quartiles and IQR.

Goal: Five-number summary.
Strategy: Median first, then medians of each half (excluding the median when $n$ is odd).

$\text{median}=7\text{th value}=26$
$\frac{13+1}2=7$ .
$Q_1:\ 12,15,18,21,23,23\Rightarrow\tfrac{18+21}2=19.5$
Lower half.
$Q_3:\ 28,30,34,35,41,47\Rightarrow\tfrac{34+35}2=34.5$
Upper half.
$\text{IQR}=34.5-19.5=15$
Spread of the middle half.

AnswerMin

12

Q_1=19.5

, median

26

Q_3=34.5

, max

47

✓ CheckAbout a quarter of the 13 values lie below

19.5

(three) ✓.

3. Box plots, outliers and skew

The same data as a box plot

The box holds the middle half of the data. The whiskers reach to the smallest and largest values. The line inside the box is the median.

Worked exampleOutlier test

Are there any outliers in the 13 scores?

Goal: Yes/no.
Strategy: Fences at $Q_1-1.5\times$ IQR and $Q_3+1.5\times$ IQR.

$1.5\times15=22.5$
Fence distance.
$19.5-22.5=-3,\quad34.5+22.5=57$
Fences.

AnswerNo value is below

-3

or above

57

: no outliers.

✓ CheckMax

47<57

✓.

Positive skew: long right tail; median nearer $Q_1$ .
Negative skew: long left tail; median nearer $Q_3$ .
Symmetric: median central.

Exam tipComparing two groups Compare a centre (median) and a spread (IQR) in context: "Group A scored higher on average and more consistently."

Pause and think

Median of $3,8,5,12,9,4$ ?

Summary

Stem-and-leaf with key.
Median $\frac{n+1}2$ ; quartiles from halves.
Outliers beyond $1.5\times$ IQR.

IQR: $Q_3-Q_1$ .
Skew: Asymmetry; direction of the long tail.
Outlier: A value unusually far from the rest.

HistoryTukey's doodles. The American statistician John Tukey invented both the box plot (around 1970) and the stem-and-leaf diagram, and promoted them in his 1977 book Exploratory Data Analysis. He wanted quick pencil-and-paper pictures that let you "look at the data" before calculating anything.

Fun factTukey named the bit. The same John Tukey coined the word "bit" (binary digit) in 1947, and is credited with the first printed use of "software" in 1958. A good word, like a good diagram, makes an idea easy to use.

Section 3 of 14

Key formulae

Median position

\text{median}=\left(\tfrac{n+1}{2}\right)\text{th value}

Interquartile range

\text{IQR}=Q_3-Q_1

A common outlier rule

x<Q_1-1.5\times\text{IQR}\quad\text{or}\quad x>Q_3+1.5\times\text{IQR}

Positive skew: the median is nearer $Q_1$ and the upper whisker is long (a tail of high values).
Negative skew: the median is nearer $Q_3$ and the lower whisker is long.
Symmetrical: the median is in the middle of the box and the whiskers are similar.

TrickCount leaves, not stems. In a stem-and-leaf diagram, write a running total of leaves at the end of each row. Then the

8

th value, say, is easy to find: go to the row whose running total first reaches

8

. In a back-to-back diagram, remember the left-hand leaves increase from right to left.

Section 4 of 14

Worked examples

Example 1 · From a stem-and-leaf diagram0 / 3 steps

A stem-and-leaf diagram (key

2\,|\,3

means

23

) has rows

1\,|\,2\ 5\ 7

\ 2\,|\,0\ 3\ 3\ 8

\ 3\,|\,1\ 4\ 6\ 9

. Find the median and the interquartile range.

Example 2 · Outliers0 / 3 steps

Data:

1,3,4,4,5,6,7,8,9,22,30

. Using the

1.5\times

IQR rule, find any outliers.

Example 3 · Comparing0 / 2 steps

Group A: median

52

, IQR

8

. Group B: median

47

, IQR

15

. Compare the groups.

Section 5 of 14

Explore the graph

Build a box plot

Loading the interactive graph…

Try this

The sliders are the five-number summary: minimum $a$ , $Q_1=q$ , median $m$ , $Q_3=u$ , maximum $b$ .
Make the plot positively skewed: median close to $Q_1$ , long right whisker.
Using the $1.5\times$ IQR rule, how large can $b$ be before the maximum counts as an outlier?

The box holds the middle half of the data; its width is the IQR.

In the real worldBox plots in medicine. Medical researchers use side-by-side box plots to compare, say, blood pressure in a treatment group and a control group. Medians show which group did better on average, and the boxes show whether the improvement was consistent or driven by a few patients.

Section 6 of 14

Common mistakes

IncorrectA stem-and-leaf diagram with no key

CorrectAlways add a key

2\,|\,3

could mean

23

2.3

230

IncorrectMedian

=\dfrac{n+1}{2}

CorrectThe

\tfrac{n+1}2

th value

That formula gives a position, not the answer.

IncorrectReading the left side of a back-to-back diagram left to right

CorrectRight to left

Its leaves grow outwards from the stem.

IncorrectComparing only averages

CorrectCompare an average and a spread

Two groups with the same median can be very different.

Box plot

the median is right here

so it's halfway along the box?

Skewed data

absolutely not

Section 7 of 14

Exam tip

Exam tipCompare in context. When asked to compare two distributions, write two sentences: one about location ("on average, the boys' times were longer, since their median is higher") and one about spread ("the girls' times were more consistent, since their IQR is smaller"). Use the context of the question, not just "A is bigger".

Section 8 of 14

Guided practice

SA-9709-P5-DR1-0014 marksL1 FoundationCalculatorQuartiles

For the data

12,\ 15,\ 18,\ 20,\ 23,\ 27,\ 31

, find the median, the lower quartile, the upper quartile and the interquartile range.

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-9709-P5-DR-0064 marksL1 FoundationCalculatorQuartiles

For the data

2,\ 4,\ 5,\ 7,\ 8,\ 10,\ 12,\ 13,\ 15,\ 18,\ 21

, find the median, the lower and upper quartiles, and the interquartile range.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P5-DR1-0023 marksL2 StandardCalculatorDisplays

A stem-and-leaf diagram (key

2\,|\,3

means

23

) has the rows

1\,|\,2\ 5\ 7

\ 2\,|\,0\ 3\ 3\ 8

\ 3\,|\,1\ 4\ 6\ 9

. Find the median and the interquartile range.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P5-DR1-0042 marksL1 FoundationCalculatorDisplays

A box plot shows minimum

5

, lower quartile

12

, median

18

, upper quartile

25

and maximum

40

. Find the interquartile range and the range.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P5-DR1-0032 marksL2 StandardCalculatorDisplays

A data set has

Q_1=20

and

Q_3=32

. An outlier is defined as a value more than

1.5\times

IQR above

Q_3

or below

Q_1

. Find the value above which a data point is an outlier.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P5-DR1-0051 markL1 FoundationCalculatorQuartiles

Find the median of

3,\ 5,\ 6,\ 9,\ 11,\ 14,\ 15,\ 20

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P5-DR-0071 markL2 StandardCalculatorQuartiles

Group A has median

34

and IQR

12

. Group B has median

30

and IQR

20

. Which statement is correct?

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P5-DR1-0081 markL1 FoundationCalculatorQuartiles

A set of

15

values is written in order. Which value (1st, 2nd, …) is the lower quartile? Give its position.

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-9709-P5-DR1-0094 marksL3 ExamCalculatorDisplays

For the data

1,\ 3,\ 4,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 22,\ 30

, an outlier is a value more than

1.5\times

IQR beyond the nearer quartile. How many outliers are there?

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Stem-and-leaf: ordered, keeps raw values, needs a key.
Median at position $\tfrac{n+1}2$ ; quartiles are the medians of each half.
Box plot: min, $Q_1$ , median, $Q_3$ , max; outliers plotted separately.
Skew: the long tail tells you the direction.
Compare with an average and a spread, in context.

Section 12 of 14

Quick quiz

Q1 · SA-9709-P5-DR-Q021 markL1 FoundationCalculatorQuartiles

The median of

3,\ 6,\ 8,\ 11,\ 14,\ 20

Salam Academy Original Practice · Proposed mark scheme

Q2 · SA-9709-P5-DR1-0061 markL1 FoundationCalculatorDisplays

On a box plot the median is close to the lower quartile and the upper whisker is much longer than the lower one. The distribution is

Salam Academy Original Practice · Proposed mark scheme

Q3 · SA-9709-P5-DR1-0071 markL1 FoundationCalculatorDisplays

Which diagram shows the shape of a distribution and keeps every original data value?

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: 9709 AS & A Level Mathematics
Where: Paper 5, Probability & Statistics 1 · 5.1 Representation of data
Expected: Select a suitable way of presenting raw data; draw and interpret stem-and-leaf diagrams (including back-to-back) and box-and-whisker plots; find the median, quartiles and interquartile range.
Calculator: Permitted. Give non-exact answers to 3 significant figures; probabilities may be fractions or decimals.
Typical style: A back-to-back stem-and-leaf diagram comparing two groups: find medians and quartiles, draw box plots, and compare.
Open in: 9709 · Paper 5

Section 14 of 14

Past paper questions

9 questions (86 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

WST01/01 · June 2025Question 514 marks
Histogram heights, estimates, median, quartiles, skewness and a normal model.
Question paper, page 12 Mark scheme
WST01/01 · January 2025Question 212 marks
Range, median, quartiles and standard deviation from a stem-and-leaf diagram, then adding data.
Question paper, page 6 Mark scheme
WST01/01 · January 2025Question 78 marks
Read a box plot, interpolation and outliers, then a normal model.
Question paper, page 22 Mark scheme
9709/52 · Oct/Nov 2024Question 610 marks
Back-to-back stem-and-leaf, median and IQR, and combined mean and standard deviation.
Question paper, page 8 Mark scheme
9709/52 · May/June 2024Question 47 marks
Median and IQR from a stem-and-leaf diagram, box plots, and mean or median.
Question paper, page 6 Mark scheme
WST01/01 · June 2024Question 113 marks
Back-to-back stem and leaf, quartiles, outliers and box plots.
Question paper, page 2 Mark scheme

Show 3 more

9709/52 · Oct/Nov 2023Question 48 marks
Back-to-back stem-and-leaf, median and IQR, and comparing spread.
Question paper, page 5 Mark scheme
9709/52 · May/June 2023Question 37 marks
Back-to-back stem-and-leaf: median, IQR, box plots and choosing an average.
Question paper, page 6 Mark scheme
9709/52 · May/June 2022Question 37 marks
Back-to-back stem-and-leaf: median, IQR, box plots and a comparison.
Question paper, page 5 Mark scheme

All YMA01 past paper questions by topic All 9709 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Cambridge International 9709, 2026–2027 (official specification).