How this instrument works
R-squared, the coefficient of determination, answers a specific question: of all the variation present in a set of actual, observed values, what fraction does a given set of predictions account for? It ranges from 0 (the predictions explain none of the variation — no better than just guessing the average every time) up to 1 (the predictions match the actual values perfectly). An R² of 0.98, for instance, means 98% of the spread in the actual values is captured by the predictions, leaving 2% unexplained.
The calculation compares two sums of squares. The residual sum of squares (SSres) totals up the squared gap between each actual value and its corresponding predicted value — the leftover error the predictions didn't capture. The total sum of squares (SStot) totals up the squared gap between each actual value and the overall mean of the actual values — the total variation there was to explain in the first place. R² is 1 minus the ratio of the two: how much of the total variation the residual error fails to explain, subtracted from 1, i.e. flipped into how much it does explain.
This instrument takes actual and predicted values directly, as two already-computed series — it does not fit a new regression line the way this site's linear-regression calculator does. That distinction matters: here, the predictions might come from any external source at all (a machine-learning model, a physical formula, someone else's forecast, even a hand-guessed set of numbers), and this page simply scores how well those specific predictions line up with reality, rather than fitting the best possible line through the data itself.
- Enter the real, observed values into Actual (observed) values, separated by commas, spaces, or new lines.
- Enter the corresponding predicted or modeled values, in the same order, into Predicted (model) values.
- Both series need the same number of entries, and at least 2 pairs, for the comparison to mean anything.
- Read Coefficient of determination (R-squared) for the headline result: the closer to 1, the better the predictions track the actual values.
- Read Residual sum of squares (SSres) and Total sum of squares (SStot) if you want the underlying components R² is built from.
Worked example — five actual values against a model's predictions
A model predicted five outcomes as 2.8, 5.4, 6.6, 9.2, 10.4, while the actual observed values turned out to be 3, 5, 7, 9, 11. Entering both series, the instrument first finds the actual values' mean, 7, and computes Total sum of squares (SStot) as the sum of squared deviations from that mean: (3−7)²+(5−7)²+(7−7)²+(9−7)²+(11−7)² = 16+4+0+4+16 = 40.
Next it compares each actual value to its own prediction: residuals of 0.2, −0.4, 0.4, −0.2, 0.6, squared and summed to give Residual sum of squares (SSres) = 0.04+0.16+0.16+0.04+0.36 = 0.76. Coefficient of determination (R-squared) then reads 0.981, computed as 1 − 0.76/40 = 1 − 0.019. That model is explaining 98.1% of the variation in the actual outcomes, leaving only a small residual gap between what was predicted and what actually happened.
Questions
How is this different from the R² reported by the linear-regression calculator?
This instrument scores predictions you already have against actual outcomes you already have — it does no fitting of its own, and the 'predictions' can come from anywhere: a machine-learning model, a manual estimate, a physical formula. This site's separate linear-regression calculator instead fits a brand-new best-fit line through your x,y data and reports that fitted line's own R² — a different question (how good is the model I'm about to fit) from the one this page answers (how good is the model I already have).
Can R² come out negative?
Yes, mathematically, though it's rare with well-behaved predictions. Because this page takes predictions from any external source rather than fitting an optimal line, a genuinely poor set of predictions — one that does worse than simply guessing the actual data's mean every time — can push SSres above SStot, making R² negative. A least-squares regression's own R² can never go negative, since by construction it fits at least as well as the mean; predictions from elsewhere carry no such guarantee.
What does an R² of exactly 0 mean here?
It means the predictions do no better, on average, than simply predicting the actual data's own mean value every single time — SSres equals SStot exactly, so the ratio subtracted from 1 is 1, leaving 0. It's the 'naive baseline' benchmark: any set of predictions worth using should generally beat this, since it means the predictions carry no more information than just guessing the average.
Does a high R² mean the predictions are unbiased?
Not necessarily. R² measures how much of the variation is explained, which is sensitive to how well the predictions track the ups and downs of the actual data, but a set of predictions that's consistently offset by a constant amount (always 2 units too high, for instance) can still have SSres be small relative to SStot if the ups and downs still track closely, keeping R² high despite the systematic bias. Checking whether the residuals are centered near zero on average is a separate, complementary check this instrument doesn't perform automatically.
Does the order of my actual and predicted values matter?
Yes — each actual value is compared against the predicted value in the same position, so the two series need to be entered in matching, corresponding order. Entering them out of order, or with mismatched counts, pairs the wrong observations together and produces a meaningless R², which is why this instrument requires both series to have exactly the same number of entries before computing anything.