Bivariate Data, Correlation & Causation
Get to grips with bivariate data, correlation and causation for NSW Year 12 Mathematics Standard 2. You will learn to tell one-variable data from bivariate data, identify the independent and dependent variables, and read a scatterplot to describe an association.
This topic shows how to judge the direction (positive or negative), strength (strong, moderate or weak) and form (linear or non-linear) of a relationship, and why a correlation never proves causation — a core Standard 2 skill for interpreting real-world data.
Theory
Bivariate data records two variables for each individual. This Year 12 Standard 2 (NSW) guide shows how to read a scatterplot — identify the independent and dependent variables, describe the association by its direction, strength and form, and tell correlation apart from causation.
One-variable (univariate) data records a single measurement per individual; bivariate data records two variables for each individual, so you can study how they are related.
The independent variable is the input you change or that comes first; the dependent variable is the response. On a scatterplot the independent variable goes on the horizontal \(x\)-axis and the dependent variable on the vertical \(y\)-axis.
A scatterplot reveals the association between the two variables. You describe it by its direction (positive, negative or none), its strength (strong, moderate or weak) and its form (linear or non-linear). Crucially, a correlation is not causation: even a strong correlation can be caused by a third lurking variable, so it never proves that one variable causes the other.
There is no single formula for reading a scatterplot — you describe the direction from how \(y\) changes as \(x\) increases:
The strength is judged by how tightly the points cluster about one trend, from strong (little scatter) through moderate to weak (much scatter). The form is linear if a straight line fits the pattern and non-linear if the points follow a curve.
How to describe the association in a scatterplot
- Identify the variables — which is independent (\(x\)) and which is dependent (\(y\)).
- Direction — do the points rise (positive), fall (negative), or drift with no pattern (none) as \(x\) increases?
- Strength — how tightly do the points cluster about a single trend? Strong, moderate or weak.
- Form — does a straight line fit (linear) or do the points follow a curve (non-linear)?
- Causation check — even a strong correlation does not prove cause; look for a lurking third variable.
Work through direction, then strength, then form.
| \(x\ \text{up},\ y\ \text{up}\) | \(\Rightarrow\) | \(\text{positive}\) |
| \(\text{points hug a line}\) | \(\Rightarrow\) | \(\text{strong}\) |
| \(\text{straight pattern}\) | \(\Rightarrow\) | \(\text{linear}\) |
A strong, positive, linear association.
Decide the direction, then how much the points scatter about the trend.
| \(x\ \text{up},\ y\ \text{down}\) | \(\Rightarrow\) | \(\text{negative}\) |
| \(\text{some scatter about it}\) | \(\Rightarrow\) | \(\text{moderate}\) |
A moderate, negative association.
Ask first whether a straight line fits the pattern.
| \(\text{points bend}\) | \(\Rightarrow\) | \(\text{non-linear}\) |
| \(y\ \text{rises with}\ x\) | \(\Rightarrow\) | \(\text{positive}\) |
A non-linear, positive association.
Look for a lurking variable before assuming cause and effect.
| \(\text{damage responds to the fire}\) | \(\Rightarrow\) | \(\text{dependent}\) |
| \(\text{bigger fire}\) | \(\Rightarrow\) | \(\text{more crews and more damage}\) |
(a) Damage is the dependent variable. (b) No — the size of the fire is a lurking variable that drives both, so the correlation does not prove causation.
Common pitfalls
Frequently asked questions
What is bivariate data?
Bivariate data records two variables for each individual, for example the height and the mass of each student. One-variable (univariate) data records only a single measurement per individual. Bivariate data lets you study how the two variables are related on a scatterplot.
How do you describe the association in a scatterplot?
Describe three things. Direction: the points rise for a positive association, fall for a negative one, or show no drift for none. Strength: how tightly the points cluster about a trend, from strong through moderate to weak. Form: linear if a straight line fits, non-linear if the points follow a curve.
What is the difference between correlation and causation?
Correlation means two variables move together. Causation means one variable actually makes the other change. A correlation can exist without causation, because a third lurking variable can drive both, so a correlation on its own never proves that one variable causes the other.
Which variable goes on the x-axis?
The independent variable, the input you change or that comes first, goes on the horizontal x-axis. The dependent variable, the response, goes on the vertical y-axis. For example, when studying how study time affects an exam mark, study time is independent and the exam mark is dependent.
Does a strong correlation prove that one variable causes the other?
No. Even a very strong correlation does not prove cause and effect. You must rule out lurking variables and give a plausible mechanism before claiming causation. Ice-cream sales and drownings are strongly correlated, but hot weather, not ice-cream, is behind both.
What is a lurking variable?
A lurking (or confounding) variable is a third variable that is not one of the two being plotted but influences both, creating a correlation that is not a direct cause. For example, town population drives both the number of churches and the number of crimes in a town.