Correlation and Regression: Key Concepts in Psychology and Statistics

Correlation

A correlation is a statistical technique used to measure and describe the relationship between two variables, usually two quantitative (continuous) variables. It tells you about the direction, form (linear vs. nonlinear), and strength/magnitude of the relationship.

Strength of a correlation

Strength (magnitude) refers to the degree to which the two variables are strongly and consistently related. It is given by the absolute value of the correlation coefficient (|r|), regardless of sign. Roughly: r ≈ .10 = weak, r ≈ .30 = moderate, r ≥ .50 = strong.

Direction of a correlation

Direction tells you how the variables move together: Positive correlation - as X increases, Y also increases. Negative correlation - as X increases, Y decreases. Zero correlation - no consistent pattern. The sign of r (+ or -) indicates direction.

Interpretation of correlation coefficient

A correlation of r = -.60 means: Direction - negative (as one variable goes up, the other goes down). Strength - |-.60| = .60, a strong relationship. So, -.60 indicates a strong, negative linear relationship.

R² (coefficient of determination)

R² is the proportion of variability in Y that is explained by X (r²). Example: if r = .40 between SAT and GPA, then R² = .16 → 16% of variance in GPA explained by SAT scores.

Purpose of a regression line

The regression line (line of best fit) 1) describes the relationship between X and Y visually, and 2) allows prediction of Y values for given X values. It is the line that minimizes residual error (least squares).

Slope and intercept of a regression line

The regression line has the form Ŷ = a + bX. Ŷ = predicted Y; a (intercept) = predicted Y when X = 0; b (slope) = rate of change in Y for each 1-unit increase in X. Positive b → Y increases as X increases; Negative b → Y decreases as X increases.

Interpretation of slope

For Y = 3 - 4X: Slope b = -4 → for every 1-unit increase in X, Y decreases by 4 units. Intercept a = 3 → when X = 0, Y = 3.

Example of slope interpretation

Example from lecture: Y = 7.7 + .009X → each $1 increase in hourly salary raises predicted happiness by .009 points.