VCE General Mathematics (Year 11) · Data Investigation · 11m read

Displaying Data — Boxplots and Histograms

Ever wonder why a single picture of data can make a pattern jump out at you instantly? Boxplots and histograms are two powerful ways to visualize groups of numbers so you can spot trends, outliers, and clusters at a glance. Mastering these tools means you'll confidently interpret real-world datasets in assignments and exams, and understand how to tell a data story that actually makes sense.

Learning objective: Construct and interpret boxplots and histograms.

By the end of this lesson you can:

• I draw a boxplot from a five-number summary.
• I construct histograms from grouped data.
• I compare distributions using displays.

A boxplot is a five-number suitcase. Min, Q1, median, Q3, max — packed neatly so you can compare groups quickly.

Reading and drawing displays

BOXPLOT (box-and-whisker):
• Shows five-number summary at a glance.
• Box = Q1 to Q3 (contains IQR).
• Line inside box = median.
• Whiskers extend to min/max OR 1.5×IQR fences.
• Individual points beyond = OUTLIERS.

CONSTRUCTING A BOXPLOT:
1. Find 5-number summary.
2. Identify outliers using 1.5×IQR rule.
3. Draw box from Q1 to Q3.
4. Mark median.
5. Draw whiskers to min/max within fences; plot outliers separately.

READING A BOXPLOT:
• SYMMETRY — median centred in box? Equal whiskers?
• SKEW — longer whisker on right → positive skew.
• OUTLIERS — dots beyond whiskers.

HISTOGRAMS:
• Continuous numerical data.
• X-axis: intervals (bins); Y-axis: frequency.
• Bars touch (no gaps — unlike bar charts).
• Bin width affects shape — too narrow = noisy; too wide = smooth.

CHOOSING BINS:
• Rule of thumb: 5-10 bins.
• Equal width.

INTERPRETING:
• Peak = mode area.
• Width = spread.
• Shape = symmetric, skewed, bimodal.

STEM-AND-LEAF PLOTS:
• Preserve actual values.
• Useful for small datasets.

DOT PLOTS:
• Good for discrete data.
• Each dot = one observation.

BAR CHARTS (categorical):
• Bars separated by gaps.
• Height = count or percentage.

COMPARING DISTRIBUTIONS:
• Use parallel boxplots.
• Compare centre, spread, shape, outliers.

COMMON MISTAKES:
• Touching bars in bar charts (should be histogram).
• No units / labels.
• Skipping outlier test.

REPORTING:
• Title, axes, units, legend.
• Annotate interesting features.

Display rules

• Labels: Every axis labelled.
• Outliers: Dots, not whiskers.
• Comparisons: Same scale.

Boxplots and histograms

• Five-number summary: \min, Q_1, Q_2, Q_3, \max
• Outlier rule: Q_1 - 1.5\,IQR,\; Q_3 + 1.5\,IQR

Quick reference

| Display | Shows |
|---|---|
| Histogram | Distribution shape |
| Boxplot | Five-number summary |
| Outlier fence | Q1 − 1.5×IQR, Q3 + 1.5×IQR |
| Parallel boxplots | Compare groups |
| Skew | Tail direction |

Worked example (Medium)

Compare two boxplots: Class A median 70, IQR 10; Class B median 72, IQR 25.

1. Centre. Very similar (70 vs 72).
2. Spread. Class B has 2.5× the IQR → much more variable.
3. Interpretation. Class A is more consistent; B has both high and low performers.

Answer: Same average, but Class A's performance is more consistent; Class B has higher variability.

Worked example (Hard)

A dataset has the five-number summary: Min=12, Q1=25, Median=40, Q3=60, Max=95. Draw a boxplot description and identify any outliers.

1. IQR. IQR = Q3 − Q1 = 60 − 25 = 35.
2. Fences. Lower fence = 25 − 1.5 × 35 = −27.5. Upper fence = 60 + 1.5 × 35 = 112.5.
3. Outliers. Min 12 > −27.5 ✓. Max 95 < 112.5 ✓. No outliers.
4. Shape. Median (40) is closer to Q1 (25) than Q3 (60) → positively skewed. Upper whisker (95 − 60 = 35) longer than lower whisker (25 − 12 = 13), confirming positive skew.

Answer: Box from 25 to 60, median at 40, whiskers to 12 and 95. No outliers. Positively skewed distribution.

Common mistakes

• Extending whiskers to outliers instead of plotting them as dots Whiskers stop at the last value within the 1.5 × IQR fence; outliers are individual points. Fix: Calculate fences first, then draw whiskers only to the last value inside each fence.
• Leaving gaps between bars in a histogram Histograms represent continuous data; gaps imply discrete categories (bar chart). Fix: Histogram bars must touch. Use gaps only for bar charts (categorical data).
• Not labelling axes on displays Unlabelled graphs lose marks and communicate nothing. Fix: Every display needs: title, labelled axes with units, and a legend if comparing groups.

Choosing the right display for your data

Selecting the correct display is as important as calculating statistics. Each display type reveals different aspects of your data, and VCAA examiners expect you to justify your choice.

Boxplots excel at comparing distributions side-by-side. Parallel boxplots on the same scale instantly reveal differences in centre, spread, and skew between groups. They also highlight outliers clearly. However, boxplots hide the shape detail — a bimodal distribution looks the same as a unimodal one with the same five-number summary.

Histograms show the shape of a distribution in detail. You can see modes, gaps, clusters, and the exact form of skewness. The trade-off is that histograms are harder to compare side-by-side, and the choice of bin width affects the impression of the shape.

Stem-and-leaf plots preserve individual data values — useful for small datasets where you want both shape and exact numbers. For large datasets they become unwieldy.

Dot plots work well for small discrete datasets, showing each observation individually.

The golden rule: always plot your data first before calculating statistics. A graph may reveal features (bimodality, outliers, clusters) that summary statistics alone would miss. In VCE SACs, examiners reward students who demonstrate this exploratory approach.

Practice questions

1. In a boxplot, the LINE inside the box represents:

   1. Mean
   2. Median
   3. Mode
   4. Range

   (1 marks)

2. Why must histogram bars TOUCH while bar chart bars DO NOT?

   (3 marks)

Summary

• Draw a boxplot from a five-number summary.
• Construct histograms from grouped data.
• Compare distributions using displays.