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Class 12 Statistics Notes · GSEB

Coefficient of Determination

Regression — learn R² = r², explained vs unexplained variation, and how to interpret the proportion of variation explained by regression. GSEB Class 12 Statistics notes.

Last updated: 22 Sep 2026

Notes

Coefficient of Determination

Linear Regression — Chapter 4, GSEB Class 12 Statistics

What is R²?

The coefficient of determination (R²) tells us what proportion of variation in Y is explained by the regression on X. The remaining variation is unexplained (due to other factors or random error).

Coefficient of Determination

R2=r2R^2 = r^2

Variation Decomposition

Total Variation in YExplained (R²)UnexplainedR² × Total(1−R²) × Total

Interpreting R²

R² = 1

Perfect Prediction

100% of Y's variation explained by X. All points lie on the line.

R² = 0.64

Moderate Fit

64% explained, 36% unexplained. E.g., r = 0.8 → R² = 0.64

R² = 0

No Prediction

0% explained. X has no predictive power over Y. r = 0.

Example: If r = 0.8, then R² = 0.64. This means 64% of the variation in marks is explained by study hours. The remaining 36% is due to other factors (sleep, aptitude, etc.).

Worked Example

If r = 0.9, interpret R²

R² = r² = (0.9)² = 0.81

Explained variation = 81% of total variation in Y

Unexplained variation = 1 − 0.81 = 19%

81% of the variation in Y is explained by its relationship with X. Only 19% is due to other factors.

Key Takeaways

Key Takeaways

  • R² = r² — the coefficient of determination is the square of the correlation coefficient.
  • R² tells us the proportion of variation in Y explained by the regression on X.
  • R² always lies between 0 and 1 (0% to 100%).
  • Higher R² = better fit = more reliable predictions.
  • Remaining variation (1 − R²) is unexplained — due to other factors or random error.
  • If r = 0.8, R² = 0.64 → 64% of Y's variation is explained by X.