VCE General Mathematics

Types of data

A variable is a characteristic that can differ from one person or object to the next, and the values it takes are the data. The first step in any analysis is to classify the variable, because the type decides which displays and summary statistics make sense.

Two points cause most of the mistakes. A number is not always a numerical variable: a postcode or a jersey number is a label, so it is nominal, and arithmetic on it means nothing. A category is ordinal when the order itself carries information, although the gaps between the categories are not measured, so the difference between "high" and "medium" is not known to equal the difference between "medium" and "low".

Continuous data are usually recorded to a limited accuracy (a height to the nearest centimetre, say), but the variable is still continuous because it could in principle be measured more finely.

Frequency tables

The frequency of a category is the number of observations in it. A frequency table lists each category with its frequency, and a relative frequency (or percentage) column divides each frequency by the total.

A survey of 80 students asks how they travel to school.

Each percentage is the frequency divided by 80, so for bus, $\frac{30}{80} = 0.375 = 37.5\%$. The percentages add to 100%, which checks the arithmetic.

Percentages matter as soon as two groups of different size are compared, because a frequency of 30 means something different out of 80 than out of 300.

Bar charts

A bar chart shows one bar for each category, with the height giving the frequency or the percentage. The bars have gaps between them because the categories are separate, and for a nominal variable they can be in any order; for an ordinal variable they are placed in the natural order.

A segmented bar chart divides a single bar into segments, one for each category, so each segment is that category's share of the whole. With 100 on the vertical axis (a percentage segmented bar chart) it shows the same information as the percentage column of a frequency table. Two or more segmented bars placed side by side compare the distribution of one variable across the groups of another, which is how associations between categorical variables are shown.

Bars must start at zero, and a vertical axis that starts higher makes small differences look large.

Describing the distribution of a categorical variable

The description of a categorical variable says which categories are most and least common, and gives the figures that show it, rather than only listing the table.

For the travel data: the most common way of travelling is the bus, at 37.5% of the students, and the least common is the car at 15%. Walking and the train are close together, at 22.5% and 25%. Together, 62.5% of students use the bus or the train, since $37.5 + 25 = 62.5$.

Where a variable is ordinal, a description can also say where most of the responses sit along the scale, for example that most students chose the two higher satisfaction categories.

Answering statistical questions from a distribution

A statistical question is answered from the data with a statement that uses the numbers and is written in the context of the situation. A table or chart is read for the relevant percentages, which are then compared and stated in a sentence.

Suppose the question is whether car travel is more common than train travel in the sample. The table shows 12 of 80 students travel by car and 20 of 80 by train, so in this sample train travel is more common, by 10 percentage points ($25\% - 15\% = 10\%$). The words "in this sample" matter: a conclusion about a wider population from a sample has to allow for the sample being one of many that could have been taken.

Two habits make answers more reliable. A percentage difference is stated as a difference in percentage points, not as a percentage of a percentage, and every statement names the variable and the category it refers to, so that it makes sense on its own.