Boxplots and the five-number summary: VCE General Mathematics Unit 3 Data analysis

What this dot point is asking

In "Investigating data distributions" the Study Design lists "summary of the distributions of numerical variables; the five-number summary and boxplots (including the use of the lower fence ($Q_1 - 1.5 \times \text{IQR}$) and upper fence ($Q_3 + 1.5 \times \text{IQR}$) to identify and display possible outliers)", and in "Investigating association between two variables" it lists parallel boxplots "and their use in identifying and describing associations between a numerical variable and a categorical variable". You need to:

1. find the five-number summary (minimum, $Q_1$, median, $Q_3$, maximum) from raw data, a dot plot, a stem plot or CAS;
2. calculate the IQR and the fences, and identify possible outliers;
3. draw a boxplot correctly, with whiskers ending at the last value inside the fences;
4. read a boxplot: quartiles, range, IQR, the percentage of data in each section, and the shape;
5. compare distributions with parallel boxplots, quoting medians and IQRs.

Boxplot questions appear in both examinations every year. In 2025 Examination 1 Questions 3 and 6 asked students to compare two boxplots ($82\%$) and count lower outliers ($79\%$); in 2025 Examination 2 Question 2 asked for the range ($85\%$) and upper fence ($82\%$); in 2023 Examination 1 Question 2 the shape of a distribution depended on the upper fence ($88\%$).

The answer

The median and quartiles

Put the data in order. For $n$ values:

• the median is in position $\frac{n + 1}{2}$ (the middle value if $n$ is odd, the mean of the two middle values if $n$ is even);
• the lower half is the values below the median position and the upper half the values above it. When $n$ is odd, the median itself is left out of both halves;
• $Q_1$ is the median of the lower half and $Q_3$ is the median of the upper half.

This is the method used by CAS one-variable statistics, so your hand answers and your calculator should agree. With a dot plot or frequency table, count cumulatively to the positions you need.

Fences and possible outliers

A data value less than the lower fence or greater than the upper fence is a possible outlier. A value exactly on a fence is not an outlier. The fences are a calculation to decide which points to plot separately; they are not drawn on a finished boxplot.

Drawing the boxplot

1. Draw a scale and label it with the variable and units.
2. Draw a box from $Q_1$ to $Q_3$ with a vertical line at the median.
3. Plot each possible outlier as a separate point (a dot or a cross).
4. Draw each whisker from the box to the most extreme data value that is not an outlier. Without outliers, the whiskers run to the minimum and maximum.

The runners' times from 2023 Examination 1 (40 runners: three at 134 s, eight at 135 s, eleven at 136 s, six at 137 s, four at 138 s, two each at 139 s and 140 s, and one each at 141, 142, 143 and 146 s) give this boxplot.

Reading a boxplot

Each of the four sections (lower whisker, lower half of the box, upper half of the box, upper whisker) holds about 25% of the data. So about $50\%$ of values lie in the box, about $75\%$ lie above $Q_1$, and about $25\%$ lie above $Q_3$. For a sample of 200, about 50 values lie in each section.

The shape follows from the positions of the median and the whiskers:

• positively skewed: the median is closer to $Q_1$ and the upper whisker (or the upper tail including outliers) is longer;
• negatively skewed: the median is closer to $Q_3$ and the lower whisker is longer;
• approximately symmetric: the median is near the centre of the box and the whiskers are of similar length.

Describe a distribution with shape (including any outliers), centre (median) and spread (IQR and range).

Comparing parallel boxplots

Parallel boxplots share one scale, one boxplot per group, so they show whether a numerical variable is associated with a categorical one. Compare:

• centre: the medians;
• spread: the IQRs (and the ranges);
• shape and outliers, if they differ.

A comparison earns marks only when it states the direction and quotes values: "The median life expectancy for Sample T (about 76.5 years) is higher than for Sample H (about 67.2 years)." To claim an association, say that the value of the numerical variable tends to differ by group (for example, life expectancy tends to be higher for T countries), supported by the medians.

Useful facts for multiple choice: "all values in T exceed $x$" is a statement about T's minimum; "at least $75\%$ of T exceed H's median" is true whenever T's $Q_1$ is above H's median; and quartiles can be compared across groups directly ($Q_3$ of H against $Q_1$ of T).

Why boxplots are built on the median and quartiles

The median and quartiles are resistant to extreme values: moving the largest runner's time from 146 s to 200 s would not change the box at all, whereas the mean and standard deviation would change. That makes the boxplot the natural display for skewed data and data with outliers, and the fence rule gives a consistent, checkable definition of "unusually far from the middle". For symmetric, bell-shaped data the mean and standard deviation (and the 68-95-99.7% rule) are the better summary.