← Unit 4 Box Plots and Data Visualisation

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MA5-DAT-C-01

Box Plots and Data Visualisation

⏱ 25 min📚 Year 9📈
Think First

A box plot shows minimum=2, Q1=5, median=8, Q3=12, maximum=20. What can you conclude about the data?

💡 Revision: Ensure you can find median, Q1, Q3, and IQR.

Learning Intentions

Know

  • Box plot (box-and-whisker)
  • Five-number summary
  • Histogram
  • Stem-and-leaf

Understand

  • What a box plot reveals about distribution shape
  • When to use each type of display

Can Do

  • Construct a box plot from data
  • Interpret box plots
  • Compare distributions using box plots
Five-number summaryWhiskerSkewDistribution
Learn Phase
1

Box Plots

Visualising the five-number summary

A box plot displays:

  • Minimum (left whisker end)
  • Q1 (left edge of box)
  • Median (line inside box)
  • Q3 (right edge of box)
  • Maximum (right whisker end)

The box spans the interquartile range (IQR)

Outliers are often shown as individual points beyond the whiskers.

2

Interpreting Box Plots

What the shape tells you

Box plot shape reveals distribution:

  • Symmetrical: median centred, whiskers roughly equal
  • Right-skewed: median closer to Q1, right whisker longer
  • Left-skewed: median closer to Q3, left whisker longer

A long whisker or outlier points indicate spread in that direction.

3

Histograms and Stem-and-Leaf

Other useful displays

A histogram shows frequency distribution using bars. The area (not just height) represents frequency.

A stem-and-leaf plot shows all data values while grouping them. The stem is the leading digit(s), the leaf is the trailing digit.

Example: 12, 15, 21, 23, 23, 30

Stem | Leaf

1 | 2 5

2 | 1 3 3

3 | 0

✓

Check Understanding

Try it yourself

Construct a box plot for: 4, 8, 10, 12, 15, 18, 22, 25, 30. Describe the shape.

Worked Example

Box Plots

1

Construct a box plot for: 5, 7, 9, 11, 13, 15, 17.

Min=5, Q1=7, Median=11, Q3=15, Max=17

Box from 7 to 15, median at 11, whiskers to 5 and 17.

2

Two classes took a test. Class A: min=40, Q1=55, med=70, Q3=80, max=95. Class B: min=50, Q1=60, med=65, Q3=75, max=90. Compare.

Class A has higher median (70 vs 65) but greater spread (IQR=25 vs 15). Class B is more consistent.

3

Data: 2, 3, 5, 7, 8, 10, 12, 15, 50. Should 50 be an outlier on the box plot?

Q1=4, Q3=13.5, IQR=9.5. Upper fence = $13.5 + 1.5(9.5) = 27.75$

50 > 27.75, so yes, plot as outlier. Whiskers extend to 15.

Common Misconceptions

✕

The whiskers always extend to min and max. No — whiskers extend to the most extreme values within 1.5×IQR of the quartiles. Outliers are plotted separately.

✕

Box plot height matters. No — box plots are one-dimensional; height is arbitrary and carries no meaning.

✕

Median is always in the middle of the box. Not necessarily — the median can be anywhere between Q1 and Q3, revealing skewness.

Your Turn

Practice — Data Displays

Work through each question in your book or digitally. Answers are in the Questions phase.

1Draw a box plot for: 6, 8, 10, 12, 14, 16, 18, 20.
2Compare two box plots: Class X (min=30, Q1=45, med=60, Q3=75, max=90) vs Class Y (min=40, Q1=50, med=55, Q3=70, max=85).
3Create a stem-and-leaf plot for: 23, 25, 28, 31, 32, 35, 38, 40, 42.
Real-World Anchor

Medicine and Public Health

Box plots are used extensively in medical research to compare treatment groups. The Australian Institute of Health and Welfare uses box plots to display life expectancy, hospital wait times, and disease incidence across regions and demographic groups.

📓 Copy Into Your Books

▼

Box plot

  • Min, Q1, Median, Q3, Max
  • Box = IQR
  • Whiskers to fence

Shape

  • Symmetric - median centred
  • Right-skew - longer right whisker
  • Left-skew - longer left whisker

Other displays

  • Histogram - frequency bars
  • Stem-and-leaf - all values shown
  • Dot plot - individual points
Questions Phase
Check Your Understanding
Answer all questions correctly to unlock the Game phase.
Box plot shows:
The box spans:
Right-skewed data has:
Outliers on box plot are:
Stem-and-leaf preserves:
In a symmetric distribution:
IQR on box plot is:
Best display for comparing two groups:
1Construct a box plot for: 8, 10, 12, 15, 18, 20, 25, 30. Identify any outliers.
2Describe the shape of a distribution where mean > median > mode.
3Why might a stem-and-leaf plot be preferable to a histogram for small datasets?

Comprehensive Answers

1Box plot for 8,10,12,15,18,20,25,30.
Min=8, Q1=11, Med=16.5, Q3=22.5, Max=30. IQR=11.5. Upper fence=22.5+17.25=39.75. No outliers.
2Mean > median > mode.
Right-skewed (positive skew) with tail to the right.
3Stem-and-leaf vs histogram.
Stem-and-leaf shows all raw values; histogram groups data and loses individual values.
MC 1Box plot shows.
Five-number summary. Answer: B
MC 2Box spans.
Q1 to Q3. Answer: B
MC 3Right-skewed.
Longer right whisker. Answer: B
MC 4Outliers on box plot.
Beyond whiskers. Answer: A
MC 5Stem-and-leaf preserves.
All individual values. Answer: B
MC 6Symmetric distribution.
Mean=Median=Mode. Answer: A
MC 7IQR on box plot.
Box width (Q3-Q1). Answer: B
MC 8Best for comparing groups.
Side-by-side box plots. Answer: B
SA 1Box plot 8,10,12,15,18,20,25,30.
Min=8, Q1=11, Med=16.5, Q3=22.5, Max=30. No outliers.
SA 2Mean > median > mode.
Right-skewed.
SA 3Stem-and-leaf vs histogram.
Preserves all raw values.
Game Phase
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Game Unlocked!
You have mastered the Check Your Understanding questions. Choose a game mode below.
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Classify & Sort
Sort mathematical objects by their properties.
⚡
Speed Challenge
Answer questions against the clock.
📈
Match Maker
Match problems to their solutions.