Coefficient of determination calculator

The coefficient of determination calculator compares observed outcomes with predictions and returns R squared, the share of variation explained by the model under the standard sum-of-squares definition. Supply two equally sized numeric arrays whose positions represent matched observations and predictions. The result includes the residual sum of squares, total sum of squares, observed mean, and pair count, making the calculation easy to inspect as well as reuse. It accepts finite numbers, rejects mismatched arrays, and reports when the observed series has no variance, because R squared is not defined in that case.

Prepare matched observed and predicted values

Enter the measured outcomes in observed and the corresponding model outputs in predicted. Position is the pairing rule: the first prediction is compared with the first observation, the second with the second, and so on. Both arrays therefore need the same number of elements, and the calculator returns an error if their lengths differ.

Use raw values expressed on the same scale and in the same units. Do not sort one array independently, remove an item from only one side, or mix probabilities with class labels, because any of those changes destroys the intended pairing. Every element must be a finite number; blank values, nulls, infinity, and text are rejected instead of being silently converted.

At least two pairs are required. If data has missing outcomes, decide how those rows should be handled before calling the calculator, then remove the complete pair consistently. That preparation keeps the resulting score tied to a clearly defined evaluation sample and makes comparisons between models meaningful.

Understand how R squared is calculated

The calculator first finds the mean of the observed values. It then computes the total sum of squares by adding each squared distance from that mean; this quantity represents the variation available to explain. Next it computes the residual sum of squares by adding each squared difference between an observation and its paired prediction.

R squared is one minus the residual sum divided by the total sum. A score of 1 indicates exact predictions for every supplied pair. A score of 0 means the predictions perform no better, by this squared-error comparison, than always using the observed sample mean.

The value can be negative when prediction errors are larger than the mean baseline errors; negative results are valid and should not be clamped to zero. When every observed value is identical, the total sum of squares is zero, so the ratio has no defined value and the calculator returns an error. The response exposes both sums and the observed mean so you can reproduce the calculation or diagnose an unexpected result.

Interpret the result in the right context

R squared summarizes fit on the particular pairs you provide, but it does not establish causation, guarantee useful future predictions, or reveal whether errors follow an acceptable pattern. A high training-set score can coexist with overfitting, so evaluate predictions from a validation or test set when measuring generalization. Compare scores only when the outcome, sample, and evaluation procedure are compatible.

Domain context also matters: a modest score may be useful for a noisy real-world outcome, while a very high score may still be inadequate for precise engineering control. Inspect residual plots and error measures such as mean absolute error or root mean squared error alongside R squared, since those tools show error scale and systematic bias that a single proportion cannot.

This implementation uses the common unweighted definition based on the observed mean; it does not add an intercept, fit a regression, adjust for the number of predictors, or calculate adjusted R squared.