Boxplots (VCE SSCE General Mathematics): Revision Notes

Boxplots

Introduction to boxplots

In addition to histograms, dot plots, and stem-and-leaf plots, we have another powerful tool for displaying numerical data: the boxplot. Boxplots are particularly valuable because they can summarise large datasets in a compact, visual form. This makes them ideal for quickly understanding the distribution of data and comparing different datasets.

The five-number summary

Before we can create a boxplot, we need to understand the five-number summary. This summary contains five key values that describe a dataset:

• Minimum: the smallest value in the dataset
• First quartile ($Q_1$): the value below which 25% of the data falls
• Median ($M$ or $Q_2$): the middle value that divides the dataset in half
• Third quartile ($Q_3$): the value below which 75% of the data falls
• Maximum: the largest value in the dataset

When we know these five values, we have a good understanding of both the centre of the data and how it spreads out. The five-number summary forms the foundation for constructing a boxplot.

Components of a boxplot

A boxplot (also called a box-and-whisker plot) is a graphical representation of the five-number summary. Let's look at the key components:

The main features of a boxplot are:

• The box: A rectangle that represents the middle 50% of the data. It extends from $Q_1$ to $Q_3$.
• The median line: A vertical line inside the box showing where the median ($M$) is located.
• The whiskers: Lines extending from each end of the box to the minimum and maximum values (or to the smallest and largest values that aren't outliers, as we'll see later).

Constructing a simple boxplot

Let's work through an example to see how to construct a boxplot step by step.

The completed boxplot gives us a clear visual summary of the rainfall distribution throughout the year.

Understanding outliers in boxplots

Sometimes a boxplot might have one extremely long whisker. This could mean either:

• The data distribution is heavily skewed, with many values in the tail, or
• The long whisker is hiding one or more outliers (unusual values that don't fit the general pattern).

To distinguish between these situations, we need a precise definition of what counts as an outlier.

Defining outliers

An outlier is a data point that lies unusually far from the rest of the data. Specifically:

Remember that the interquartile range (IQR) is calculated as:

$\text{IQR} = Q_3 - Q_1$

This 1.5 × IQR rule gives us a mathematical way to identify outliers objectively.

Upper and lower fences

To identify outliers, we use imaginary boundaries called fences:

• Lower fence: $Q_1 - 1.5 \times \text{IQR}$
• Upper fence: $Q_3 + 1.5 \times \text{IQR}$

Any data value that falls outside these fences is classified as an outlier.

When we draw a boxplot with outliers:

• Outliers are shown as individual dots or circles
• The whiskers extend only to the smallest and largest values that are NOT outliers
• The box still represents the middle 50% of the data, just as before

Constructing a boxplot with outliers

Let's work through a complete example that involves identifying and displaying outliers.

The boxplot shows:

• The box extending from $Q_1 = 25.5$ to $Q_3 = 109.5$
• A median line at 71
• The lower whisker extending to the minimum (2)
• The upper whisker extending to 226 (the largest non-outlier value)
• A dot at 264 representing the outlier

Conclusion: There is one possible outlier - the student who spent 264 hours on the project.

Interpreting boxplots: estimating percentages

Boxplots aren't just for displaying data - they're also useful for estimating what percentage of values fall within certain ranges. This is because the quartiles divide the data into specific percentage groups.

Key percentages to remember

• 25% of values lie below $Q_1$
• 50% of values lie below the median
• 75% of values lie below $Q_3$
• The box contains the middle 50% of all values
• 25% of values lie above $Q_3$

Using technology to create boxplots

Creating boxplots by hand, especially those displaying outliers, can be time-consuming. Fortunately, CAS calculators can generate boxplots automatically.

General process for CAS calculators

Most graphing calculators follow a similar process:

1. Enter your data into a list or spreadsheet column
2. Access the statistics/graphing function
3. Select boxplot as the graph type
4. Enable the "show outliers" option if available
5. Display the graph and use the trace function to read key values

The calculator will automatically:

• Calculate the five-number summary
• Determine the fences
• Identify any outliers
• Draw the boxplot with outliers shown as separate points

You can trace along the boxplot to read the values of the minimum, $Q_1$, median, $Q_3$, maximum, and any outliers.

Remember!