Karl Pearson r and coefficient of determination

Karl Pearson's coefficient of correlation

It is a numerical measure of the strength and direction of the linear relationship between two quantitative variables X and Y. It is defined as the ratio of covariance to the product of the standard deviations:

r = Cov(X, Y)/(σx·σy) = Σ(x − x̄)(y − ȳ) / √[Σ(x − x̄)² · Σ(y − ȳ)²]

Computing form: r = [nΣxy − ΣxΣy] / √{[nΣx² − (Σx)²][nΣy² − (Σy)²]}

• r lies between −1 and +1.
• r > 0: positive (direct) correlation; r < 0: negative (inverse) correlation; r = 0: no linear correlation; r = ±1: perfect correlation.

Coefficient of determination

It is the square of the correlation coefficient, r², and gives the proportion (or percentage) of the total variation in the dependent variable that is explained by its linear relationship with the independent variable.

r² = Explained variation / Total variation, 0 ≤ r² ≤ 1.

Example: if r = 0.8, then r² = 0.64, so 64% of the variation in Y is explained by X and 36% is unexplained (due to other factors and chance).

Importance in analysis

1. r shows how closely two variables are related, helping to decide whether regression is worth fitting.
2. It gives the direction of the relationship (direct or inverse).
3. r² measures the goodness of fit of a regression line and the predictive power of the model.
4. It is the basis for partial and multiple correlation and for the regression coefficients (r = ±√(b_yx·b_xy)).
5. It is free of units and independent of change of origin and scale, so variables measured in different units can be compared.
6. Used in engineering, economics and science to identify related factors (for example, load and deflection, or temperature and resistance). Note that a high r does not prove causation.