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An R-squared of 0.85 does not mean the model is “85% correct,” and an R-squared of
0.10 does not mean the study failed. R-squared (R²) answers exactly one question:
what proportion of the variance in the outcome does this specific model, fit to this specific
sample, account for? It is silent on whether the model’s functional form is right, whether the
relationship is causal, whether any individual coefficient is meaningful, or whether the model
will predict well on new data. Those are four separate questions, and treating R² as an
answer to any of them is the single most common misreading of a regression results section. This
guide covers what R² actually measures, the specific traps that misinterpretation takes, and
when adjusted R² is the number to report instead.
What R-squared actually measures
R² is the proportion of variance in the outcome variable explained by the model, relative
to a baseline model that just predicts the mean of the outcome for every observation:
R² = 1 − (SSE / SST) = SSR / SST
where SST (total sum of squares) is the total variance in the outcome around its mean, SSR
(regression sum of squares) is the variance the model’s predictions account for, and SSE (error
sum of squares, i.e. the residuals) is what’s left over. An R² of 0.40 means the model
accounts for 40% of the variance in the outcome across the sample used to fit it — the
remaining 60% is residual variance the model does not explain, which can come from measurement
error, omitted predictors, or genuine randomness in the outcome itself.
In simple linear regression with one predictor, R² is exactly the square of the Pearson
correlation coefficient (r) between the predictor and the outcome — see CASRAI’s
correlation coefficient guide for what r itself
measures and how it differs from R² once a model has more than one predictor. In multiple
regression, R² is the squared correlation between the observed outcome values and the
model’s fitted values, which no longer decomposes into any single predictor’s correlation with
the outcome.
The three things R-squared cannot tell you
R² is a summary of fit, not a certificate of validity. Three specific claims it cannot
support, no matter how high the number is:
1. That the model is correctly specified. A high R² is fully compatible
with a misspecified functional form, an omitted confounding variable, or a relationship that only
looks linear over the observed range of the data. A model can fit the sample well and still be the
wrong model — R² measures how much variance is explained, not whether the explanation
is structurally correct. Checking the model’s assumptions (linearity, homoscedasticity,
independence of errors) is a separate step R² does not substitute for; see CASRAI’s
regression analysis guide for how to check each one and
how to read a residual plot for
the diagnostic that actually catches functional-form problems R² misses.
2. That the relationship is causal. R² is a property of the joint
distribution of the fitted values and the observed outcome — it has no mechanism for
distinguishing a causal relationship from a purely associative one, a reverse-causal relationship,
or a relationship that is fully or partly generated by a confounder driving both variables. An
R² of 0.90 from an observational cross-sectional dataset carries exactly the same causal
information as an R² of 0.05 from the same dataset: none, on its own. Causal claims require a
causal-inference design (randomization, an instrumental variable, a regression-discontinuity or
difference-in-differences setup) layered on top of the regression, not a larger R².
3. That any individual coefficient is meaningful. R² is a property of the
whole model, not of any one predictor. A model can have a high overall R² while most of that
explanatory power comes from one dominant predictor and the others contribute almost nothing
individually — or a model can have a modest R² while containing one predictor with a
precisely estimated, practically important coefficient. Whether a specific predictor matters is a
question for that predictor’s own coefficient, standard error, and confidence interval, not for
the model’s aggregate R².
Misinterpretation traps to watch for specifically
Beyond the three claims above, several narrower traps show up repeatedly in how R² gets
reported and read:
Treating a low R² as a failed study. In fields where the outcome is
shaped by many small, partly unmeasured influences — psychology, education, epidemiology,
most social and behavioral research — an R² of 0.10–0.30 is routinely the
field-normal range for a genuinely useful, statistically and substantively meaningful finding.
In fields with fewer, tightly controlled inputs — physics, engineering, analytical chemistry
— a comparably low R² usually does signal a real problem. There is no universal
threshold for a “good” R² independent of the field and the outcome being modeled; read it
against what’s typical for that literature, not against an arbitrary round number.
Treating in-sample fit as predictive performance. R² is calculated on the
same data used to fit the model, so it measures how well the model fits the data it has already
seen — not how well it will predict new, unseen observations. Adding predictors, including
ones with no real relationship to the outcome, mechanically pushes R² up or leaves it
unchanged; it can never go down. A model can therefore reach a high in-sample R² largely
through overfitting — fitting noise specific to that sample — and then perform
considerably worse on a held-out test set or a new dataset. If the goal is predictive accuracy
rather than describing the sample at hand, cross-validated or out-of-sample R² (or a proper
scoring rule such as RMSE on held-out data) is the relevant number, and it is a different quantity
from the in-sample R² a regression output reports by default.
Comparing R² across models with different outcome variables. R² is
only comparable across models that predict the same outcome variable in the same form. A model
predicting log(income) and a model predicting raw income are not comparable on R² even if
both are otherwise reasonable specifications — the total variance being explained (SST) is
different in each case because the outcome itself is different. The same applies to comparing
R² across different samples, different subsets of the same data, or different units of
measurement for the outcome.
Assuming R² is always between 0 and 1. That range holds for ordinary
least squares regression with an intercept. It does not hold generally: R² calculated for a
model without an intercept, for a nonlinear model, or on out-of-sample data can come out negative
— meaning the model fits worse than simply predicting the outcome’s mean for every
observation. A negative R² is a real, informative result (the model is worse than the naive
baseline), not evidence of a calculation error.
Extrapolating fit beyond the observed range. R² describes how well the
model fits within the range of predictor values actually observed in the sample. It says nothing
about how the model will perform for predictor values outside that range — a high R²
inside the observed range is no guarantee the relationship stays linear, or holds at all, beyond
it.
Adjusted R-squared: correcting for the number of predictors
Because R² mechanically rises (or at worst stays flat) every time a predictor is added to
the model — regardless of whether that predictor has any real relationship to the outcome
— comparing raw R² across models with different numbers of predictors always favors
the larger model, even when the added predictors are noise. Adjusted R² corrects for this by
penalizing additional predictors that don’t improve the fit enough to justify their inclusion:
Adjusted R² = 1 − [(1 − R²)(n − 1) /
(n − p − 1)]
where n is the sample size and p is the number of predictors (not counting the intercept). A
worked example: with n = 100 observations and an R² of 0.50, adding a predictor moves R²
from 0.50 (p = 4) to 0.52 (p = 5). Adjusted R² for the first model is 1 − [(0.50)(99) /
95] ≈ 0.479; for the second, 1 − [(0.48)(99) / 94] ≈ 0.494. The adjusted value
still rose here, meaning the new predictor earned its place. If the raw R² had only ticked up
to 0.505 instead, adjusted R² for the five-predictor model would actually fall below 0.479
— the signal that the added predictor isn’t pulling its weight relative to the complexity it
adds.
Practical rules that follow directly from the formula: adjusted R² is always at or below
R², and it can be negative even when R² is positive, if the model explains less variance
than would be expected by chance given the number of predictors relative to the sample size.
Report adjusted R² — not raw R² — whenever comparing models with different
numbers of predictors, and always report it alongside raw R² when a model has more than a
small handful of predictors relative to the sample size, since that is exactly the situation where
the two numbers diverge most and the raw figure is most likely to overstate genuine model
improvement. See CASRAI’s multicollinearity
and VIF guide for a related problem that specifically affects models with several correlated
predictors: R² and adjusted R² can both look fine even when the individual coefficient
estimates behind them are unstable.
A worked interpretation
Illustrative example: a model predicts graduate students’ first-year research productivity
(publications and conference presentations combined) from three predictors — prior research
experience, advisor meeting frequency, and cohort size — using n = 140 students, and reports
R² = 0.28, adjusted R² = 0.26. The accurate sentence for a results section is: “The model
explained 28% of the variance in first-year research productivity (adjusted R² = 0.26),
consistent with the range typically reported for productivity models in this literature; the
remaining variance likely reflects factors not captured here, such as project-specific
circumstances and unmeasured mentoring quality.” What that sentence deliberately avoids: claiming
the model “proves” what drives productivity, treating 0.28 as low in an absolute sense without a
field benchmark, and treating the close agreement between R² and adjusted R² (a two-point
gap, not a collapse) as anything other than a sign that three predictors is a reasonable model size
for this sample.
Frequently asked questions
What counts as a “good” R-squared?
There is no fixed threshold. It depends on the field and how much of the outcome’s variance is
plausibly attributable to measurable, included predictors versus inherent noise or unmeasured
factors. Compare a reported R² against what similar studies in the same literature typically
report, not against a round number like 0.70 imported from an unrelated field.
Can R-squared be negative?
Yes, in specific circumstances: models fit without an intercept, some nonlinear model
specifications, and R² calculated on out-of-sample or held-out data (where the model can
genuinely predict worse than simply guessing the training-sample mean every time). It cannot be
negative for standard OLS regression with an intercept evaluated on its own training data.
Is a higher adjusted R-squared always better?
Higher adjusted R² means the added complexity was justified by the improvement in fit, but
it is still only a fit statistic — it doesn’t validate the model’s assumptions, causal
claims, or out-of-sample performance. Use it to compare nested models with different predictor
counts, not as a standalone measure of whether a model is “good.”
How is R-squared different from correlation?
In simple regression with one predictor, R² is the square of the Pearson correlation
coefficient (r) between that predictor and the outcome, so it loses the sign — R² can’t
tell you whether the relationship is positive or negative, only how much variance is explained. In
multiple regression, R² is the squared correlation between the observed and fitted values and
no longer maps onto any single predictor’s bivariate correlation. See CASRAI’s
correlation coefficient guide for how to read r
itself, including the Pearson-versus-Spearman distinction.
Does R-squared apply to logistic regression the same way?
No. Logistic regression doesn’t have a residual sum of squares in the same sense, so there is no
single agreed R² analogue — several pseudo-R² measures exist (McFadden’s,
Nagelkerke’s, Cox & Snell’s), they are not interchangeable with each other or with linear
regression’s R², and they tend to run lower than a linear-model R² would for a
comparably well-fitting model. See CASRAI’s logistic
regression interpretation guide for how model fit is actually reported for that model type.
