This tool helps you estimate the adjusted R-square (sometimes referred to as population R²) for your regression model. By accounting for the number of predictors and sample size, adjusted R-square provides a more accurate measure of model fit that doesn’t inflate with the addition of extra predictors. For critical analyses, verify results with professional statistical software or consult a statistician.

Adjusted R-Squared Calculator

Compute the adjusted (population) $R^2$ given the original $R^2$, sample size $n$, and number of predictors $p$.

$$ R_{\text{adj}}^2 = 1 \;-\; \frac{(1 – R^2) \cdot (n – 1)}{n – p – 1}\,. $$

Step 1: Enter Known Values

Original $R^2$:

A number between 0 and 1, e.g. 0.85.

Sample Size ($n$):

E.g., 100 data points.

Number of Predictors ($p$) (excluding intercept):

E.g., 3 independent variables.

Adjusted R-Squared Calculator (In-Depth Explanation)

Adjusted R-Squared Calculator

In multiple linear regression, we often use the coefficient of determination $(R^2)$ to measure how well the model explains the variability of the response variable. However, Adjusted R-Squared refines this measure by penalizing unnecessary complexity— preventing overestimation when you add more predictors. An Adjusted R-Squared Calculator automates this computation, ensuring you balance model fit against model complexity.

What Is R-Squared?

The R-Squared ( $R^2$ ) metric, sometimes called the coefficient of determination, is defined as:

\[ R^2 = 1 – \frac{\text{SS}_\text{res}}{\text{SS}_\text{tot}}, \] where:

$\text{SS}_\text{res}$: The sum of squared residuals (sum of squared errors)
$\text{SS}_\text{tot}$: The total sum of squares (proportional to the variance of the response variable)

$R^2$ ranges from 0 to 1. A higher $R^2$ indicates that more variation in the data is explained by the model. However, simply adding more predictors—relevant or not—can inflate $R^2$.

Why Adjust R-Squared?

Adjusted R-Squared compensates for the number of predictors ($p$) used, relative to the number of data points ($n$). This penalization helps:

Prevent overfitting and identify more parsimonious models.
Compare different regression models with varying numbers of predictors on a fair basis.

It is particularly useful when iterating through potential predictor sets or building stepwise regression models.

The Adjusted R-Squared Formula

The Adjusted R-Squared is calculated as:

\[ R_{\text{adj}}^2 = 1 – \left(1 – R^2\right)\times \frac{n-1}{n – p – 1}, \]

$R^2$: the regular R-squared
$n$: the number of observations (data points)
$p$: the number of predictors (not counting the intercept)

Notice that when $p$ increases, the fraction $\frac{n-1}{n – p – 1}$ becomes larger, inflating the penalty. Thus, Adjusted $R^2$ can decrease if the new predictor doesn’t improve the model sufficiently relative to its cost in complexity.

How the Calculator Works

A typical Adjusted R-Squared Calculator steps through:

Inputs: asks for the sample size ($n$), the total number of predictors used ($p$), and the model’s $R^2$ value.
Applies the Formula: it plugs these inputs into the adjusted formula: \[ R_{\text{adj}}^2 = 1 – (1 – R^2)\frac{n-1}{n-p-1}. \]
Outputs the Adjusted $R^2$: presenting it as a decimal or percentage.

Practical Examples

Example 1: Small Linear Model

Scenario: You have 20 data points $(n=20)$. Your regression model uses 2 predictors $(p=2)$, and you computed $R^2=0.80$.

Calculation:

\[ R_{\text{adj}}^2 = 1 – (1 – 0.80)\frac{20-1}{20-2-1} = 1 – (0.20)\frac{19}{17} \approx 1 – 0.20 \times 1.1176 \approx 1 – 0.2235 = 0.7765 \]

So the adjusted $R^2$ is roughly 0.7765—slightly less than the raw 0.80.

Example 2: Adding a Predictor

Scenario: Suppose the same dataset with $n=20$ but now you add another predictor (so $p=3$) and find $R^2=0.82$. Is this truly an improvement?

Calculation:

\[ R_{\text{adj}}^2 = 1 – (1 – 0.82)\frac{20-1}{20-3-1} = 1 – 0.18 \times \frac{19}{16} = 1 – 0.18 \times 1.1875 = 1 – 0.21375 = 0.78625 \]

The new adjusted $R^2$ is about 0.7863, which is higher than the 0.7765 from before. This suggests the added predictor meaningfully improves the model, not just artificially.

Key Takeaways

Adjusted vs. Raw $R^2$: Adjusted $R^2$ accounts for added parameters, making it a better yardstick for comparing models with different numbers of predictors.
Overfitting Control: If a new predictor doesn’t genuinely improve the model, adjusted $R^2$ may decrease.
Sample Size Dependence: The penalty factor relies on $(n-p-1)$, so small datasets with many predictors may see larger penalties.

Conclusion

An Adjusted R-Squared Calculator is an indispensable tool for anyone doing multiple regression modeling. It takes the raw $R^2$ value, the sample size, and the number of predictors, producing a more honest measure of model fit. By using adjusted $R^2$, analysts can better avoid inflated fit indicators, ensuring that each predictor justifies its inclusion in the model.