Display and describe the distribution of a numerical variable using a histogram, dot plot, stem plot or boxplot, summarise it with measures of centre and spread, and identify outliers using the lower and upper fences

What this dot point is asking

VCAA wants you to take a single variable, choose an appropriate display, and describe its distribution clearly. For a numerical variable you describe four things in order: shape, centre, spread, and outliers. You must also be able to construct a five-number summary and a boxplot, and to test for outliers using the fence rule. This is the foundation of the entire Data analysis core, and it appears every year on both exams.

Choosing a display

A categorical variable (e.g. eye colour, transport mode) is summarised with a frequency table and shown with a bar chart. A numerical variable (e.g. height, time) is shown with a dot plot, stem plot, histogram or boxplot.

• Dot plots and stem plots keep every value and suit small data sets.
• Histograms group data into intervals and suit large data sets.
• Boxplots compress the data into five numbers and are ideal for comparing groups.

Describing shape

A distribution is symmetric if it balances about its centre. It is positively skewed if it has a tail to the right (toward larger values) and negatively skewed if it has a tail to the left. Skew matters because it tells you which summary statistics to trust: for skewed data or data with outliers, report the median and IQR; for roughly symmetric data with no outliers, the mean and standard deviation are appropriate.

The five-number summary and the boxplot

The five-number summary is the minimum, first quartile $Q_1$, median $M$, third quartile $Q_3$, and maximum. The interquartile range is

The IQR measures the spread of the middle 50 percent of the data and is resistant to outliers. The boxplot draws a box from $Q_1$ to $Q_3$ with the median marked inside, and whiskers extending to the most extreme values that are not outliers.

Centre and spread for symmetric data

For roughly symmetric numerical data the mean is

and the standard deviation $s$ measures the typical distance of values from the mean. When data is symmetric and bell-shaped, the 68 to 95 to 99.7 percent rule applies: about 68 percent of values lie within one standard deviation of the mean, about 95 percent within two, and about 99.7 percent within three. This rule is examined directly and is a fast source of marks on Exam 1.

Reading the standard deviation

The standard deviation is best read from a calculator in this course, but you should know its behaviour. A larger $s$ means more spread. If every value is increased by a constant, the mean shifts by that constant but $s$ is unchanged. If every value is multiplied by a constant $k$, both the mean and $s$ are multiplied by $|k|$. These transformation rules are tested on Exam 1 where a calculator may help on Exam 2 but the reasoning must be by hand.

Why this matters for the exams

Univariate analysis is assumed knowledge for the rest of Data analysis. Boxplots reappear when you compare two or more groups, the 68 to 95 to 99.7 percent rule underpins standardised scores, and the idea of a resistant summary returns when you fit regression lines. Master the four-part description and the fence rule first, because the rest of the core is built on them.