Constructing and Interpreting Scatterplots

Bivariate Data and the Role of Variables

Bivariate data involves the simultaneous collection of two numerical variables from the same individual or unit. Unlike univariate data, which describes a single characteristic, bivariate data allows us to explore whether and how two quantities are related. In the QCAA General Mathematics course, the analysis of bivariate data begins with a clear understanding of the two variable roles: the explanatory variable and the response variable.

The explanatory variable (sometimes called the independent variable) is the variable believed to explain, predict, or cause changes in the other. It is plotted on the horizontal axis (x-axis) of a scatterplot. The response variable (sometimes called the dependent variable) is the one whose values are expected to respond to or depend upon the explanatory variable. It is plotted on the vertical axis (y-axis).

Correctly identifying which variable is explanatory and which is the response is essential before constructing any scatterplot. Ask yourself: Which variable do I think might predict or influence the other? This is a practical, context-driven decision rather than a purely mathematical one.

• Example A: A Queensland researcher collects data on daily maximum temperature (degrees C) and ice cream sales (units) at a Gold Coast kiosk over 20 summer days. Here, temperature is the explanatory variable (x-axis) and ice cream sales is the response variable (y-axis), because we expect temperature to influence how many ice creams are sold.
• Example B: A school records the number of hours per week students spend on revision and their end-of-semester exam score (%). Study hours is the explanatory variable and exam score is the response variable.

Note carefully: the fact that one variable is explanatory does not automatically mean it causes changes in the response variable. Association does not imply causation. This distinction is critical in the QCAA course and is frequently tested in extended-response questions.

Constructing a Scatterplot

A scatterplot (also called a scatter diagram or scatter graph) is a graph that displays the values of two quantitative variables as points on a Cartesian plane. Each point represents a single observation: its x-coordinate is the value of the explanatory variable and its y-coordinate is the value of the response variable.

Steps to construct a scatterplot:

• Step 1: Identify which variable is explanatory (x-axis) and which is the response (y-axis).
• Step 2: Draw and label both axes with the variable name and its units. Choose axis scales that allow all data points to be plotted clearly without excessive white space.
• Step 3: For each observation, locate its x-value on the horizontal axis and its y-value on the vertical axis, then plot the point at the intersection.
• Step 4: Give the scatterplot a descriptive title that references both variables.

Worked Example: Body Mass and Resting Heart Rate in Brisbane Adults

A health researcher collected data from 10 Brisbane adults. The explanatory variable is body mass (kg) and the response variable is resting heart rate (bpm).

Each pair is plotted as a single point: for body mass 55 kg and resting HR 62 bpm, place a point at (55, 62). Repeat for all 10 pairs. The horizontal axis should run from approximately 50 to 115 kg, and the vertical axis from approximately 60 to 95 bpm. Both axes must be clearly labelled with variable names and units.

In QCAA assessments, you may be asked to add a single additional point to an existing scatterplot or identify which point corresponds to a given data pair. Always check both coordinates carefully before plotting.