Types of data (categorical and numerical), appropriate graphical displays, and describing a numerical distribution in terms of shape, centre and spread using the mean, median, range, interquartile range and standard deviation

What this dot point is asking

VCAA wants you to look at a dataset, decide whether each variable is categorical or numerical, choose a graph that suits that data type, and then describe a numerical distribution in plain language using shape, centre and spread. This is the foundation of the whole data analysis module, so the exam rewards getting the vocabulary exactly right and choosing the correct measure of centre and spread for the shape you see.

Choosing a display

The display you choose depends on the data type.

• Categorical data is summarised with a frequency table and shown with a bar chart (the bars have gaps between them). A two-way frequency table is used when comparing two categorical variables.
• Numerical data is shown with a histogram, a dot plot, or a stem-and-leaf plot. A histogram has no gaps between bars because the horizontal axis is a continuous number line.

A common exam instruction is to read frequencies or percentages off a histogram, so be comfortable with the fact that the area pattern of the bars shows where the data clusters.

Describing shape

The shape of a numerical distribution is described as one of:

• Symmetric: roughly a mirror image about the centre.
• Positively skewed: a tail stretching to the right (towards larger values).
• Negatively skewed: a tail stretching to the left (towards smaller values).

You also note any outliers, which are values sitting well away from the main body of data.

Centre and spread

The centre is a single typical value. The two measures are the mean $\bar{x}$ and the median $M$.

The median is the middle value when the data is ordered. For $n$ values, the median is at position $\frac{n+1}{2}$.

The spread describes how scattered the data is. The three measures are:

• Range $= \text{maximum} - \text{minimum}$.
• Interquartile range $\text{IQR} = Q_3 - Q_1$, the spread of the middle 50 percent.
• Standard deviation $s$, the typical distance of a value from the mean.

The standard deviation is found with technology in this course, but you must know what it means. A larger $s$ means the values are more spread out from the mean. For symmetric, bell-shaped data the 68-95-99.7 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.

Putting it together in the exam

A full description answers four things: shape, centre, spread, and outliers. A strong sentence reads: the distribution is positively skewed with a median of 5 messages, an IQR of 4 messages, and an outlier at 28 messages. Stating the units every time is a frequent source of easy marks. When the data is symmetric instead, swap to the mean and standard deviation and report both with units.