Exam mark predicted from revision minutes and a baseline assessment. R = 0.96469, R² = 0.93062, and the model is not the interesting part. Revision minutes correlates +0.84202 with the mark on its own and −0.47910 once the baseline is held fixed — the same predictor, the same data, opposite signs. Its unique contribution to R² is 0.02067 against the baseline score’s 0.22162. Reporting the simple correlation would make revision look like the stronger of the two predictors. It is the weaker one, and it points the other way.
40 cases · 2 predictors
R = 0.96469, R² = 0.93062
Adjusted R² is 0.92687, and the F test gives p = 0.000e+0 on 2 and 37 degrees of freedom. The largest variance inflation factor is 8.0493.
R
0.96469
never negative
R²
0.93062
adjusted 0.92687
F
248.1514
p = 0.00e+0
Largest VIF
8.0493
predictors are distinct enough
What each predictor actually adds
Simple, partial and unique contribution for every predictor, with variance inflation
Predictor
Simple r
Partial
Semi-partial
Unique R²
VIF
revision minutes
0.84202
-0.47910
-0.16887
0.02067
8.0493
baseline score
0.95391
0.87269
0.30760
0.22162
8.0493
One predictor changes sign between the first two columns. The simple correlation and the partial correlation are different quantities, and when they disagree, quoting the first while fitting the second is how a predictor gets described backwards.
The unique contributions sum to 0.24229 against an R² of 0.93062. The gap is variation the predictors explain jointly and none of them can claim alone, which is why a table of unique contributions never adds up to the model.
With a single predictor R is exactly the absolute value of that predictor’s correlation with the outcome, and adding a predictor can never lower R². The verification suite asserts both on generated data — the first on 150 datasets, the second on 120.
Unique contributions Variance inflation Adjusted R² shown Linear combinations only
What this tool shows
On the shipped preset one predictor correlates +0.84202 with the outcome on its own and −0.47910 once the other is held fixed. Its unique contribution to R² is 0.02067 — two points out of 93 — against the other predictor’s 0.22162. A table of simple correlations would rank them the wrong way round and give the weaker one the wrong sign. This page prints all three columns next to each other, because the disagreement between them is the finding.
Multiple R and R² with the adjusted figure beside them, and an F test with both degrees of freedom
Simple, partial, semi-partial and unique R² for every predictor, in one table
Variance inflation factors, so predictors measuring the same thing announce themselves
The sum of unique contributions against R², which shows how much is shared rather than attributable
Presets where the signs disagree, where two predictors duplicate each other, and where nothing is real at all
Why R² rising is not evidence that a predictor belongs in the model
Unique contributions Variance inflation Adjusted R² shown Simple and partial together
R measures the best linear combination. Only linear ones.
Updated 13 September 2026 · Works in any browser, no installation
The multiple correlation R is the correlation between an outcome and the best linear combination of a set of predictors, and R² is the share of the outcome’s variance that combination explains. R is never negative, because the combination is chosen to point the right way. The useful output is not R itself but the breakdown underneath it: what each predictor contributes on its own, and what it still contributes once the others are present.
At a glance
Formula shown
R² = 1 − SS_residual/SS_total from the least-squares fit of y on the predictors; R = √R². Adjusted R² = 1 − (1 − R²)(n − 1)/(n − k − 1), which penalises each added predictor and can fall. F = [R²/k] / [(1 − R²)/(n − k − 1)] on k and n − k − 1 degrees of freedom. A predictor’s unique contribution is R² for the full model minus R² for the model without it — equal to its squared semi-partial correlation.
Scenario support
Deciding which of several candidate predictors is worth keeping, checking whether a model is explaining anything beyond what its size guarantees, diagnosing predictors that duplicate one another, and reporting how much each variable adds rather than how much it correlates.
Educational estimate
Planning support from the values you enter — not professional advice.
The three columns disagree, and that is the output
Most tools report R² and a list of coefficients. The interesting information is in the gap between what a predictor looks like alone and what it does in company.
Revision minutes correlates +0.84202 with exam marks on its own. Held against a baseline score, its partial correlation is −0.47910.
Its unique contribution to R² is 0.02067. The baseline score contributes 0.22162 — more than ten times as much.
A ranked table of simple correlations would put them the wrong way round, and would give the weaker one the wrong sign into the bargain.
Which is why all three columns are printed together. When they agree, nothing is lost. When they disagree, the disagreement is what you came for — and it is the same quantity a partial correlation reports one pair at a time.
The contributions do not add up, and should not
A natural reading of the unique-contribution column is that the entries should sum to R². They never do, and the shortfall is informative rather than an error.
On the shipped preset the unique contributions sum to 0.24229 against an R² of 0.93062. Three quarters of the explained variance belongs to no single predictor.
That shared portion is variation the predictors explain jointly. Either could account for it alone; neither can claim it once the other is present.
The more correlated the predictors, the larger the shared part grows. On the second preset the two unique contributions sum to 0.03691 out of an R² of 0.79951.
So “this variable explains x% of the outcome” needs qualifying with whether that is on its own or on top of everything else, because the two numbers can differ by an order of magnitude.
Variance inflation, and what it is really telling you
The VIF column is usually skimmed as a threshold check. The second preset shows what it is actually measuring, which is a warning about stability rather than about fit.
Two predictors correlating 0.983738 give a VIF of 30.998 each. The usual rule of thumb starts worrying above 5 and treats 10 as serious.
The model is fine. R² is 0.79951, F is highly significant, and predictions from it would be perfectly good.
The coefficients are not. Adding the second predictor raises R² by 0.0887 percentage points over the first alone, and its partial correlation is −0.06636 against a simple correlation of +0.87377.
Collinearity damages interpretation, not prediction. If the model is for forecasting, a high VIF is survivable. If anyone is going to read a coefficient as an effect, it is not — and the fix is to drop one of the pair rather than to report both.
R² always rises. Adjusted R² does not.
The third preset exists because “R² went up” is used as evidence for keeping a variable, and it is not evidence of anything at all.
Adding a predictor of pure noise took R² from 0.74066 to 0.74112. It rose, because it always rises.
Adjusted R² fell from 0.72938 to 0.71759. That is the figure that noticed the predictor was worthless.
The mechanism is arithmetic rather than statistical. Least squares will use any column it is given, and a column of random numbers always has some accidental alignment with the residuals.
Which is why adjusted R² is the one to quote, and why a rising R² across a sequence of larger models tells you nothing about whether the models are getting better.
Six predictors, twenty cases, nothing real
The fourth preset is a table of independent random numbers, and the output looks like a finding. That is the whole lesson.
R² is 0.36048. A third of the outcome apparently explained, which would read as a modest but real model in a report that stopped at that line.
Adjusted R² is 0.06532 and the F test gives p = 0.3562. Both say what is actually true: nothing.
The strongest single predictor correlates 0.37031 with the outcome — by chance alone, from six tries at twenty cases.
R² climbs towards 1 as the predictor count approaches the sample size whether or not anything is real, reaching exactly 1 when they are equal. Any model with a predictor count near its sample size should be read with that in mind.
What R cannot see
R is the correlation with the best linear combination, and that qualifier does more work than it appears to.
A relationship that is not linear is invisible to it, exactly as it is to an ordinary correlation — a perfect U-shape can give an R near zero.
Interactions are invisible too unless the product term was included as its own column, which is a modelling decision rather than something the arithmetic discovers.
A low R is therefore not evidence of no relationship. It is evidence of no linear relationship, and distance correlation is the measure that distinguishes the two.
And a high R says nothing about direction of causation, nor about whether the predictors would still work on data collected somewhere else.
Reporting a multiple correlation
Four items, and the third is the one that keeps a reader from over-reading a single predictor.
Give adjusted R², not just R², with the number of predictors and the number of cases. All three, or none of them mean anything.
Give the F and both degrees of freedom. That is what says whether the fit beat what its size guarantees.
Give unique contributions rather than simple correlations. And when the two disagree in sign, say so explicitly rather than choosing one.
Give the largest VIF. It tells a reader whether the individual coefficients can be interpreted at all, which is separate from whether the model fits.
Sources and methodology
References for the coefficient and its adjustments.
Method. The fit is by normal equations with a matrix inverse, and every predictor’s unique contribution is computed by refitting the model without it rather than by reading a coefficient, so the figure is exactly the R² that predictor adds when entered last. Variance inflation factors come from regressing each predictor on the others, which is the definition. The verification suite anchors the whole thing on two identities: with a single predictor, R must equal the absolute value of that predictor’s correlation with the outcome, asserted on 150 generated datasets; and adding a predictor can never lower R², asserted on 120 more alongside the requirement that every variance inflation factor be at least 1. Predictor sets that are exactly collinear are refused rather than returned with an arbitrary solution. That engine is verified on every change against 90 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Partial CorrelationPartial and semi-partial correlation with any number of controls, computed from residuals, with the raw correlation and the share each control already explains.
Coefficient of DeterminationR-squared across five models at once with adjusted R-squared beside it, because adding a term can never lower it and the value most tools report for a curve fit was computed in log space.
Linear RegressionThe least-squares line with r and r² — and the regression of x on y beside it, because those are two different lines rather than one line rearranged.
Correlation CoefficientReports Pearson, Spearman and Kendall together with the scatter plot, and ships Anscombe's quartet built in — four datasets with an identical r of 0.816 that Spearman tells apart.
Distance CorrelationDistance correlation with a permutation test and Pearson's r on the same data, detecting curved and variance-based dependence that a correlation scores as zero.
Covariance MatrixSample covariance and correlation matrices with eigenvalues, determinant and condition number, so redundancy spread across several variables cannot hide from a pairwise scan.
An educational tool. R measures the best linear combination of the predictors, so a strong non-linear relationship can give a low R and a high R² says nothing about causation. With a predictor count close to the sample size, R² approaches 1 whether or not any relationship exists — the adjusted figure and the F test are what distinguish the two cases.