Coefficient of Determination Calculator
Compute coefficient of determination from paired data. Enter X and Y lists to get correlation, R-squared percentage, and calculation steps.
Equation Preview
Helping Notes
- Guide method: compute correlation \(r\) via sums, then \(R^2 = r^2\) and optionally report as a percentage. :contentReference[oaicite:1]{index=1}
- \(r=\dfrac{n\sum xy-\sum x\,\sum y}{\sqrt{\left(n\sum x^2-(\sum x)^2\right)\left(n\sum y^2-(\sum y)^2\right)}}\) ; finally square \(r\) to get \(R^2\). :contentReference[oaicite:2]{index=2}
- \(0\le R^2\le 1\) with this definition from paired \((x,y)\) data; higher means better linear fit. :contentReference[oaicite:3]{index=3}
Results
Coefficient of Determination (R²)
Correlation (r)
Sums Used
Error
What is Coefficient of Determination Calculator?
A Coefficient of Determination (R²) Calculator quantifies how well a model explains variability in observed responses. For a dataset of actual values \(y_i\) and model predictions \(\hat y_i\), R² measures the proportion of total variation explained by the model. Core sums of squares are:
In simple linear regression with an intercept, R² equals the squared Pearson correlation between \(x\) and \(y\):
Because adding predictors can inflate R², modelers often consult adjusted R²:
About the Coefficient of Determination Calculator
This tool accepts paired data (actuals and predictions) or raw \((x,y)\) for simple regression. It computes SST, SSE, SSR, R², and optional adjusted R², then presents step‑by‑step calculations. Domain checks warn when inputs contain missing values, identical responses (SST = 0), or when evaluating on data different from the model’s fit set. For teaching and audits, each formula renders responsively and intermediate totals are shown so you can reconcile results with spreadsheets or statistical software.
How to Use this Coefficient of Determination Calculator
- Choose input mode: Actuals & Predictions or Raw (x,y) with a fitted simple regression.
- Paste values (comma/space/newline separated). Confirm units and consistent lengths.
- Click calculate to obtain SST, SSE, SSR, R², and (optionally) adjusted R².
- Review the step table and interpretation guidance; investigate outliers or high residuals if R² is low.
- Export the computation steps for documentation or homework.
Examples
Example 1: Small dataset
\(y=[2,4,6],\; \hat y=[2,5,7]\). Mean \(\bar y=4\); totals:
Example 2: Perfect fit
\(\hat y=y\Rightarrow\text{SSE}=0\Rightarrow R^2=1\).
Example 3: Worse than the mean
\(y=[1,2,3],\; \hat y=[3,3,3]\):
FAQs
What does R² measure?
The proportion of variance in responses explained by the model relative to a mean‑only baseline.
How is R² computed from data?
Calculate SST and SSE, then use R² = 1 − SSE/SST with predictions from your model.
What is adjusted R² and when should I use it?
Adjusted R² penalizes extra predictors; use it to compare models with different numbers of predictors.
Can R² be negative?
Yes, when predictions perform worse than the mean baseline or when there’s no intercept in the model.
Is a higher R² always better?
Not necessarily. Overfitting can inflate R²; evaluate residuals and validate on new data.
What’s the difference between SST, SSE, and SSR?
SST is total variation, SSE is unexplained error, and SSR is explained variation by the model.
Does R² indicate causation?
No. It is descriptive of fit and does not imply causal relationships.
Can I use R² for nonlinear models?
Sometimes, but interpretation varies. Consider appropriate goodness‑of‑fit metrics for the model family.
Why does my R² differ from software output?
Differences may arise from rounding, missing‑value handling, weights, or definitions (e.g., centered vs uncentered).
Is R² valid for time‑series data?
Autocorrelation can mislead R². Use cross‑validation, adjusted metrics, or information criteria.
What if SST equals zero?
All responses are identical; variance is zero and R² is undefined. Any prediction is equally good.
How many observations do I need?
At least p+1 for estimation with an intercept; more are needed for reliable assessment and validation.
Is R² the same as correlation squared?
Only for simple linear regression with an intercept; in multiple regression, R² generalizes beyond a single correlation.