Measuring linear association with Pearson's correlation coefficient, fitting a simple linear regression line, interpreting its slope and intercept, and using it to predict while distinguishing interpolation from extrapolation.

What this dot point is asking

The SQA wants you to measure the strength and direction of a linear relationship between two variables using Pearson's correlation coefficient, fit and interpret a regression line, use it to predict, and recognise that correlation does not prove causation. Software produces the numbers; your job is to interpret and apply them.

Pearson's correlation coefficient

Pearson's correlation coefficient, written $r$, measures how closely points lie to a straight line. You obtain it from software (for example the PEARSON function in a spreadsheet or cor in R); the skill examined is interpretation.

State both the strength (strong, moderate, weak) and the direction (positive or negative) in context. Remember $r$ only detects a linear pattern: a strong curved relationship can still give an $r$ near zero, so a scatter plot should accompany the number.

The regression line

When the relationship is roughly linear, the least-squares regression line is the straight line that best fits the data by minimising the total squared vertical distance from the points. Software gives it in the form $y = a + bx$.

The regression is of $y$ on $x$, predicting $y$ from $x$, so identify which variable is the explanatory one (the input) and which is the response (the output you predict).

Prediction, interpolation and extrapolation

To predict, substitute an $x$-value into the line. The reliability depends on where the $x$-value sits.

For the heating line above, predicting cost at $x = 8$ degrees (within typical data) is interpolation, while predicting at $x = 40$ degrees is an extrapolation well outside any realistic Scottish data and should be treated with caution.

Correlation is not causation

A strong correlation shows two variables move together, but not that one causes the other. The link may be coincidence, reverse, or driven by a confounding variable that affects both. The classic example is ice cream sales and drownings, both driven by hot weather. The SQA expects you to state this caution and suggest a plausible confounder when relevant.

Try this

Q1. A correlation coefficient is $r = -0.91$. Describe the relationship. [1 mark]

• Cue. A strong negative linear relationship: as one variable increases, the other decreases.

Q2. A regression line is $y = 50 + 3x$. Interpret the slope and predict $y$ when $x = 12$. [2 marks]

• Cue. Each unit of $x$ adds about $3$ to $y$; at $x = 12$, $y = 50 + 36 = 86$.

Q3. Sales of sunglasses and visits to outdoor pools are strongly correlated. Explain why one need not cause the other. [2 marks]

• Cue. A confounding variable, hot sunny weather, raises both, so the correlation reflects a common cause, not causation.