Maths › Statistics › Correlation and regression
Correlation and regression
A scatter graph asks whether two variables move together; the correlation coefficient scores the answer and the regression line turns it into predictions. The skill is knowing what each number licenses you to say, and what it never can.
Builds on Representing and interpreting data and Straight lines.
IN THIS TOPIC
- Describe correlation from a scatter diagram and interpret a product moment correlation coefficient r between −1 and 1.
- Use a calculated regression line y = a + bx to make predictions, interpreting a and b in context.
- Say when a prediction is trustworthy: interpolation within the data, the right variable as the explanatory one, and correlation never proving causation.
WHAT YOU PROBABLY THINK
A correlation of r = 0.9 proves that changing one variable causes the other to change.
Reading a scatter graph
Plot one variable against another and the cloud of points has a story. Rising together is positive correlation; one falling as the other rises is negative; a shapeless cloud is no correlation at all. The product moment correlation coefficient r puts a number on it, always between −1 and 1: the sign gives the direction, the size gives the strength, and the extremes mean the points lie on a perfect straight line.
Your calculator produces r from the data; the exam asks you to interpret it. “r = 0.96 shows strong positive linear correlation: taller plants do tend to carry more fruit.” Two ingredients every time: strength and direction, tied to the actual context.
The regression line
The least squares regression line y = a + bx is the straight line that makes the sum of the squared vertical distances from the points as small as it can be. It always passes through the mean point. The letters earn marks when read in context: b is the change in y for each extra unit of x, and a is the predicted y when x is zero, which may or may not be a meaningful situation.
WORKED EXAMPLE
Interpreting a and b
For ice cream sales s (pounds) against temperature t (°C), a calculator gives s = 42 + 15.3t. Interpret the numbers.
The gradient: each extra degree is associated with about £15.30 of extra daily sales.
The intercept: £42 of sales predicted at 0 °C, plausible as a baseline, though 0 °C may sit outside the data collected.
The wording earns the marks: “associated with”, not “causes”, and always in the units of the problem.
What a prediction is worth
Predicting inside the range of the observed x values is interpolation and is usually safe. Predicting beyond it is extrapolation: the line will happily produce a number, but no data supports the model there, and the honest comment is that the prediction is unreliable.
Direction matters too. The regression line of y on x is built to predict y from x, with x the explanatory variable under some control and y the response. Using it backwards, feeding in y to fish out x, is not valid. And however strong r is, correlation is not causation: ice cream sales and drowning rates rise together because summer drives both.
THE EXAM BIT
- Interpret r with strength, direction and context in one sentence; a number repeated back scores nothing.
- Interpret b as “for each extra unit of x, y changes by b” in the question's own units; interpret a as the prediction at x = 0 and say whether that is meaningful.
- Only predict for x values inside the data range, with the line of y on x used to predict y. Anything else earns the word “unreliable” and a reason.
- “Correlation does not imply causation” needs the third-factor idea in context to score: name the lurking variable.
CHECK YOURSELF
A regression line for crop yield y (tonnes) against rainfall x (cm), from data with 20 ≤ x ≤ 60, is y = 1.2 + 0.08x. A farmer asks for the predicted yield when x = 100. What do you say?
Show a hint
Where does 100 sit relative to the data?
Show the answer
The line gives 1.2 + 0.08 × 100 = 9.2 tonnes, but x = 100 is far outside the observed range 20 to 60.
This is extrapolation, so the prediction is unreliable and should not be trusted: the linear pattern may not continue.
r scores direction and strength of a linear link, never cause.
Regress y on x, predict only inside the data, and read a and b in the question's own units.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Describe correlation from a scatter diagram and interpret a product moment correlation coefficient r between −1 and 1.
- Use a calculated regression line y = a + bx to make predictions, interpreting a and b in context.
- Say when a prediction is trustworthy: interpolation within the data, the right variable as the explanatory one, and correlation never proving causation.
No animated video for this topic yet; these notes stand alone.