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 $p$ .

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

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{{SS}_{res}}{{SS}_{tot}},

where:

${SS}_{res}$ : The sum of squared residuals (sum of squared errors)
${SS}_{tot}$ : 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_{adj}^{2} = 1 - (1 - R^{2}) \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: $R_{adj}^{2} = 1 - (1 - R^{2}) \frac{n - 1}{n - p - 1} .$
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_{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_{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.