Mr. Ward's Teaching Materials

111i - Creating Data Displays

9 min read

Lesson 111 — Creating Numerical Data Displays, Including Stem-and-Leaf Plots

Strand: Statistics | Descriptor: AC9M7ST02 | Duration: 45 minutes

Learning Intentions

  • To construct stem-and-leaf plots, dot plots and column graphs for numerical data.
  • To choose a display suited to the data and the question.

Success Criteria

I can:

  1. Construct a stem-and-leaf plot with a key.
  2. Construct a dot plot and a column graph accurately.
  3. Choose an appropriate display for a given data set.
  4. Read individual values back out of a display.

Warmup

(6 minutes — what does this show? projected)

Display this stem-and-leaf plot without explanation:

 Stem | Leaf
    1 | 2 5 7
    2 | 0 3 3 6 8 9
    3 | 1 4 4
    4 | 2

Key: 2 | 3 means 23
  1. How many values are in the data set?
  2. What is the smallest value? The largest?
  3. Which value appears twice?
  4. What was the point of the key?

Answers: 1. Thirteen; 2. and ; 3. and both appear twice; 4. Without it, "" could mean , or — the key fixes the place value.

The plot’s virtue, named: it shows the shape of the data and keeps every original value. No other display does both.

Activities

Activity 1 — Explicit Instruction: the Stem-and-leaf Plot (14 min)

How it works. Each value splits into a stem (all but the last digit) and a leaf (the last digit). Values sharing a stem share a row.

I do — build one from raw data. Test scores out of :

Step 1 — find the range of stems. Smallest , largest : stems , , .

Step 2 — place each leaf, unordered first:

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

Step 3 — order the leaves within each row:

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

Key: 3 | 4 means 34

Step 4 — add the key. Never omit it.

What the plot gives you immediately:

  • Every value is recoverable —
  • The shape — the s row is longest, so the data bunches there.
  • The median — count to the th of values: .
  • The range — .

Empty stems must still appear. If no value falls in the s, the row is written with no leaves — otherwise the gap in the data is invisible, and the shape lies.

We do — build together: heights in cm: .

 Stem | Leaf
   14 | 7 8 9
   15 | 2 2 5 6 8
   16 | 1 3

Key: 15 | 2 means 152 cm

(Stems are the tens-and-hundreds here — the split point is a choice, made so the plot has a useful number of rows.)

Activity 2 — Three Displays, One Data Set (14 min)

Pairs, grid paper. Same data, three ways.

Goals scored by a team in matches:

Task 1 — frequency table.

GoalsTallyFrequency
…

Task 2 — dot plot. One dot per match, stacked above each value on a number line.

Task 3 — column graph. Bars with gaps between them (the data is discrete), height = frequency, axes labelled.

Then answer:

  1. Which display shows the mode most obviously?
  2. Which shows individual values best?
  3. Which would you use in a newsletter? Why?
  4. Why do the columns have gaps between them?

Socratic scaffolding:

PromptPurpose
Count carefully — do your three displays agree on the totals?Cross-checking between representations.
Q4: what would touching bars imply?Continuous data, where categories run into each other. Discrete counts get gaps.
Would a stem-and-leaf plot suit this data?Poorly — all values are single digits, so there would be one stem. Displays have ranges of usefulness.

(Answers: frequencies , , , , , ; total ✓. 1. Dot plot or column graph — the tallest stack is instant; 2. All three are equivalent here since values repeat, but the dot plot shows each match as a dot; 3. Column graph — familiar and quick to read; 4. Gaps signal discrete data: there is no ” goals” category.)

Activity 3 — Inquiry: Which Display When? (9 min)

Pairs.

For each situation, choose a display and justify. There may be more than one defensible answer.

  1. Twenty students’ exact test marks out of , and the teacher wants to see individual marks.
  2. The number of siblings of students.
  3. Heights of students, to the nearest centimetre.
  4. Daily rainfall over a month, showing which days were wettest.

Socratic scaffolding:

PromptPurpose
Q1: what does the teacher need that a column graph would lose?The individual marks — a stem-and-leaf keeps them.
Q2: how many distinct values?About to — few categories, so a dot plot or column graph is clean.
Q3: fifty different heights — what happens in a dot plot?Fifty separate columns of one — useless. Group into intervals (Lesson 107).
Q4: what makes rainfall different?It is time-ordered — the sequence matters, so a column graph by day, or a line graph.
The general rule?Match the display to the number of distinct values and to the question being asked.

The decision guide to record:

DataBest display
Few distinct values, discreteDot plot or column graph
Many values, want individuals keptStem-and-leaf plot
Many values, continuousGrouped column graph (histogram-style)
Values over timeColumn or line graph in time order

Checks for Understanding

(5 minutes — exit ticket)

Data: .

  1. Construct a stem-and-leaf plot with a key.
  2. From your plot, state the smallest and largest values.
  3. From your plot, find the median.
  4. Why must a stem-and-leaf plot include a key?
  5. Reasoning. Why would a dot plot be a poor choice for this data?

Answers: 1.

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

Key: 2 | 3 means 23
  1. and ; 3. Ten values, so the median is between the th () and th (): ; 4. Without it the place value is ambiguous; 5. Ten values spread across possible numbers would give a nearly flat row of single dots — no shape visible.

Common Misconceptions

MisconceptionHow to pre-empt it
Omitting the key.Warmup Q4 and every construction; marked.
Leaves left unordered.Step 3 is explicit; unordered leaves make the median hard to find.
Skipping empty stems.Named directly — the gap is information.
Using more than one digit as a leaf.The leaf is a single digit; the stem takes the rest.
Column graphs with touching bars for discrete data.Task 3’s Q4.
Choosing a display without considering the question.The decision guide, built from the inquiry.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A stem-and-leaf plot has stems with , and leaves. How many values, and what is the median’s position?

Answer

Twelve values; the median lies between the th and th, both in the stem.

E2 (AMC Junior style). In the plot below, find the range and the mode.

 Stem | Leaf
    1 | 4 7 7
    2 | 0 2 5 5 5 8
    3 | 1 6

Key: 2 | 0 means 20
Answer

Values to : range . The mode is (three times).

E3 (Challenge). A stem-and-leaf plot of values has a median of . The stems are , and . What is the minimum number of leaves the stem must have?

Answer

The median is the th value and equals , so at least one leaf is in the stem. If the stem held leaves, the th value is the stem’s first — one leaf suffices. So the minimum is one.

E4 (Challenge). Two classes’ marks are shown in a back-to-back stem-and-leaf plot (leaves extending left and right of a shared stem). What is the advantage over two separate plots?

Answer

The shared stem aligns the two distributions on identical scales, so shape, centre and spread can be compared at a glance — separate plots invite scale mismatches, exactly the trap from Lesson 83.

Homework

  1. Construct a stem-and-leaf plot with a key for: .
  2. From your Q1 plot, state: (a) the range (b) the median (c) the mode, if any.
  3. Construct a stem-and-leaf plot for these masses (kg): . (Hint: use the whole number as the stem and the decimal digit as the leaf.)
  4. Draw a dot plot for the number of pets owned by students: .
  5. Draw a column graph for the same data, with labelled axes and gaps between the bars.
  6. Which display would you choose for each, and why: (a) exam marks out of for students, keeping individual marks (b) the number of siblings of students (c) the heights of adults to the nearest centimetre?
  7. Read the values out of this plot and list them in order:
 Stem | Leaf
    6 | 2 5 9
    7 | 0 0 3 8
    8 | 1

Key: 7 | 0 means 70
  1. Reasoning. Explain why an empty stem must still be shown.
  2. Reasoning. Explain what a stem-and-leaf plot can do that a column graph cannot.
  3. Challenge. Construct a back-to-back stem-and-leaf plot for Class A: and Class B: . Comment on which class performed better.

Answers: Q1 — stems (), (), (); key means . Q2 — (a) (b) twelve values: median between the th () and th () (c) no mode. Q3 — stems (), (), (); key means kg. Q6 — (a) stem-and-leaf (b) dot plot or column graph (c) grouped column graph. Q7 — . Q8 — the empty row shows a genuine gap in the data; omitting it distorts the shape. Q9 — it retains every original value, so exact figures and the median can be read directly. Q10 — Class A: (median ); Class B: (median ). Class A performed slightly better on both median and minimum.

Graph View

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