Regression & correlation

R-Squared Calculator

The share of the variation in y that the model explains — and the adjustment for how many predictors it took.

Also called coefficient of determination calculator, R2 calculator, adjusted R squared calculator.

Updated August 2026Runs in your browser — nothing is uploadedVerified against R

How to read this result

R² is the proportion of variance explained. Total variation in y is the sum of squared deviations from ȳ; the model accounts for some of it and leaves the rest as residual. R² is the first divided by the total, so it runs from 0 to 1.

It never falls when you add a predictor — even a column of random numbers raises it slightly. That is why adjusted R² exists: it charges a penalty per predictor, can decrease, and is the version to compare models with.

A high R² does not mean the model is right. A curved relationship fitted with a straight line can produce R² above 0.95 while being systematically wrong at both ends. The residual plot is the check that R² cannot provide.

A low R² does not mean the model is useless. In fields where outcomes are genuinely noisy — individual human behaviour, market returns — an R² of 0.1 with a well-estimated slope can be an important finding. R² measures noise as much as it measures the model.

The formula

R^{2} = 1 - \frac{S S_{r e s}}{S S_{t o t}} = 1 - \frac{\sum (y_{i} - {\hat{y}}_{i})^{2}}{\sum (y_{i} - \bar{y})^{2}}

$S S_{r e s}$: the residual sum of squares — what the model does not explain
$S S_{t o t}$: the total sum of squares — variation around the mean of y

Adjusted for the number of predictors

R^{2_{a d j}} = 1 - (1 - R^{2}) \frac{n - 1}{n - k - 1}

$k$: the number of predictors
$n$: the sample size

Unlike R², this can fall when a predictor earns less than its cost in degrees of freedom.

Worked example

R² for the advertising line

The linear regression example: ten months of spend against sales, fitted with y = 0.30 + 2.015x.

Total variation in y around its mean of 11.38.
SS_tot = 335.656
Variation left over after the line.
SS_res = 0.8385
The model accounts for the difference.
SS_reg = 335.656 − 0.8385 = 334.818
R² is the explained share.
334.818 / 335.656 = 0.99750
Adjusted for one predictor and ten observations.
1 − (1 − 0.99750) × 9/8 = 0.99719

R² 0.99750adjusted R² 0.99719r 0.99875residual SS 0.8385

An R² of 0.9975 is the kind of number that only appears with tidy data over a wide range of x. Real observational data rarely gets close, and an R² this high on messy data is usually a sign that x and y are two measurements of the same thing.

Checked against R's summary(lm(y ~ x))$r.squared.

The calculator above is loaded with these numbers by the Load the worked example button.

Assumptions, and when to use something else

R² inherits every assumption of the model it summarises. For linear regression that means linearity, independence and constant variance; when those fail, R² is still computable and no longer means what it appears to.

Two specific traps. R² has no meaning for a model fitted without an intercept — some software reports a very high value that is not comparable. And R² computed on the same data the model was fitted to is optimistic; the honest version is out-of-sample, which needs a held-out set.

You want the full regressionLinear regressionCoefficients, standard errors, p-values and the residual plot.
You have several predictorsMultiple regressionWhere adjusted R² starts to matter.
You want the correlation insteadCorrelation coefficientR² is r² for a single predictor.

Questions people ask

What is the difference between R² and adjusted R²?

R² rises whenever a predictor is added, even a useless one. Adjusted R² subtracts a penalty for each predictor and can fall, which makes it the right one for comparing models with different numbers of terms.

Is R² the same as r squared?

For simple linear regression with one predictor, yes exactly. In multiple regression R² is the squared correlation between the observed and fitted values, which generalises the same idea.

Can R² be negative?

Not for an ordinary least-squares fit with an intercept — it is bounded at 0. It can go negative when a model is evaluated on data it was not fitted to, and means the model predicts worse than simply using the mean.

What R² do I need for a good model?

There is no threshold. Compare against what is typical in your field, look at whether the coefficients are sensible, and check the residual plot. A model with R² = 0.2 and a well-estimated, theoretically expected slope beats one with R² = 0.9 and a curve in its residuals.