R-Squared Calculator

Compute R-squared (R²), Pearson correlation coefficient (r), slope, intercept, and regression line equation from your (x, y) data.Visualize the scatter plot and best‑fit line instantly. Essential for goodness-of-fit analysis, predictive modeling, and statistical learning.

Format: numeric values separated by comma. At least 2 non‑identical x values required.
? Perfect Positive (r=1)
? Strong Correlation
? Moderate Correlation
? Weak / Noisy
? Negative Correlation
Regression Results
R² (Coefficient of Determination)
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Correlation r (Pearson)
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Regression Equation: y = bx + a
Slope (b): —  |  Intercept (a): —
Data points (n): 0  |  Interpretation: —
Load or enter data above.
Privacy first: All calculations happen locally in your browser. No data is transmitted to any server.
Observed data
Regression line (ŷ = bx + a)

Understanding R‑Squared (Coefficient of Determination)

R² measures the proportion of variance in the dependent variable (Y) that is predictable from the independent variable (X) using a linear regression model. It ranges from 0 to 1—closer to 1 indicates that the model explains a large fraction of the data's variability. In simple linear regression, R² equals the square of the Pearson correlation coefficient (r).

R² = 1 - (SSres / SStot),   where SSres = Σ(yᵢ - ŷᵢ)², SStot = Σ(yᵢ - ȳ)²

Why Use an Interactive R² Calculator?

  • Educational Depth: Visualize how regression line fits data, strengthen intuition for goodness‑of‑fit.
  • Research & Analytics: Quickly assess linear relationships in survey data, economics, or scientific experiments.
  • Model Validation: Benchmark predictive performance before advanced modeling.
  • Interactive Learning: Modify data points and instantly see how R² and regression parameters change.

Computation Methodology & Assumptions

The ordinary least squares (OLS) method minimizes the sum of squared residuals. Slope (b) = Cov(X,Y)/Var(X), intercept a = ȳ - b·x̄. The tool validates numeric inputs, checks for sufficient variation (non‑constant X), and prevents division by zero. Linear regression assumes linearity, independence, homoscedasticity, and normality of residuals for inference — yet R² remains a descriptive metric even when assumptions are mildly violated.

Step‑by‑step walkthrough

  1. Enter pairs (X,Y) in the text area (each line: x,y).
  2. Click “Compute Regression & Draw” to calculate slope, intercept, R², and r.
  3. The scatter plot displays original points and regression line dynamically.
  4. Interpret R²: 0.80 means 80% of Y variance explained by X.

Interpretation Benchmarks & Real‑World Examples

R² Value Interpretation Example Context
> 0.80 Very strong linear relationship Height vs. shoe size (population average)
0.50 – 0.79 Moderate predictive power Study hours vs. exam scores
0.20 – 0.49 Weak linear association Daily temperature vs. stock returns
< 0.20 Little to no linear fit Randomized controlled noise
Case Study: Marketing Spend vs. Revenue

A digital agency tracks monthly ad spend (X, in $1000) and revenue (Y, in $1000) for 8 months. Using our calculator they obtain R² = 0.87, meaning 87% of revenue variation is explained by ad spend. Slope = 3.2 indicates each $1K ad spend yields $3.2K additional revenue. The scatter plot reveals linear trend and few outliers, confirming reliable forecasting.

Common Misconceptions

  • High R² implies causation? No, R² only measures correlation strength, not causality.
  • R² near 1 guarantees good predictions? Not if overfitting or extrapolating beyond data range.
  • Low R² = useless model? In some fields (social sciences), low R² is still meaningful for identifying trends.
  • R² always increases with more predictors? Adjusted R² penalizes extra variables — but simple linear uses only one X.

Practical Applications Across Disciplines

  • Economics: Quantify how GDP growth explains unemployment changes.
  • Medicine: Dosage vs. drug effectiveness (dose‑response).
  • Machine Learning: Baseline linear model performance metric.
  • Environmental Science: CO₂ emissions vs. temperature anomaly.

Rooted in statistical theory – This implementation follows standard least‑squares principles introduced by Legendre (1805) and Gauss. References include Montgomery, Peck & Vining's "Introduction to Linear Regression Analysis" and authoritative resources from the American Statistical Association. Verified by GetZenQuery tech team, last updated April 2026.

Frequently Asked Questions

It depends on your field. In physical sciences, >0.9 is expected; in social sciences, 0.3–0.5 is often acceptable. The context and research question matter most.

In simple linear regression with OLS, R² is always between 0 and 1. Negative R² can occur only when fitting a non‑linear model or without an intercept — our tool enforces standard linear regression.

Adjusted R² penalizes additional predictors; for simple linear regression (one X), R² = adjusted R². This calculator focuses on the bivariate case.

At least 2 non‑identical X values. However, for reliable inference, n ≥ 10 is recommended.
References: Wolfram MathWorld: Correlation; NIST Engineering Statistics Handbook.