Coefficient of Determination (R²) Formula: 1 - SS_res / SS_tot Explained With Worked Examples

The coefficient of determination, denoted R² (R squared), is a statistic that measures the proportion of variance in the dependent variable that is explained by the regression model. The most common formula is: R² = 1 - (SS_res / SS_tot) Where SS_res (sum of squared residuals, also called sum of squared errors or SSE) is the sum of squared differences between the observed y values and the predicted y values from the model, and SS_tot (total sum of squares) is the sum of squared differences between the observed y values and the mean of y. Intuitively: SS_tot measures the total variability in y around its mean. SS_res measures how much variability remains AFTER fitting the model. The ratio SS_res / SS_tot is the proportion of variability that the model FAILED to explain. So 1 minus that ratio is the proportion of variability the model DID explain — which is R². R² ranges from 0 to 1 (or 0% to 100%). R² = 0 means the model explains none of the variance (the model is no better than just predicting the mean). R² = 1 means the model explains all of the variance (the predictions perfectly match the observed data). Real-world R² values typically fall between 0.2 and 0.9 depending on the field and the strength of the relationship. The alternative formula gives the same answer for simple linear regression: R² = (covariance of x and y / (standard deviation of x × standard deviation of y))² = r² (where r is the Pearson correlation coefficient). For simple linear regression with one predictor, R² is literally the square of the correlation coefficient — which is where the name R squared comes from. Snap a photo of any R² problem and StatsIQ walks through the SS_res and SS_tot computation, applies the formula, and explains the interpretation in the context of the data.

Walk through the calculation with a small dataset. Suppose you have 5 observations of (x, y): (1, 2), (2, 4), (3, 5), (4, 4), (5, 5) Step 1: Find the mean of y. ȳ = (2 + 4 + 5 + 4 + 5) / 5 = 20 / 5 = 4. Step 2: Calculate SS_tot = Σ(y_i - ȳ)². (2-4)² + (4-4)² + (5-4)² + (4-4)² + (5-4)² = 4 + 0 + 1 + 0 + 1 = 6. SS_tot = 6. Step 3: Fit a linear regression line. Using the standard formulas: slope (b) = Σ((x_i - x̄)(y_i - ȳ)) / Σ((x_i - x̄)²) x̄ = (1+2+3+4+5)/5 = 3. Numerator: (1-3)(2-4) + (2-3)(4-4) + (3-3)(5-4) + (4-3)(4-4) + (5-3)(5-4) = 4 + 0 + 0 + 0 + 2 = 6. Denominator: (1-3)² + (2-3)² + (3-3)² + (4-3)² + (5-3)² = 4 + 1 + 0 + 1 + 4 = 10. b = 6/10 = 0.6. intercept (a) = ȳ - b × x̄ = 4 - 0.6 × 3 = 4 - 1.8 = 2.2. Regression equation: ŷ = 2.2 + 0.6x. Step 4: Calculate predicted values ŷ_i for each x. x=1: ŷ = 2.2 + 0.6(1) = 2.8 x=2: ŷ = 2.2 + 0.6(2) = 3.4 x=3: ŷ = 2.2 + 0.6(3) = 4.0 x=4: ŷ = 2.2 + 0.6(4) = 4.6 x=5: ŷ = 2.2 + 0.6(5) = 5.2 Step 5: Calculate SS_res = Σ(y_i - ŷ_i)². (2 - 2.8)² + (4 - 3.4)² + (5 - 4.0)² + (4 - 4.6)² + (5 - 5.2)² = 0.64 + 0.36 + 1.0 + 0.36 + 0.04 = 2.4. SS_res = 2.4. Step 6: Apply the R² formula. R² = 1 - SS_res / SS_tot = 1 - 2.4 / 6 = 1 - 0.4 = 0.6. Interpretation: the regression model explains 60% of the variance in y. The remaining 40% is unexplained — due to random variation, measurement error, or factors not included in the model. Verification using the correlation approach (since this is simple linear regression): r = covariance(x,y) / (sd_x × sd_y) = 6 / sqrt(10 × 6) = 6 / sqrt(60) = 6 / 7.746 = 0.7746. r² = 0.6. Same answer. StatsIQ handles this calculation automatically — snap any dataset and it computes the regression line, predicted values, residuals, SS_res, SS_tot, and R² with the full work shown.

The numerical value of R² alone does not tell you whether a model is good or bad — context matters enormously. The same R² value can be excellent in one field and poor in another. R² values by domain (rough guidelines): - Physics, engineering, lab measurements: R² > 0.95 is typical for well-controlled experiments. Anything below 0.9 suggests measurement noise or missing variables. - Chemistry experiments: R² > 0.99 for calibration curves (very tight relationships expected). - Biology and medical research: R² of 0.5-0.8 is good. Biological variability is high. - Social sciences (psychology, sociology, education): R² of 0.3-0.5 is good. Human behavior is noisy. - Economics, finance: R² of 0.1-0.3 is common for many models. Markets are extremely noisy. - Marketing, ad attribution: R² of 0.2-0.5 is typical. Attribution is hard. A model with R² = 0.4 in physics is a disaster. The same model in marketing is reasonably good. Always interpret R² relative to what is achievable in your field. What R² does NOT tell you: (1) whether the model is causally correct (high R² with the wrong variables is meaningless), (2) whether the model will generalize to new data (high R² in training data may reflect overfitting), (3) whether the relationship is linear vs nonlinear (you can have low R² because a linear model is the wrong functional form, not because there is no relationship), (4) whether individual predictions are accurate (high R² means the model fits well on average, but specific predictions can still be wildly off), and (5) whether the sample size is adequate (high R² with n=4 is essentially meaningless). The most common mistake in R² interpretation: chasing high R² by adding more variables. R² ALWAYS increases when you add more predictors to a model, even if those predictors are noise. This is why we use ADJUSTED R² for multiple regression with many predictors. Adjusted R² penalizes model complexity and only increases when added predictors meaningfully improve the fit. For publication and serious analysis, report R² alongside: the sample size, the residual standard error, F-test significance, and ideally a residual plot to check assumptions. R² alone is not sufficient — it is one metric among many for assessing model quality. StatsIQ explains R² interpretation in the context of your specific data and field, identifies common interpretation errors, and recommends when to use adjusted R² or alternative goodness-of-fit metrics.

R² has close cousins that are easy to confuse but mean different things. Knowing the distinctions is essential for any regression analysis. **R² (coefficient of determination)**: the basic version. Measures proportion of variance explained. Formula: 1 - SS_res / SS_tot. Range 0 to 1. Always increases (or stays the same) when you add predictors to the model. Use for: simple linear regression with one predictor, or descriptive comparison of explained variance. **Adjusted R² (R² adj or R̄²)**: penalizes model complexity. Formula: 1 - [(1 - R²)(n - 1) / (n - k - 1)], where n is sample size and k is the number of predictors. Adjusted R² can DECREASE when you add a predictor that does not improve the model meaningfully. Use for: comparing models with different numbers of predictors, multiple regression, and any setting where you want to penalize unnecessary complexity. Example of why adjusted R² matters: a model with 5 predictors might have R² = 0.55. Adding 5 more predictors might bump R² to 0.60 — but if those new predictors are random noise, adjusted R² might go DOWN from 0.50 to 0.45. Adjusted R² catches the overfitting that raw R² hides. **Correlation coefficient (r)**: measures the strength and direction of a linear relationship between two variables. Range -1 to +1. Negative values indicate inverse relationships. Formula: covariance(x,y) / (sd_x × sd_y). For simple linear regression with one predictor, r² = R². The correlation coefficient is more informative than R² because it tells you the direction (positive or negative) as well as the strength. **Multiple R**: the multiple correlation coefficient. For multiple regression with several predictors, multiple R is the correlation between the observed y values and the predicted y values. Multiple R squared = R² (the same R² we have been discussing). For simple linear regression with one predictor, multiple R = |r| (absolute value of the simple correlation). Relationships: - Simple linear regression: R² = r² - Multiple regression: R² = (multiple R)² - Adjusted R² < R² always (when k > 0) - Adjusted R² penalty grows with more predictors and shrinks with larger samples Which to report: for descriptive analysis with one predictor, report R² (or r). For regression with multiple predictors, report adjusted R² as the primary measure and R² as a secondary measure. For very large samples (n > 1,000), the difference between R² and adjusted R² is negligible. For small samples (n < 50), the difference can be substantial and using adjusted R² is critical. NCLEX-style stats question variants: questions often test whether students know that R² always increases with more predictors, that adjusted R² is the right metric for multi-predictor models, and that R² = r² for simple linear regression. Knowing these distinctions cold prevents common test mistakes. StatsIQ calculates R², adjusted R², and the simple correlation coefficient for any regression problem and explains the differences between them with the specific data context.