Coefficient of Determination Calculator

R² can never fall, so “higher R²” proves nothing

This is the property that makes R² useless for model selection when used on its own, and it follows from the definition rather than from anything about your data.

Adding a term to a model can never decrease its R². The larger model contains the smaller one as a special case — set the new coefficient to zero and you recover it exactly — so least squares can always do at least as well. It will do strictly better unless the new term is perfectly useless, and randomness guarantees it is not.

So a quadratic always beats a line, whether or not the curvature is real. Add enough terms and R² reaches exactly 1: with n points, a polynomial of degree n − 1 passes through every one of them. That model has learned the noise and will predict nothing.

Adjusted R² is the number that can fall. It applies a penalty of (n − 1)/(n − p − 1) for each parameter, so a term only raises it if it earns more than it costs. The tool shows both, and where the adjusted value drops while R² rises, the extra term is not paying for itself.

Adjusted R² can itself go negative, which is its way of saying the model is worse than useless once the parameter count is accounted for. That is informative rather than an error.

The general rule this points at: R² is a description of fit on the data you fitted, not a measure of a model’s quality. To compare models honestly, use the adjusted version, an information criterion, or — best — performance on data the model has not seen.

The R² you were shown for a curve fit was computed in log space

Exponential, power and logarithmic regressions have no closed-form least-squares solution on their own scale. So every spreadsheet and calculator fits them the same way: take logarithms until the relationship is straight, run ordinary least squares, transform back.

What almost none of them say is that the R² comes from the transformed regression. The fit minimises squared error in log space, so the R² describes how well a line fits the logged data — not how well the curve fits your actual values.

The gap is real and usually flattering. On the tool’s default dataset the exponential fit reports 0.995 in log space and achieves 0.931 on the original scale — 6.4 percentage points of fit that does not exist. On a power fit the gap can exceed 14 points. Measured across random exponential data, the log-space figure is the higher of the two about three times in four, and the average gap is clearly positive — so it is a systematic bias rather than a universal law, and the tool shows both numbers whenever they differ rather than asserting which is larger.

The underlying difference is what is being minimised. Least squares on logs minimises proportional error, so a 10% miss on a small value counts as much as a 10% miss on a large one. Least squares on the original scale minimises absolute error, so large values dominate. Neither is wrong — on data spanning orders of magnitude the proportional version is usually what you want — but they are different fits, and one of them is being reported with the other’s goodness-of-fit statistic.

Which to report depends on what the error costs. If a 10% error matters equally at every scale, keep the log fit and say so. If large absolute errors are what hurt, the log-space fit is not the right one and its R² is doubly misleading — wrong statistic on the wrong fit.

The logarithmic model is the exception: y = a + b⋅ln x is linear in ln x with no transform on y at all, so its R² is already on the original scale and there is no gap.

R² of zero does not mean no relationship

R² measures how well a specified model fits. A low value can mean there is no relationship, or that you fitted the wrong shape, and those need opposite responses.

Load the tool’s “A curve” preset. Seven points on a perfect parabola. The linear R² is exactly 0.000000 — a straight line accounts for none of the variation, so on that evidence x tells you nothing about y. The quadratic R² is exactly 1.000000. y is completely determined by x.

Two conclusions from identical data, and the linear one is the confident one. This is the same failure the correlation coefficient has, for the same reason: both measure fit to a straight line and neither can see any other shape.

The remedy is the residual plot, not a bigger R². Plot the residuals against x. If the model is right they scatter without pattern. Curvature in the residuals means a missing term; a funnel shape means the error variance changes with x. Both are invisible in R² and obvious in the plot.

Range restriction cuts the other way. R² depends on how much x varies in your sample. Fit the same true relationship on a narrower range of x and R² drops sharply, because SST shrinks while the noise does not. Two studies of the same phenomenon can report R² of 0.7 and 0.2 with neither being wrong — which is one reason R² values are not comparable across datasets.

It can be negative

The name says “R squared”, which suggests a number that cannot go below zero. The definition says otherwise, and the cases where it does are the ones that matter most.

R² = 1 − SSE/SST. When SSE exceeds SST — when the model’s errors are larger than simply predicting the mean of y every time — the ratio exceeds 1 and R² goes negative.

It cannot happen for an ordinary linear fit with an intercept, because that model contains the flat line at ȳ as a special case and least squares will never do worse. That special case is why the impossibility is usually taught as a general rule.

It happens readily in three situations. A regression forced through the origin, where the intercept is not available to absorb the mean. A curve fit whose parameters were chosen in log space, which is not optimising the quantity R² measures. And — most importantly — a model evaluated on data it was not fitted to.

That last case is where a negative R² is genuinely useful information. An out-of-sample R² below zero says the model performs worse than a constant prediction on new data, which is the clearest possible signal that it has fitted noise. A model with an in-sample R² of 0.95 and an out-of-sample R² of −0.3 is not a good model with a validation problem; it is not a model.

So a negative R² is not an error to be clamped away. Software that reports 0 instead is discarding the finding.

What to report instead

R² is a ratio with no units, which is why it travels well and why it says so little about whether a model is useful. Three alternatives carry more.

The residual standard error is in the data’s own units. “Predictions are typically within ±3.2 kg” is something a reader can weigh against what they need. “R² = 0.82” is not, because whether 0.82 is good depends entirely on the field and the spread of y. The tool reports it beside every fit.

Out-of-sample error is the only honest measure of prediction. Hold data back, or cross-validate. In-sample R² describes how well the model reproduces data it has already seen, and a model with enough parameters reproduces it perfectly while predicting nothing.

The residual plot answers questions no summary can. Curvature means a missing term; a funnel means the error variance changes with x; a lone point far from the rest means one observation is driving the fit. A high R² is compatible with all three.

And state the range of x. R² depends on how much x varies in your sample, so the same true relationship gives very different values on wide and narrow ranges. Two studies reporting 0.7 and 0.2 can be measuring the same effect with neither being wrong, which is the single strongest reason not to compare R² across datasets.

Sources and methodology

References for the definition and its failure modes.

• NIST/SEMATECH e-Handbook — assessing a fit, including why R² alone is insufficient
• Anscombe, The American Statistician 27(1), 1973 — four datasets with identical R² of 0.667 and nothing else in common
• Chicco, Warrens & Jurman, PeerJ CS 7, 2021 — on R² against alternative regression scores, and its behaviour out of sample
• Correlation Coefficient Calculator — whose square this is, for simple linear fits
• Linear Regression Calculator — for the fitted line and its residuals
• How we calculate — methodology

Method. R² is computed as 1 − SSE/SST throughout, never as the squared correlation — the two agree only for a simple linear fit with an intercept, and the difference is precisely where the interesting cases live. For exponential and power models, the fit is obtained by linearising and the R² is then recomputed against the original y values, with the log-space figure shown separately so the gap is visible rather than silent. Points the transform cannot take are excluded and counted rather than dropped quietly. The suite asserts that quadratic R² never falls below linear R² across a thousand random datasets, that a perfect parabola gives a linear R² of exactly zero and a quadratic R² of exactly one, that the log-space and original-scale figures differ by 6.4 points on the default data, and that an out-of-sample evaluation produces a negative R² rather than a clamped zero. It also asserts that the log-space figure is the higher of the two in only 70–85% of random datasets — an assertion written specifically to stop this page claiming “always”, which an earlier draft did. That engine is verified on every change against 37 assertions. The count and the per-case breakdown are published on the formula verification page.

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