You found r = 0.70 and called it a strong correlation. But r² = 0.49 means 51% of the variation is still unexplained. Here is what that means for your research.
R-squared measures the proportion of variance in one variable that is explained by another variable. It ranges from 0 to 1, or 0% to 100%.
X explains 49% of the variance in Y. The remaining 51% is due to other factors, measurement noise, or variables not captured in this simple relationship. The original correlation is positive, but r squared reports magnitude rather than direction.
Effect size: Large (Cohen's benchmark: r² >= .25)
R-squared (r²), also called the coefficient of determination, measures the proportion of variance in one variable that is explained by another variable. It ranges from 0 to 1, or from 0% to 100%.
| r² value | Meaning |
|---|---|
| r squared = 0 | X explains none of the variation in Y. The model is no better than the mean. |
| 0 < r squared < 1 | X explains part of the variation in Y. Higher values mean more explained variance. |
| r squared = 1 | X perfectly explains all variation in Y. This is rare outside controlled systems. |
Even a strong correlation can leave a large part of the outcome unexplained. That is why research often needs multiple variables, not one single predictor.
r is not a percentage. r² is. When you say r = 0.70, you are not saying 70% explained; you are saying 49% explained. The jump from r to r² is where most people's intuition breaks down.
| r value | Common intuition | r² value | Actual explanatory power |
|---|---|---|---|
| 0.30 | Weak correlation | 0.09 | Only 9% of variance explained |
| 0.50 | Moderate correlation | 0.25 | Only 25% of variance explained |
| 0.70 | Strong correlation | 0.49 | Only 49% of variance explained |
| 0.90 | Very strong correlation | 0.81 | 81% of variance explained |
| 0.95 | Nearly perfect | 0.90 | 90% of variance explained |
It depends on your field. The current value is closest to the benchmark band for Education research, but field norms matter more than universal cutoffs.
X explains a substantial share of Y, but there is still meaningful residual variation.
| Research field | Typical r² range | Why |
|---|---|---|
| Physics and engineering | 0.90 to 0.99 | Systems are controlled and measurement noise is often low. |
| Medicine and physiological measures | 0.50 to 0.80 | Biological systems are patterned but still contain person-level variation. |
| Psychology and behavior | 0.10 to 0.40 | Human behavior usually has many interacting causes. |
| Social science and economics | 0.05 to 0.30 | Important causes are numerous and hard to control. |
| Financial markets | 0.01 to 0.15 | Markets are noisy, adaptive, and heavily affected by shocks. |
| Education research | 0.15 to 0.50 | Outcomes are moderately predictable but still shaped by many factors. |
| r² value | Cohen effect size | Corresponding r |
|---|---|---|
| 0.01 or higher | Small | 0.10 or higher |
| 0.09 or higher | Medium | 0.30 or higher |
| 0.25 or higher | Large | 0.50 or higher |
Use r² for a simple correlation or simple regression. In APA style, report decimals without a leading zero and usually round to two decimal places.
A simple linear regression was performed to examine the relationship between [X] and [Y]. The results indicated that [X] significantly predicted [Y], β = [β value], t([df]) = [t value], p = [p value]. The model explained a significant proportion of variance in [Y], r² = [value], F([df1], [df2]) = [F value], p = [p value].There was a [positive/negative] correlation between [X] and [Y], r([df]) = [r value], p = [p value], r² = [value], indicating that [X] accounted for [r² × 100]% of the variance in [Y].There are two common paths. If you already have a correlation coefficient, square it. If you are working from a regression model, compare the unexplained residual variation with the total variation in Y.
Example: r = 0.70 gives r² = 0.70 × 0.70 = 0.49.
is residual variation the model missed. is the total variation in Y.
For simple linear regression with one predictor, both formulas give the same result. Formula 2 generalizes to multiple regression, where squaring a single correlation no longer describes the full model.
A biased model can have high r². If the regression line systematically over-predicts at low values and under-predicts at high values, the model can still be wrong. Always check residual plots.
An r² of .06 in human behavior can be meaningful when the effect is reliable and scalable. Statistical significance and practical importance are different judgments.
r² = .81 between two variables does not prove one causes the other. It measures explanatory power inside your model, not a causal mechanism in the world.
Every variable you add to a regression model increases regular R² or leaves it the same, even if the variable is noise. Adjusted R² penalizes unhelpful complexity.
Here is the number of predictors and is the sample size.
| Situation | Use which value? |
|---|---|
| Simple linear regression with one predictor | r² and adjusted R² are usually close; use r² for plain explanation |
| Multiple regression with two or more predictors | Use adjusted R² so extra variables are penalized |
| Comparing models with different predictor counts | Use adjusted R² because regular R² always rises or stays flat |
R-squared tells you what percentage of the variation in Y can be explained by X. For example, r² = 0.64 means 64% of the differences in Y are accounted for by X, while 36% remains unexplained.
No. r is the correlation coefficient and ranges from -1 to +1. r² is its square and ranges from 0 to 1. r tells you direction and strength; r² tells you the proportion of variance explained.
It depends on the field. In physics, r² below 0.90 may be weak. In psychology, r² around 0.15 can be meaningful because behavior is influenced by many variables.
When calculated as r squared from a correlation coefficient, no. In regression, R² = 1 - SSres/SStot can be negative if a model fits worse than a mean-only model.
Regular r² always increases or stays the same when you add predictors. Adjusted R² penalizes model complexity, so it only increases when a new predictor improves the model enough.
r² is one part of the interpretation. Pair it with the original r value, the p-value, and the assumptions behind the correlation method.