R-Squared & Adjusted R-Squared Calculator

R-squared, the coefficient of determination, answers one question: how much of the variation in your dependent variable y is explained by the model? It is built from three sums of squares defined in the NIST/SEMATECH e-Handbook of Statistical Methods.

Let n be the number of observations and ȳ the mean of the actual y values. The total sum of squares is the spread of y around its own mean:

SST = Σ(yᵢ − ȳ)²

The model produces a predicted value ŷᵢ for each point. In paired mode the tool fits an ordinary least-squares simple linear regression, choosing the slope b and intercept a that minimise the squared errors:

b = Σ(xᵢ − x̄)(yᵢ − ȳ) / Σ(xᵢ − x̄)² ; a = ȳ − b·x̄ ; ŷᵢ = a + b·xᵢ

In actual-vs-predicted mode you supply the ŷᵢ yourself — for example, the fitted values from a multiple regression run elsewhere. Either way, the residual (error) sum of squares measures what the model misses, and the regression (explained) sum of squares is the rest:

SSE = Σ(yᵢ − ŷᵢ)² ; SSR = SST − SSE

The coefficient of determination is then the explained fraction:

R² = SSR / SST = 1 − SSE / SST

Because R² never decreases when you add another predictor — even an irrelevant one — it can flatter a multi-variable model. Adjusted R² corrects for this by penalising the predictor count k against the sample size n:

R²_adj = 1 − (1 − R²)·(n − 1)/(n − k − 1)

Two edge cases matter. When every y value is identical, SST is zero and R² is undefined — the tool says so instead of dividing by zero. And when n − k − 1 is zero or negative there are too few observations for the number of predictors, so adjusted R² is undefined and the tool shows a message rather than a misleading number. As a cross-check in paired mode, the calculator also reports the Pearson correlation r and confirms the identity r² = R² that holds for any simple linear regression.

• NIST/SEMATECH e-Handbook §1.4.1 — sums of squares (SST, SSR, SSE) and the coefficient of determination R²
• NIST/SEMATECH e-Handbook §4.4.1 — adjusted R² and the number of parameters
• NIST/SEMATECH e-Handbook of Statistical Methods — full handbook

The formulas on this page were last cross-checked against the NIST e-Handbook on 2026-07-09. R² and adjusted R² are universally-agreed mathematical definitions, so they are stable; this page is reviewed if the cited handbook sections are revised.