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Coefficient of Determination Calculator (R-squared)

Free R-squared calculator. Compute R², adjusted R², RMSE, F-statistic from data, correlation, or sum

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Why This Statistical Analysis Matters

Why: Statistical calculator for analysis.

How: Enter inputs and compute results.

R²
STATISTICSRegression & Fit

Coefficient of Determination — How Much Variance Does Your Model Explain?

R², adjusted R², RMSE, F-statistic. From raw data, correlation, or sum of squares. Step-by-step breakdown with charts.

Linear Regression →Correlation →

Real-World Scenarios — Click to Load

Input Mode

xy
For adjusted R² and F-statistic
r_squared_results.sh
CALCULATED
$ r_squared --mode="raw" --n=5 --p=1
R²
0.9897
Strong fit
Adjusted R²
0.9863
RMSE (SE)
0.2449
F-statistic
289.0000
Share:
Coefficient of Determination
R² = 0.990
Strong fit
Adj R² = 0.986RMSE = 0.245F = 289.00
numbervibe.com/calculators/statistics/coefficient-of-determination-calculator

Calculation Breakdown

COMPUTATION
Mean of y (ȳ)
5.4000
SS_tot
29.2000
\text{Sigma} (yᵢ - ȳ)^{2}
SS_res
0.3000
\text{Sigma} (yᵢ - ŷᵢ)^{2}
SS_reg
28.9000
SS_tot - SS_res
R²
0.9897
1 - SS_res/SS_tot
Adjusted R²
0.9863
1 - (1-R^{2}) rac{n-1}{n-p-1}
RMSE
0.2449
√(SS_res/n)
MODEL TEST
F-statistic
289.0000
(SS_reg/p)/(SS_res/(n-p-1))

R² Interpretation

Scatter with Regression Line

SS Decomposition (Explained vs Unexplained)

For educational and informational purposes only. Verify with a qualified professional.

Key Takeaways

  • • R²: Proportion of variance in Y explained by the model. R² = 1 − SS_res/SS_tot = SS_reg/SS_tot.
  • • Adjusted R²: Penalizes extra predictors. R²_adj = 1 − (1−R²)(n−1)/(n−p−1). Use when comparing models with different numbers of predictors.
  • • From correlation: For simple linear regression, R² = r².
  • • Interpretation: R² = 0.7 means 70% of variance in Y is explained by the model.
  • • RMSE: √(SS_res/n). Root mean squared error — average prediction error magnitude.
  • • F-statistic: (SS_reg/p)/(SS_res/(n−p−1)). Tests overall model significance.

Formulas Reference

R² = 1 − SS_res / SS_tot = SS_reg / SS_tot

SS_res = Σ(yᵢ − ŷᵢ)² (residual sum of squares)

SS_tot = Σ(yᵢ − ȳ)² (total sum of squares)

SS_reg = Σ(ŷᵢ − ȳ)² (regression sum of squares)

R²_adj = 1 − (1−R²)(n−1)/(n−p−1)

RMSE = √(SS_res / n)

F = (SS_reg / p) / (SS_res / (n−p−1))

Interpretation Guide

R²Interpretation
0Model explains nothing
0.1 - 0.3Weak fit
0.3 - 0.7Moderate fit
0.7 - 1.0Strong fit
1Perfect fit

When to Use Adjusted R²

R² always increases when you add predictors, even if they are useless. Adjusted R² penalizes model complexity. Use adjusted R² when comparing models with different numbers of predictors. If adjusted R² decreases when adding a predictor, that predictor may not improve the model.

Did You Know?

📊R² is also called the coefficient of determination. It was popularized in regression analysis.Source: NIST Handbook
📈In simple linear regression, R² equals the square of the Pearson correlation r.Source: OpenIntro Statistics
🔬Adjusted R² can be negative when the model fits worse than predicting the mean.Source: Penn State STAT 501
📐SS_reg + SS_res = SS_tot. R² = SS_reg/SS_tot.Source: Khan Academy
🎯R² does not indicate causation. A high R² can occur with spurious correlations.Source: Rice Virtual Lab
📉Outliers can inflate or deflate R². Always inspect residuals.Source: NIST Handbook

Frequently Asked Questions

What is the difference between R² and adjusted R²?

R² measures fit but increases with more predictors. Adjusted R² penalizes extra predictors and can decrease when adding irrelevant variables.

Can R² be negative?

Yes. R² = 1 − SS_res/SS_tot. If the model is worse than predicting the mean, SS_res > SS_tot, so R² < 0.

What does R² = 0.7 mean?

70% of the variance in Y is explained by the model. The remaining 30% is unexplained (residual) variance.

How do I get R² from correlation?

For simple linear regression, R² = r². So if r = 0.8, R² = 0.64.

What is the F-statistic in regression?

F tests whether the model explains significant variance. F = (SS_reg/p)/(SS_res/(n−p−1)). Large F suggests the model is useful.

When should I use RMSE vs R²?

R² is scale-free (0–1). RMSE is in the same units as Y — use it to interpret prediction error magnitude.

Step-by-Step: Compute R² from Data

Step 1: Fit the model (e.g., linear regression) to get predicted values ŷᵢ.

Step 2: Compute ȳ (mean of y). SS_tot = Σ(yᵢ − ȳ)².

Step 3: SS_res = Σ(yᵢ − ŷᵢ)².

Step 4: R² = 1 − SS_res/SS_tot.

Residuals and Model Fit

Residuals = observed y − predicted ŷ. A good fit has residuals scattered randomly around zero with no pattern. If residuals show a curve (e.g., U-shape), the relationship may be nonlinear. RMSE summarizes the average magnitude of residuals.

Disclaimer: This calculator is for educational purposes. R² does not indicate causation. For research, use established statistical software (R, Python, SAS, SPSS). Always check regression assumptions: linearity, homoscedasticity, independence, normality of residuals.

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