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

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:

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.

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.

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.

Finding the quartiles by hand

To locate the quartiles, first find the median, which splits the ordered data into a lower half and an upper half. When the number of values $n$ is odd, the median is the middle value and is excluded from both halves; when $n$ is even, the median is the average of the two middle values and the data divides cleanly. The lower quartile $Q_1$ is the median of the lower half and the upper quartile $Q_3$ is the median of the upper half. Getting this convention right matters because VCAA marks the quartile values directly, and an off-by-one slip (for instance, including the median in a half when $n$ is odd) shifts every later calculation, including the IQR and the fences.

Reading skew from a boxplot

The position of the median inside the box and the relative whisker lengths reveal the shape of the distribution. If the median sits closer to $Q_1$ and the right whisker is longer, the distribution is positively skewed (a tail of larger values). If the median sits closer to $Q_3$ and the left whisker is longer, it is negatively skewed. A median roughly central with similar whiskers indicates an approximately symmetric distribution. VCAA expects you to use the words positively skewed, negatively skewed or approximately symmetric, and to justify the choice by pointing to the median position or whisker lengths rather than just asserting a shape.

Why boxplots beat raw data for comparison

A boxplot compresses a whole distribution into five robust summary numbers, and quartiles and the median are resistant to extreme values in a way the mean and range are not. This is exactly why VCAA favours parallel boxplots for comparing groups: with one shared scale you can read differences in centre, spread and shape at a glance, and outliers are shown explicitly rather than distorting a summary statistic. When you write a comparison, always pair a feature with a numerical value and a direction, for example stating that group A has a higher median (12 against 9) and a smaller IQR (4 against 6), so it tends to be larger and more consistent.