Lesson 4 ~25 min Unit 4 · Data & Probability +85 XP

Dot Plots and Stem-and-Leaf

Display every individual data value without losing any information. Spot clusters, gaps, outliers, and the overall shape of a distribution in seconds.

Today’s hook: A frequency table tells you how many values fall in each group. But it destroys the individual values. A dot plot and a stem-and-leaf plot show you every single value — and you can still see the distribution shape at a glance. Best of both worlds.

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Think First

warm-up

What is the fastest way to show every individual value in a data set while still being able to see the overall shape of the distribution?

Record in your workbook.

The Big Idea

+5 XP

Dot plots and stem-and-leaf plots are display-all tools. No information is lost. Every individual value appears in the diagram, making it easy to spot patterns.

A dot plot places one dot above a number line for each data value. A stem-and-leaf plot splits each value into a stem (leading digit/s) and a leaf (last digit), preserving the actual values while showing the shape.

One dot = one data value, stacked above the number line

What You’ll Master

objectives

Know

How to draw and read a dot plot
How to construct an ordered stem-and-leaf plot
How to interpret a back-to-back stem-and-leaf plot

Understand

How to identify mode, median, and range from these displays
How to describe shape: symmetric, skewed left, skewed right
What clusters, gaps, and outliers tell us about a data set

Can Do

Construct a dot plot and stem-and-leaf from raw data
Find mode, median, and range from each display
Compare two distributions using a back-to-back stem-and-leaf

Words You Need

vocabulary

Dot plotA graph where each data value is represented by a dot above a number line. Values stack vertically.

Stem-and-leaf plotA display that splits each value into a stem (leading digit/s) and a leaf (last digit), preserving all original values.

StemThe leading digits of a value (usually tens or hundreds). Shared across a row.

LeafThe last digit of a value. Written beside its stem in a row.

OutlierA value that sits well apart from the rest of the data. Appears as an isolated dot or leaf.

ClusterA group of values bunched closely together, showing a concentration in the data.

SkewA distribution is skewed left if the tail extends left, or skewed right if the tail extends right.

Back-to-backA stem-and-leaf plot showing two data sets sharing the same stems, leaves going left and right.

Spot the Trap

heads-up

Wrong: Leaving the leaves unordered in a stem-and-leaf plot. “3 | 7 2 5 1” makes it hard to find the median.

Right: Always write leaves in ascending order: “3 | 1 2 5 7”. Then finding the median is just counting to the middle.

Wrong: In a back-to-back plot, reading both sides left-to-right. The left group’s leaves read outward (right-to-left away from the stem).

Right: Left-side values are always read from the stem outward. The value “4 | 9” on the left means 94, not 49.

Drawing and Reading Dot Plots

+5 XP

Steps to draw a dot plot:

Draw a horizontal number line covering the range of the data.
For each data value, place a dot directly above that value on the line.
If a value repeats, stack the dots vertically.
Label the axis with the variable name and add a title.

Reading a dot plot:

Mode: the value with the tallest stack of dots
Range: maximum value − minimum value
Outlier: a dot isolated well away from the main group
Shape: symmetric (balanced), skewed right (long tail right), skewed left (long tail left)

Stem-and-Leaf Plots

+5 XP

For two-digit values: the stem = tens digit, the leaf = units digit. Always order leaves from smallest to largest within each row.

Example: Data: 23, 25, 31, 34, 38, 42, 44, 47, 51

Stem		Leaves
2	\|	3 5
3	\|	1 4 8
4	\|	2 4 7
5	\|	1

Key: Stem | Leaf, so 2 | 3 = 23 and 4 | 7 = 47

Median: with 9 values, the median is the 5th value (counting left to right through leaves): 31, 34, 38 → the 5th value overall is 34.

Back-to-Back Stem-and-Leaf

+5 XP

A back-to-back stem-and-leaf plot shares a central stem column. One group’s leaves go left, the other’s go right. It is the best tool for comparing two distributions.

Example: comparing Group A and Group B quiz scores.

Group A (leaves)	Stem	Group B (leaves)
9 7 4	3	2 5 8
8 5 3 1	4	0 3 6 9
6 2	5	1 4 7
3	6	0

Key: Group A reads outward (right to left). “3 | 4 | 9” means Group A has 43 and Group B has 49. Left leaves are ordered smallest closest to stem.

Describing Shape, Clusters, Gaps, and Outliers

+5 XP

When describing a distribution from a dot plot or stem-and-leaf, address these four features:

Shape

Symmetric (roughly equal on both sides), skewed right (tail extends right), or skewed left (tail extends left).

Clusters

Where are values bunched together? A cluster shows where data is concentrated.

Gaps

Empty sections on the number line. Large gaps can suggest subgroups in the population.

Outliers: isolated values far from the rest. Report the value and consider whether it might be a data entry error or a genuine extreme value.

Watch Me Solve It · Stem-and-leaf median

Watch Me Solve It · Find the median from a stem-and-leaf

+15 XP per step

PROBLEM

Data: 34, 38, 41, 45, 45, 48, 52, 56, 59, 63, 67. Construct an ordered stem-and-leaf plot and find the median.

1

Identify stems and sort

Stems: 3, 4, 5, 6. Order the data — already sorted here.

The stem is the tens digit. For values 30s, 40s, 50s, 60s → stems 3, 4, 5, 6.
2

Write each leaf beside its stem in order

3 | 4 8 4 | 1 5 5 8 5 | 2 6 9 6 | 3 7
3

Find the median (11 values → 6th value)

Count through leaves: 34, 38, 41, 45, 45, 48 → 6th value = 48

For $n = 11$ values, the median is at position $\frac{11+1}{2} = 6$.

AnswerMedian = 48. Stem-and-leaf: 3|4 8 4|1 5 5 8 5|2 6 9 6|3 7

Brain Trainer · 4 problems

Brain Trainer · Dot Plots & Stem-and-Leaf

4 problems

1 A dot plot shows values: 5 (1 dot), 6 (2 dots), 7 (4 dots), 8 (3 dots), 9 (1 dot), 12 (1 dot). What is the mode, range, and are there any outliers?

Mode = 7 (tallest stack, 4 dots). Range = 12 − 5 = 7. Outlier: 12 sits well apart from the main cluster of 5–9.Mode=7, Range=7, Outlier=12
2 Write the stem-and-leaf entry for these values: 61, 67, 73, 79, 75.

Stem 6: leaves 1, 7 (ordered). Stem 7: leaves 3, 5, 9 (ordered). Written: 6 | 1 7 7 | 3 5 96 | 1 7 7 | 3 5 9
3 From the stem-and-leaf: 4 | 2 5 8 5 | 0 3 3 7 6 | 1 4. How many values are there, and what is the range?

Count leaves: 3 + 4 + 2 = 9 values. Min = 42, Max = 64. Range = 64 − 42 = 22.n = 9, Range = 22
4 A back-to-back stem-and-leaf has Group A leaves on the left: “8 6 3 | 5 | 1 4 7”. What are all the values for Group A in the stem-5 row?

Read Group A outward from the stem: stem 5, leaves 3, 6, 8 (reading right-to-left but the values are formed stem+leaf): 53, 56, 58.53, 56, 58

Common Pitfalls

heads-up

Unordered leaves

Writing leaves in the order they appear in the data (e.g. 4 | 8 2 6 1) instead of sorted order (4 | 1 2 6 8). Makes it impossible to find the median quickly.

Fix: Always write an unordered draft first, then rewrite with ordered leaves.

Forgetting to include a key

Without a key, “3 | 5” could mean 35, 3.5, or 350. A key makes the plot unambiguous.

Fix: Always write “Key: 3 | 5 = 35” below every stem-and-leaf plot.

Copy This Into Your Book

Dot Plot

One dot per data value, stacked above a number line
Mode = tallest stack
Range = max − min
Describe: shape, clusters, gaps, outliers

Stem-and-Leaf

Stem = leading digit(s); Leaf = last digit
Order leaves smallest to largest
Always include a Key
Median: count to the middle position

Median Position

$n$ odd: median at position $\frac{n+1}{2}$
$n$ even: average of positions $\frac{n}{2}$ and $\frac{n}{2}+1$

Back-to-Back

Shared central stems
Left group’s leaves read outward (right-to-left)
Used to compare two distributions

Quick Check · 5 questions

A dot plot has stacks: 3 has 1 dot, 4 has 2 dots, 5 has 4 dots, 6 has 3 dots, 7 has 1 dot. What is the mode?

+10 XP

In a stem-and-leaf plot, the stem is 4 and the leaf is 7. What is the original value?

+10 XP

A stem-and-leaf plot shows: 2|3 5 3|1 4 8 4|2 4 7 5|1. What is the range?

+10 XP

A dot plot shows values 12, 13, 14, 14, 15, 16, 17, 18, 42. Which value is the outlier?

+10 XP

A back-to-back stem-and-leaf plot is most useful for:

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 3 MARKS

Q6. The following times (minutes) to complete a puzzle were recorded for 15 students: 8, 12, 9, 11, 14, 10, 8, 13, 9, 11, 15, 10, 12, 9, 7. Draw a dot plot (in your workbook). State the mode, range, and identify any outliers.

Draw the dot plot in your workbook.

Apply Medium 3 MARKS

Q7. Construct an ordered stem-and-leaf plot for these 12 values: 43, 27, 55, 48, 31, 62, 35, 51, 29, 46, 58, 37. Include a key. Find the median.

Draw in your workbook.

Analyse & Compare Hard 3 MARKS

Q8. A back-to-back stem-and-leaf shows test results for Class A (left) and Class B (right):
7 5 2 | 4 | 1 3 6
9 6 4 1 | 5 | 0 2 5 8
8 3 | 6 | 2 4 7
Describe TWO differences between the distributions of Class A and Class B.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. C — Value 5 has 4 dots, the tallest stack.

2. A — Stem 4, leaf 7 → 47.

3. B — Range = 51 − 23 = 28.

4. D — 42 is isolated far from the cluster 12–18.

5. A — Back-to-back compares two group distributions.

Model Answers

Q6 (3 marks): Values range from 7 to 15. Mode = 9 (appears three times: positions verified from data). Range = 15 − 7 = 8 [1]. No outliers — all values are clustered in a continuous range 7–15 [1]. Shape: roughly symmetric, slight cluster around 9–12 [1].

Q7 (3 marks): Sorted: 27, 29, 31, 35, 37, 43, 46, 48, 51, 55, 58, 62. Stem-and-leaf: 2|7 9 3|1 5 7 4|3 6 8 5|1 5 8 6|2. Key: 2|7 = 27 [1 for plot; 1 for correct ordered leaves]. Median of 12 values: average of 6th and 7th = (43 + 46) ÷ 2 = 44.5 [1].

Q8 (3 marks): [2 marks for two valid comparisons, 1 for using specific values]. Possible answers: (1) Class B has higher scores overall — its median is in the 50s stem while Class A’s median is also in the 50s but the upper cluster differs; (2) Class A has more scores in the high 50s and low 60s, while Class B has more spread through the 40–60 range; (3) Class A’s lowest value is 42 while Class B’s lowest is 41 — similar minimums, but Class B has higher maximum of 67 vs Class A’s 68.

Stretch Challenge · +25 XP, +10 coins

Heights Back-to-Back

Heights of 10 boys (cm): 152, 158, 160, 161, 165, 167, 170, 172, 175, 181.
Heights of 10 girls (cm): 148, 152, 155, 158, 160, 162, 163, 165, 168, 172.
Construct a back-to-back stem-and-leaf plot. Compare the median, range, and shape of each group’s distribution.

Reveal solution

Back-to-back:
Boys (right-to-left) | Stem | Girls (left-to-right)
                   | 14 | 8
8 2                 | 15 | 2 5 8
7 5 1 0            | 16 | 0 2 3 5 8
5 2 0               | 17 | 2
1                   | 18 |
Key: 15|2 = 152.
Boys: Median = average of 5th & 6th = (165+167)/2 = 166. Range = 181−152 = 29.
Girls: Median = average of 5th & 6th = (160+162)/2 = 161. Range = 172−148 = 24.
Shape: Boys are slightly higher overall (median 166 vs 161) and have a greater spread (range 29 vs 24). Both distributions are roughly symmetric with a slight skew toward lower values (more data in the 150s–170s).

Quick Review

Dot plot

One dot per value, stacked above a number line

Mode from dot plot

Value with the tallest stack of dots

Stem-and-leaf

Stem = leading digit(s), leaf = last digit

Always order leaves

Smallest to largest within each row; include key

Back-to-back

Shared stems; left group reads outward

Describe distributions

Shape, clusters, gaps, outliers

Badges This Lesson

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Dot Plotter

Stem Spotter

Leaf Reader

Outlier Observer

Shape Describer

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Unit overview · Maths subject page