Ways of Representing Data

Overview

Data representations — tables and graphs — allow us to see patterns, comparisons and trends that are invisible in raw numbers. Foundation Mathematics requires you to both construct and interpret these displays.

KEY TAKEAWAY: Every graph type has a purpose. Match the graph to the data type and the message you want to communicate.

Tables

Tables organise data into rows and columns for easy reading and comparison.

Frequency tables add a count (frequency) column:

Column Graphs

Used for numerical discrete data or comparing categories over time.

• Bars are vertical
• Each bar represents one category/time period
• Bars should have equal width and gaps between them
• Y-axis starts at zero

Example use: Monthly rainfall totals, weekly attendance.

Bar Graphs

Similar to column graphs but bars are horizontal.

• Often used when category names are long (fit better horizontally)
• Used for categorical comparisons

Example use: Survey results comparing preferences.

EXAM TIP: The only structural difference between a column graph and bar graph is orientation. Column = vertical, bar = horizontal.

Line Graphs

Used for continuous data or data collected over time (time series).

• Points are plotted and joined with a line
• The line implies that values between plotted points are meaningful (continuity)
• X-axis is usually time (hours, days, months, years)

Example use: Temperature over a day, stock prices over a year, population growth.

COMMON MISTAKE: Joining points on a graph of categorical data (e.g. favourite subjects). Categories are separate — a connecting line implies there are values between them, which doesn’t make sense.

Pie Charts

Used to show parts of a whole as proportional sectors.

• Each sector’s angle = $\frac{\text{frequency}}{\text{total}} \times 360°$
• All sectors must add to $360°$ (or $100\%$)
• Best when there are fewer than 8 categories (more makes it hard to read)

Example: $\frac{45}{180} \times 360° = 90°$ for a category with $45$ out of $180$ responses.

Constructing a Graph — Checklist

• [ ] Title — descriptive, tells reader what the graph shows
• [ ] Axis labels — variable name + units (e.g. “Height (cm)”)
• [ ] Scale — consistent intervals, starts at zero (for column/bar)
• [ ] Accurate plotting — values match the data
• [ ] Legend — if multiple datasets or colours used

Worked Example — Sector Angle Calculation

$90$ people chose pizza, $54$ chose pasta, $36$ chose salad (total $180$).

$$\text{Pizza angle} = \frac{90}{180} \times 360° = 180°$$
$$\text{Pasta angle} = \frac{54}{180} \times 360° = 108°$$
$$\text{Salad angle} = \frac{36}{180} \times 360° = 72°$$
$$\text{Check: } 180 + 108 + 72 = 360° \checkmark$$

VCAA FOCUS: VCAA tasks may ask you to draw a graph from a table or read specific values from a given graph. Accuracy of plotting and labelling are both assessed.