What are not ordering the leaves?

A finished stem-and-leaf plot must have the leaves in ascending order along each row; an unordered plot loses the "ordered" mark and makes the median hard to read.

NSWMaths Standard 2Syllabus dot point

How are dot plots and stem-and-leaf plots used to display a data set and reveal its clusters, gaps, outliers and shape?

Display and interpret numerical data using dot plots and stem-and-leaf plots, including back-to-back stem-and-leaf plots, and describe the clusters, gaps, outliers and shape of the data

A focused answer to the HSC Maths Standard 2 dot point on dot plots and stem-and-leaf plots. How to construct and read each display, how to build a back-to-back stem-and-leaf plot to compare two groups, and how to describe clusters, gaps, outliers and the shape of a distribution, with worked Australian examples.

Generated by Claude Opus 4.814 min answerUpdated 2026-06-21

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking

NESA wants you to display a set of numerical data in two simple but powerful ways - the dot plot and the stem-and-leaf plot - and then to read meaning out of the picture. Building each display is the easy half. The marks live in the second half: describing the data's clusters (where values bunch up), gaps (empty stretches), outliers (lone values far from the rest) and shape (symmetric, or skewed to one side). You also need the back-to-back stem-and-leaf plot, which lines two groups up on a shared stem so you can compare them at a glance. These displays are the entry point to the whole Data Analysis module, because the words you learn here - cluster, outlier, skew - are the same words every later question wants.

The answer

Both displays keep every original data value visible while showing the overall pattern. A dot plot stacks one dot per value above a number line, so the height of each stack is that value's frequency. A stem-and-leaf plot splits each number into a stem (the leading digit or digits) and a leaf (the last digit), groups the leaves by stem, and so sorts the data while keeping every figure readable. Choose the dot plot for small data sets over a short range of whole numbers; choose the stem-and-leaf plot once the numbers spread over many values (typically two-digit data), where a dot plot would become a forest of stacks.

In the dot plot above, the stack heights are the frequencies: one game with $0$ goals appears as a column of dots, and the value $1$ has the tallest stack, so $1$ goal is the mode. The lone dot above $7$ , cut off from the rest by an empty stretch, is an outlier.

Constructing a dot plot

A dot plot is built on a horizontal number line. The steps are short:

Draw a number line that covers the full range of the data, marking every whole value from the smallest to the largest (including values with no data, so gaps show up).
For each data value, place one dot above its position on the line.
Stack repeated values directly on top of one another, keeping the dots the same size and evenly spaced, so the stack heights can be compared fairly.

Reading it back, the height of a stack is the frequency of that value, the tallest stack marks the mode, and you can spot clusters, gaps and outliers by eye. Dot plots suit small data sets of whole numbers over a short range - the number of pets per household, goals per game, siblings per student. Once the data spreads over dozens of values, switch to a stem-and-leaf plot.

Constructing a stem-and-leaf plot

A stem-and-leaf plot (sometimes called a stem plot) splits each number into two parts. For two-digit numbers the stem is the tens digit and the leaf is the units digit, so $74$ becomes stem $7$ , leaf $4$ . To build one:

List the stems in a vertical column, in order, from smallest to largest, drawing a vertical line to their right.
Write each value's leaf in the row of its stem.
Re-write each row with the leaves in ascending order (smallest first), which sorts the whole data set.
Add a key, for example $7\,|\,4$ means $74$ , so a reader knows how to decode the plot.

The plot below shows the resting heart rates $58, 62, 65, 67, 71, 72, 72, 74, 75, 78, 80, 81, 84, 85, 88, 96$ beats per minute.

Stem | Leaf
  5  | 8
  6  | 2 5 7
  7  | 1 2 2 4 5 8
  8  | 0 1 4 5 8
  9  | 6

Key: 5 | 8 means 58 beats per minute

Turned on its side, the rows of leaves are exactly the stacks of a dot plot, so the longest row (the $70$ s) is where the data clusters, and the shape of the data is the shape of the leaf lengths. Because every original value is preserved, you can read the smallest value ( $58$ ), the largest ( $96$ ) and the mode ( $72$ , the repeated leaf in the $70$ s) straight off the plot.

Back-to-back stem-and-leaf plots

A back-to-back stem-and-leaf plot compares two data sets that share the same stems. The stems sit in a central column; one group's leaves run to the left and the other group's to the right. The one thing to watch is direction: on the left side the leaves increase outward from the stem, so the smallest leaf sits nearest the stem and the row reads right to left. The example below compares two squads' push-up counts.

Reading it, both squads have $8$ players. The junior leaves bunch in the $20$ s and the senior leaves bunch in the $30$ s, so the seniors completed more push-ups on average. Comparing two groups always comes back to the same two ideas: which group has the higher centre (typical value) and which has the larger spread (how stretched out the values are).

Describing clusters, gaps, outliers and shape

Once a display is drawn, a "describe the data" question wants a short paragraph hitting these features:

Cluster. A range where values bunch together. State where the data piles up ("most values lie between $1$ and $3$ goals").
Gap. A stretch of the scale with no data. Name it ("there is a gap from $3$ to $7$ "); a gap before a lone value is what marks it as an outlier.
Outlier. A value sitting well away from the rest, usually past a gap. Name the value and say it is an outlier.
Shape. Describe the overall outline of the stacks or leaves:
- Symmetric - the two sides mirror each other about a central peak.
- Positively skewed (skewed to the right) - a peak on the left with a long tail trailing to the higher values.
- Negatively skewed (skewed to the left) - a peak on the right with a long tail trailing to the lower values.
- Bimodal - two separate peaks (two modes).

A useful tip on skew: the skew is named for the side the long tail points to, not the side of the peak. A long tail stretching towards the big numbers is positive skew.

How exam questions ask about dot plots and stem-and-leaf plots

The command words map straight onto the method:

"Construct / draw a dot plot (or stem-and-leaf plot)" - build the display: a number line with stacked dots, or stems with sorted leaves and a key. For stem-and-leaf plots the key is worth a mark on its own, so never omit it.
"How many ... ?" - count dots or leaves; the total must match the number of items given (use this as your check).
"State the mode / median / range" - read them off: the tallest stack or repeated leaf is the mode; the values are already in order for the median and range.
"Identify any outliers" - name the lone value (or values) past the gap and call it an outlier.
"Describe the shape / distribution" - give the cluster, any gap, any outlier, and the shape word (symmetric or the direction of skew).
"Compare the two data sets" (a back-to-back plot) - address BOTH centre (which group is higher on average) AND spread (which group's values vary more), each justified from the plot.

Building and reading dot plots and stem-and-leaf plots

Three tasks, one for each display type: construct a dot plot and describe it, build a stem-and-leaf plot, and read a back-to-back stem-and-leaf comparison. Each finishes by reading meaning out of the picture, which is where the marks sit.

Construct a dot plot and describe it

The number of pets owned by $20$ students is recorded: $0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 8$ . Construct a dot plot and describe the data.

Tally each value first. Counting the occurrences of each number of pets:

0:2,\quad 1:4,\quad 2:6,\quad 3:4,\quad 4:2,\quad 5:1,\quad 8:1

The counts total $2 + 4 + 6 + 4 + 2 + 1 + 1 = 20$ , matching the $20$ students.

Draw the plot. On a number line from $0$ to $8$ , stack that many dots above each value (two dots above $0$ , four above $1$ , six above $2$ , and so on), leaving values $6$ and $7$ empty.

Describe it. The data clusters between $0$ and $5$ pets, with the tallest stack at $2$ pets, so the mode is $2$ . There is a gap from $5$ to $8$ where no student appears, and the value $8$ sits alone past that gap, so $8$ is an outlier. The main cluster rises to a peak at $2$ and falls away fairly evenly on each side, so apart from the outlier the shape is roughly symmetric.

Build a stem-and-leaf plot

The masses (in kilograms, rounded) of $15$ dogs at a vet are: $7, 12, 14, 18, 21, 23, 23, 25, 28, 31, 34, 35, 42, 46, 53$ . Construct an ordered stem-and-leaf plot and state the mode and the range.

Choose stems and place the leaves. The values run from single digits to the $50$ s, so use the tens digit as the stem (treating $7$ as stem $0$ , leaf $7$ ) and the units digit as the leaf. Placing each value and ordering the leaves:

\begin{array}{r|l} 0 & 7 \\ 1 & 2\ 4\ 8 \\ 2 & 1\ 3\ 3\ 5\ 8 \\ 3 & 1\ 4\ 5 \\ 4 & 2\ 6 \\ 5 & 3 \\ \end{array}

with the key $2\,|\,1$ means $21$ kilograms.

Read off the mode. The only repeated value is $23$ (the leaf $3$ appears twice in the $20$ s row), so the mode is $23$ kg.

Read off the range. The smallest value is $7$ and the largest is $53$ , so

53 - 7 = 46

the range is $46$ kg. (Check: the leaves number $1 + 3 + 5 + 3 + 2 + 1 = 15$ , matching the $15$ dogs.)

Compare two groups with a back-to-back stem-and-leaf plot

Two Year $11$ classes sit the same quiz out of $50$ . Class A: $22, 27, 31, 33, 35, 38, 41, 44$ . Class B: $18, 24, 26, 29, 33, 37, 42, 45$ . Build a back-to-back stem-and-leaf plot and compare the classes.

Set up the shared stems and split the leaves. Using the tens digit as the stem, Class A leaves go left and Class B leaves go right, each ordered outward from the stem:

Class A     | Stem |  Class B
            |   1  |  8
          7 2 |   2  |  4 6 9
      8 5 3 1 |   3  |  3 7
          4 1 |   4  |  2 5

with the key Class A $2\,|\,2$ means $22$ and Class B $1\,|\,8$ means $18$ .

Compare centre. Class A's marks bunch in the $30$ s while Class B's are more spread between the $20$ s and $40$ s, with Class A's typical mark a little higher. Class A runs $22$ to $44$ and Class B runs $18$ to $45$ .

Compare spread. Class A's range is $44 - 22 = 22$ and Class B's range is $45 - 18 = 27$ , so Class B's marks are more spread out. In short, Class A is slightly higher on average and more consistent, while Class B is more variable. (Check: $8$ leaves on each side, matching the $8$ students per class.)

Common traps

Leaving out the key on a stem-and-leaf plot: Without a key the reader cannot tell whether $5\,|\,8$ means $58$ , $5.8$ or $580$ . The key is a marked item, so always write one (for example $5\,|\,8$ means $58$ ).
Not ordering the leaves: A finished stem-and-leaf plot must have the leaves in ascending order along each row; an unordered plot loses the "ordered" mark and makes the median hard to read.
Reading the left side of a back-to-back plot the wrong way: On the left the leaves grow outward from the stem, so the smallest leaf is nearest the stem and the row reads right to left. Reading it left to right reverses the values.
Naming the skew by the peak instead of the tail: Skew is named for the side the long tail points to. A peak on the left with a tail running right is positive skew, not negative.
Forgetting empty values on a dot plot: Leave a space on the number line for values with no data, otherwise a gap (and any outlier beyond it) is hidden.
Comparing only the centre: "Compare" questions want centre AND spread. Mentioning only which group is higher on average drops the spread mark.

In plain English

Think of a dot plot as everyone in a class lining up over their answer, one person standing on another's shoulders when they pick the same number; the tallest tower shows the most popular answer, and a single person standing far off on their own is the odd one out. A stem-and-leaf plot is just a tidy way of filing numbers: the front digits become labelled drawers and you drop the last digit of each number into the right drawer, so the numbers sort themselves while you can still see every single one. The fullest drawer is where the data crowds together. When you want to compare two teams, you stand them on opposite sides of the same set of drawers and see whose numbers sit higher and whose are more spread out. The "shape" is simply whether the pile is even on both sides or has a long tail leaning off to one end.

Exam-style practice questions

Practice questions written in the style of NESA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

2021 HSC-style3 marksThe dot plot below shows the number of times

20

people visited a gym in a week. The stacks are:

0

visits -

4

dots;

1

5

;

2

4

;

3

3

;

4

2

;

5

1

;

9

1

. (a) Find the number of people who visited at least

3

times. (b) Identify the outlier. (c) Describe the shape of the data.

Show worked answer →

Part (a): "at least $3$ " means $3$ or more, so add the stacks at $3, 4, 5$ and $9$ : $3 + 2 + 1 + 1 = 7$ people. One mark for the correct total; a common slip is to omit the outlier at $9$ or to misread "at least" as "more than".

Part (b): the value $9$ is the outlier, sitting alone well above the rest with a clear gap after $5$ .

Part (c): the main body of the data is clustered at the low end ( $0$ to $5$ ) with a peak at $1$ visit and a long tail trailing out to the right towards $9$ , so the distribution is positively skewed (skewed to the right) with an outlier at $9$ .

Markers reward the correct count, naming the outlier by its value, and a shape description that uses the words "cluster", "skew" (with the correct direction) and "outlier".

2022 HSC-style4 marksA back-to-back stem-and-leaf plot compares the ages of members at two community clubs, where

3\,|\,4

means

34

. Club A leaves (left):

1\,|\,8\ 9

;

2\,|\,3\ 5\ 7

;

3\,|\,1\ 6

;

4\,|\,2

. Club B leaves (right):

2\,|\,1\ 4

;

3\,|\,0\ 5\ 5\ 8

;

4\,|\,3\ 7

;

5\,|\,2

. (a) State how many members are in each club. (b) State the youngest member overall. (c) Compare the ages of the two clubs, referring to centre and spread.

Show worked answer →

Part (a): count leaves on each side. Club A has $2 + 3 + 2 + 1 = 8$ members; Club B has $2 + 4 + 2 + 1 = 9$ members.

Part (b): the youngest is $18$ (Club A, the smallest leaf in the $10$ s row); Club B's youngest is $21$ , so $18$ is the youngest overall.

Part (c): Club A's ages cluster in the late teens to thirties and Club B's cluster in the thirties to forties, so Club B has the higher centre (older typical member). For spread, Club A runs $18$ to $42$ (range $24$ ) and Club B runs $21$ to $52$ (range $31$ ), so Club B is also more spread out.

Markers reward both correct counts, the correct youngest value read from the right row, and a comparison that explicitly addresses BOTH centre (which club is older on average) AND spread (which club's ages vary more), each justified from the plot.

2023 HSC-style3 marksA student writes the stem-and-leaf plot for the data

32, 35, 41, 41, 44, 50, 58

but forgets the key and leaves one row blank. Their plot reads:

3\,|\,2\ 5

;

4\,|\,1\ 1\ 4

;

5\,|\,0\ 8

. (a) Write a suitable key. (b) State the mode. (c) The student claims the data has a gap. Explain whether this is correct.

Show worked answer →

Part (a): a suitable key is $3\,|\,2$ means $32$ (any correct stem-leaf example earns the mark).

Part (b): the mode is $41$ , since the leaf $1$ repeats in the $40$ s row ( $41$ appears twice) and no other value repeats.

Part (c): there is no genuine gap. Every stem from $3$ to $5$ has at least one leaf, so the values run fairly continuously from $32$ to $58$ with no empty stretch and no isolated point. The student is incorrect; the data is reasonably evenly spread, not split by a gap.

Markers reward a valid key, the correct mode read from the repeated leaf, and a justified yes/no on the gap that refers to the rows all being populated (no empty stem, no isolated value).

Practice questions

Original practice questions graded from foundation to exam level, each with a full worked solution. Try them before revealing the solution.

foundation2 marksA class records the number of siblings each of

15

students has:

0, 1, 1, 2, 2, 2, 1, 0, 3, 2, 1, 4, 2, 1, 2

. (a) State the most common number of siblings. (b) State how many students have exactly

1

sibling.

Show worked solution →

Tally each value. Counting how many times each number appears:

0:\ 2,\quad 1:\ 5,\quad 2:\ 6,\quad 3:\ 1,\quad 4:\ 1

The counts add to $2 + 5 + 6 + 1 + 1 = 15$ , which matches the $15$ students, so nothing is missed.

Part (a) - most common value. The tallest count is for $2$ siblings (it occurs $6$ times), so the most common number is $2$ .

Part (b) - students with exactly one sibling. The value $1$ occurs $5$ times, so $5$ students have exactly $1$ sibling. (On a dot plot these are the heights of the stacks above $2$ and above $1$ .)

foundation3 marksThe dot plot shows the number of goals scored by a netball player in each game of a season. The stacks above the values are:

0

goals -

3

dots;

1

goal -

5

dots;

2

goals -

4

dots;

3

goals -

2

dots;

7

goals -

1

dot. (a) How many games were played? (b) Identify the outlier. (c) Describe the cluster.

Show worked solution →

Part (a) - total games. Each dot is one game, so add the stack heights:

3 + 5 + 4 + 2 + 1 = 15 \text{ games}

Part (b) - the outlier. Most games sit between $0$ and $3$ goals, but one lonely dot sits out at $7$ goals with a clear gap before it. The outlier is the game of $7$ goals.

Part (c) - the cluster. The bulk of the data is bunched between $0$ and $3$ goals, so there is a cluster from $0$ to $3$ goals, with most games at $1$ goal. (The gap from $3$ to $7$ is what makes the $7$ stand out as the outlier.)

core3 marksThese are the resting heart rates (beats per minute) of

16

people:

58, 62, 65, 67, 71, 72, 72, 74, 75, 78, 80, 81, 84, 85, 88, 96

. Construct a stem-and-leaf plot using the tens digit as the stem, then state the lowest and highest rates.

Show worked solution →

Choose the stem. The values run from the $50$ s to the $90$ s, so use the tens digit as the stem and the units digit as the leaf.

Sort each value into its row, leaves in order. Reading along each ten:

Stem | Leaf
  5  | 8
  6  | 2 5 7
  7  | 1 2 2 4 5 8
  8  | 0 1 4 5 8
  9  | 6

A key is essential: $5\,|\,8$ means $58$ beats per minute.

Read off the extremes. The first leaf in the top row gives the lowest rate, $58$ , and the last leaf in the bottom row gives the highest, $96$ . (Check the leaf count: $1 + 3 + 6 + 5 + 1 = 16$ , which matches the $16$ people.)

core3 marksA stem-and-leaf plot of test marks (out of

100

) is shown, where

6\,|\,3

means

63

. Stems and leaves:

4\,|\,2\ 8

;

5\,|\,1\ 5\ 5\ 9

;

6\,|\,0\ 3\ 3\ 4\ 7

;

7\,|\,2\ 6\ 8

;

8\,|\,5

. (a) How many students sat the test? (b) State the modal mark. (c) State the range.

Show worked solution →

Part (a) - number of students. Count every leaf:

2 + 4 + 5 + 3 + 1 = 15 \text{ students}

Part (b) - the modal mark. The mode is the value that appears most often. The leaves show two $5$ s in the $50$ s row ( $55$ twice) and two $3$ s in the $60$ s row ( $63$ twice). Both $55$ and $63$ occur twice and no value occurs more, so the data is bimodal with modes $55$ and $63$ .

Part (c) - the range. The smallest mark is the first leaf of the top row, $42$ , and the largest is the single leaf of the bottom row, $85$ :

85 - 42 = 43

so the range is $43$ marks. (A stem-and-leaf plot keeps every original value, which is why the mode and range can be read straight off it.)

core4 marksTwenty households are surveyed for the number of devices connected to their home internet. The counts are:

2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 8, 9, 9, 14, 15

. (a) Construct a frequency tally for each value from

2

9

. (b) Identify any outliers and the gap that separates them. (c) Describe the shape of the main cluster.

Show worked solution →

Part (a) - tally the values $2$ to $9$ . Counting each:

2:1,\ 3:2,\ 4:3,\ 5:4,\ 6:3,\ 7:2,\ 8:1,\ 9:2

These $18$ values, plus the two large counts $14$ and $15$ , give $18 + 2 = 20$ households.

Part (b) - outliers and the gap. The values $14$ and $15$ sit far above the rest. There is a wide gap from $9$ up to $14$ where no household appears, so $14$ and $15$ are outliers separated from the main group by that gap.

Part (c) - shape of the main cluster. Ignoring the two outliers, the counts rise to a peak of $4$ at the value $5$ and then fall away roughly evenly on each side ( $1, 2, 3, 4, 3, 2, 1, 2$ ). The main cluster is therefore roughly symmetric, mounded around $5$ devices. (Plotting these as a dot plot makes the single mound and the two stray dots at $14$ and $15$ easy to see.)

exam5 marksA coach records the number of push-ups completed in one minute by the players in two squads. Junior squad:

18, 21, 22, 24, 25, 27, 28, 31

. Senior squad:

26, 29, 32, 33, 35, 37, 38, 41

. (a) Construct a back-to-back stem-and-leaf plot using the tens digit as the stem (junior leaves on the left, senior on the right). (b) State the highest score in each squad. (c) Compare the two squads' performance, referring to centre and spread.

Show worked solution →

Part (a) - build the back-to-back plot. Use the tens digit as the shared stem. Junior leaves are written to the LEFT of the stem (read right to left, smallest nearest the stem); senior leaves to the RIGHT (read left to right):

Junior      | Stem |  Senior
          8 |   1  |
8 7 5 4 2 1 |   2  |  6 9
          1 |   3  |  2 3 5 7 8
            |   4  |  1

A key is required: for the seniors $2\,|\,6$ means $26$ ; for the juniors $5\,|\,2$ means $25$ .

Part (b) - highest in each squad. Reading the largest value in each side, the junior best is $31$ (the lone leaf $1$ in the $30$ s row, left side) and the senior best is $41$ (the leaf $1$ in the $40$ s row, right side).

Part (c) - compare centre and spread. Both squads have $8$ players. The senior values cluster in the $30$ s while the junior values cluster in the $20$ s, so the seniors have the higher centre (a higher typical number of push-ups). For spread, the juniors run from $18$ to $31$ , a range of $31 - 18 = 13$ , and the seniors run from $26$ to $41$ , a range of $41 - 26 = 15$ , so the spreads are similar with the seniors slightly more spread out. Overall the seniors perform better on average, by a margin of roughly ten push-ups. (Check: $8$ leaves on each side, matching the $8$ players per squad.)

exam5 marksThe waiting times (in minutes, rounded) at a medical centre are recorded for

20

patients:

4, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 9, 9, 10, 11, 12, 24, 26

. (a) Describe the shape of the data and name the type of skew. (b) Identify the outliers. (c) Explain, with reference to the plot, why the mode of

7

minutes is a fairer summary of a typical wait than the mean would be.

Show worked solution →

Part (a) - shape and skew: Tallying the main group: the counts climb to a peak at $7$ minutes (which occurs $4$ times) and then trail off slowly towards the larger times. A distribution with a short side on the left and a long tail stretching to the right is positively skewed (skewed to the right).
Part (b) - the outliers: The values $24$ and $26$ minutes sit far above the main cluster, separated from it by a long gap after $12$ minutes, so $24$ and $26$ are outliers.
Part (c) - why the mode is fairer here: The mean is pulled upward by the two large outliers ( $24$ and $26$ ), so it would report a "typical" wait longer than almost anyone actually experienced. The plot shows the data piled up between $4$ and $12$ minutes with a clear peak at $7$ , so the mode of $7$ minutes describes the most common, and most representative, waiting time without being dragged up by the two unusual long waits. (This is the general rule: a long right tail or outliers make the mean an over-estimate of the centre, so a resistant measure is fairer.)

What this dot point is asking

The answer

Constructing a dot plot

Constructing a stem-and-leaf plot

Back-to-back stem-and-leaf plots

Describing clusters, gaps, outliers and shape

How exam questions ask about dot plots and stem-and-leaf plots

Exam-style practice questions

Practice questions

Related dot points