Free Coefficient of Determination Calculator (R-squared)

What Is R² (Coefficient of Determination)?

The coefficient of determination, commonly denoted as R² (R-squared), is a key metric in linear regression. It quantifies the proportion of variance in the dependent variable that can be explained by the independent variable(s). In simpler terms, R² indicates how well the regression predictions approximate the real data points. Because it measures the goodness of fit of a model, a higher R² suggests a stronger linear relationship between the variables.

This R-squared calculator (also referred to as a coefficient of determination calculator or goodness of fit calculator) lets you quickly compute the R² value for any two-variable dataset using simple linear regression (Y ~ aX + b). It is a handy tool for students, analysts, and researchers who need to evaluate the strength of a linear relationship.

How to Use This Online R² Calculator

Using the R2 calculator is straightforward:

1. Enter your paired data into the designated rows. Additional rows will appear as you input more points.
2. Once you have at least three data points, the calculator automatically computes the R-squared value and displays it along with an interpretation.
3. You can choose to view a basic summary or the full details of the computation, including intermediate sums of squares.

No manual calculations are required — the tool handles everything from fitting the regression line to computing the coefficient of determination.

Interpreting the Coefficient of Determination

R² always falls between 0 and 1 (or between 0% and 100% when expressed as a percentage):

• R² = 1 indicates a perfect fit: all data points lie exactly on the regression line, and 100% of the variance in Y is explained by X.
• R² = 0 means the model performs no better than simply using the mean of Y as a predictor; the independent variable explains none of the variability.
• Values in between represent the fraction of the variance accounted for by the model. For example, if R² = 0.75, then 75% of the variance in Y is predictable from X, while the remaining 25% remains unexplained.

To get the explained variance percentage, multiply R² by 100%. The unexplained proportion is given by $(1 - R^{2}) \times 100\%$.

The R² Formula and Key Sums of Squares

For a dataset $(x_{1}, y_{1}), \dots, (x_{n}, y_{n})$, let $\bar{y}$ be the mean of the observed $y$ values, and $\hat{y}_{i}$ be the predicted values from the simple linear regression model $\hat{y} = aX + b$. The calculation of R² relies on three sums of squares:

• Total Sum of Squares (SST): measures the total variability in Y.

  $$SST = \sum_{i=1}^{n} (y_{i} - \bar{y})^{2}$$

• Regression Sum of Squares (SSR): measures how much of the variability is explained by the regression model.

  $$SSR = \sum_{i=1}^{n} (\hat{y}_{i} - \bar{y})^{2}$$

• Sum of Squared Errors (SSE): also called the residual sum of squares, measures the discrepancy between observed and predicted values.

  $$SSE = \sum_{i=1}^{n} (y_{i} - \hat{y}_{i})^{2}$$

These three sums satisfy: $SST = SSR + SSE$. Consequently, the coefficient of determination can be expressed in several equivalent forms:

$$R^{2} = \frac{SSR}{SST} = 1 - \frac{SSE}{SST} = \frac{SSR}{SSR + SSE}$$

Step-by-Step Example: Finding R² by Hand

Consider the data points: $(0, 1)$, $(2, 4)$, $(4, 4)$.

1. Calculate the mean of Y: $\bar{y} = (1+4+4) / 3 = 3$.
2. Fit the simple regression line using the least squares method; the resulting equation is $\hat{y} = 0.75x + 1.5$.
3. Compute the predicted values:
   • For $x=0$: $\hat{y}_{1} = 0.75(0) + 1.5 = 1.5$
   • For $x=2$: $\hat{y}_{2} = 0.75(2) + 1.5 = 3$
   • For $x=4$: $\hat{y}_{3} = 0.75(4) + 1.5 = 4.5$
4. Calculate SST: $$(1-3)^{2} = 4,\quad (4-3)^{2} = 1,\quad (4-3)^{2} = 1 \quad \Rightarrow \quad SST = 4+1+1 = 6$$
5. Calculate SSR: $$(1.5-3)^{2} = 2.25,\quad (3-3)^{2} = 0,\quad (4.5-3)^{2} = 2.25 \quad \Rightarrow \quad SSR = 2.25+0+2.25 = 4.5$$
6. Apply the R² formula: $$R^{2} = \frac{SSR}{SST} = \frac{4.5}{6} = 0.75$$ This means 75% of the variance in Y is explained by X.

Relationship Between R² and Pearson Correlation

In the context of simple linear regression (one independent variable), the coefficient of determination equals the square of the Pearson correlation coefficient $r$ between the two variables. Therefore, if you already know the correlation, you can find R squared online simply by squaring $r$.

• If $r = 0.9$, then $R^{2} = 0.81$, meaning that 81% of the variability in Y is accounted for by its linear relationship with X.
• This equivalence makes the linear regression calculator a dual-purpose tool: it provides both the correlation coefficient and the coefficient of determination.

Whether you are using a dedicated goodness of fit calculator or a full regression tool, understanding R² helps you assess how well your model fits the data and whether additional variables might improve the prediction.