Correlation and the regression line
Read the strength of a correlation from r, find the mean point, set the regression line with its gradient and intercept, and predict inside the data.
📖 Start here: read the steps of the “Lesson” in order (1, 2, 3 …) and answer each question. Then continue with “Practice”.
- UK: Year 10
- France: grade 11
- Germany: grade 11
- Afghanistan: grade 11
- Syria: grade 7
- Palestine: grade 10
- Russia: grade 11
Guide
correlation describes how two variables move together. The correlation coefficient r is a number from −1 to +1: near +1 the points rise in a tight line, near −1 they fall in a tight line, and near 0 there is no straight-line pattern. The sign gives the direction, the size gives the strength.
The regression line is the best straight line for predicting y from x. It always passes through the mean point (x̄, ȳ). Its gradient is m = Σ(x − x̄)(y − ȳ) ÷ Σ(x − x̄)² and its intercept is c = ȳ − m·x̄, so the line is y = mx + c.
Use the line to predict y for an x inside the range of the data (interpolation). Far outside the data (extrapolation) the pattern may not continue, so the prediction cannot be trusted. And a strong correlation never proves causation (cause and effect).