Correlation and regression

Correlation describes how strongly two variables move together. The correlation coefficient r runs from −1 to 1. The least squares regression line is the line of best fit worked out exactly, used for predictions.

Key facts

  • r near 1: strong positive. Near −1: strong negative. Near 0: none.
  • Least squares line: y = a + bx with b = Sxy/Sxx and a = ȳ − bx̄.
  • Predict within the data range (interpolation); predicting outside it (extrapolation) is unreliable.
  • Correlation doesn’t prove one thing causes the other.

Worked examples

Example 1

The regression line of score on hours revised is y = 32 + 6.5x. Predict the score for 4 hours. Is it reliable for 20 hours?

  1. y = 32 + 6.5 × 4 = 58
  2. 20 hours is likely outside the data (extrapolation), so that prediction is unreliable.

Answer: 58; 20 hours is unreliable

Example 2

r = −0.92 between temperature and hot-chocolate sales. Describe it.

  1. Close to −1: strong negative.
  2. As temperature rises, sales tend to fall.

Answer: Strong negative correlation