VICGeneral MathematicsSyllabus dot point

How do you build a boxplot from a five-number summary, test for outliers, and compare two groups using parallel boxplots?

The five-number summary, construction and interpretation of boxplots, the use of the lower and upper fences to identify outliers, and comparison of distributions using parallel boxplots

A focused answer to the VCE General Mathematics Unit 3 data analysis key knowledge on the five-number summary, constructing and reading boxplots, applying the 1.5 IQR fence rule for outliers, and comparing groups with parallel boxplots.

Generated by Claude Opus 4.76 min answerUpdated 2026-05-29

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

Have a quick question? Jump to the Q&A page

Jump to a section

What this dot point is asking
Building a boxplot
The fence rule for outliers
Comparing distributions with parallel boxplots

What this dot point is asking

VCAA wants you to summarise a numerical distribution with five numbers, draw the matching boxplot, test for outliers with the fence rule, and compare two distributions using parallel boxplots. Boxplots appear in almost every General Mathematics exam, so the fence calculation and the language of comparison are high-value skills.

Building a boxplot

A boxplot is drawn against a numerical scale. The box runs from $Q_1$ to $Q_3$ , with a line at the median $M$ . The whiskers extend to the smallest and largest values that are not outliers. Outliers are plotted as separate dots or crosses.

The box itself spans the interquartile range:

\text{IQR} = Q_3 - Q_1

This middle 50 percent of the data is the most important part of the picture. A median line sitting nearer $Q_1$ signals positive skew, and a median nearer $Q_3$ signals negative skew.

The fence rule for outliers

An outlier is any value beyond the fences.

\text{lower fence} = Q_1 - 1.5 \times \text{IQR}

\text{upper fence} = Q_3 + 1.5 \times \text{IQR}

Any value below the lower fence or above the upper fence is an outlier and is plotted as a point. The whisker then stops at the most extreme value that is still inside the fences.

Five-number summary, fences and a boxplot

Problem

A dataset has the ordered values:

4,\ 7,\ 8,\ 9,\ 10,\ 11,\ 12,\ 13,\ 14,\ 30

Find the five-number summary, test for outliers, and describe the boxplot.

Solution

There are $n = 10$ values. The median sits between the 5th and 6th values:

M = \frac{10 + 11}{2} = 10.5

The lower half is $4, 7, 8, 9, 10$ , so $Q_1 = 8$ (the middle value).

The upper half is $11, 12, 13, 14, 30$ , so $Q_3 = 13$ .

\text{IQR} = 13 - 8 = 5

Now the fences:

\text{upper fence} = 13 + 1.5 \times 5 = 13 + 7.5 = 20.5

\text{lower fence} = 8 - 1.5 \times 5 = 8 - 7.5 = 0.5

The value $30$ is above $20.5$ , so it is an outlier. No value is below $0.5$ .

The five-number summary is $4,\ 8,\ 10.5,\ 13,\ 30$ . The right whisker stops at $14$ (the largest non-outlier), and $30$ is plotted as a separate point. The box spans $8$ to $13$ .

Comparing distributions with parallel boxplots

When two boxplots share one scale, you compare them on three fronts.

Centre: compare the two medians. A higher median means typically larger values.
Spread: compare the IQRs (box widths) and the overall ranges.
Shape and outliers: note skew and any outlier points.

A strong comparison sentence pairs a measure with a number and a direction. For example: the median for group A (12) is higher than for group B (9), so group A tends to be larger, and group A is also more spread out with an IQR of 6 against 4.

Exam-style practice questions

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

2025 VCAA1 marksThe following information relating to life expectancy comes from a sample of nations in the Oceania region. The first quartile is 74.9 years. The third quartile is 78.5 years. The lowest five values recorded are 68.5, 68.6, 69.0, 70.1 and 74.8 years. How many outliers would be displayed at the lower end of a boxplot showing this sample of Oceania data? A. 1 B. 2 C. 3 D. 4

Show worked answer →

An outlier is any value below the lower fence, where lower fence = Q1 - 1.5 x IQR.

IQR = Q3 - Q1 = 78.5 - 74.9 = 3.6.

lower fence = 74.9 - 1.5 x 3.6 = 74.9 - 5.4 = 69.5.

Any value below 69.5 is an outlier. From the lowest five values, 68.5, 68.6 and 69.0 are all below 69.5, while 70.1 and 74.8 are not. That gives 3 outliers, so the answer is C.