Coefficient of Determination

Statistics • Simple Linear Regression

Written by STEM Calculators Team Published December 27, 2025 Updated February 24, 2026

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Coefficient of Determination (r²)

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Data pairs (x, y)

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The file should contain two columns: x and y (a header row is ok).

Delimiter

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Show total errors (to \(\bar{y}\))

Vertical segments visualize \(y_i - \bar{y}\) (total variation, SST).

Show residuals (to \(\hat{y}\))

Vertical segments visualize \(e_i = y_i - \hat{y}_i\) (unexplained variation, SSE).

What does r² mean?

The coefficient of determination \(r^2\) is the proportion of the total variation in \(y\) explained by the regression on \(x\): \[ r^2 = \frac{\text{SSR}}{\text{SST}} = 1 - \frac{\text{SSE}}{\text{SST}}, \quad 0 \le r^2 \le 1. \]

Total variation (SST): mean line \(\bar{y}\)

Click a point to inspect its contributions.

Unexplained variation (SSE): regression line \(\hat{y}=a+bx\)

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Decomposition: \(\text{SST}=\text{SSR}+\text{SSE}\)

r²: —

Calculation steps

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Coefficient of Determination (r²)

The coefficient of determination, denoted by r², measures the proportion of the total variation in the dependent variable y that is explained by a simple linear regression on x. It always satisfies:

\[ 0 \le r^2 \le 1 \]

In a simple linear regression model \( \hat{y} = a + bx \), the total variation in \(y\) can be split into: the part explained by the regression and the part left as random error (residuals).

Key sums of squares

Define the mean of the \(y\)-values as \( \bar{y} \), and the fitted (predicted) value at \(x_i\) as \( \hat{y}_i \). Then:

\[ \text{SST} = \sum (y_i - \bar{y})^2 \] \[ \text{SSE} = \sum (y_i - \hat{y}_i)^2 \] \[ \text{SSR} = \sum (\hat{y}_i - \bar{y})^2 \]

These satisfy the decomposition:

\[ \text{SST} = \text{SSR} + \text{SSE} \]

Definition of r²

The coefficient of determination is the fraction of total variation explained by the regression:

\[ r^2 = \frac{\text{SSR}}{\text{SST}} \]

An equivalent and commonly used form is:

\[ r^2 = 1 - \frac{\text{SSE}}{\text{SST}} \]

Interpretation:

If \(r^2\) is close to \(1\), the regression line explains most of the variability in \(y\).
If \(r^2\) is close to \(0\), the regression line explains little of the variability in \(y\).

How to use the calculator

Paste your data as two columns of numbers (x, y) into the input box. You can use comma, tab, semicolon, or spaces. A header row is allowed.
Alternatively, load a CSV file using the Load CSV input.
Click Calculate to compute the regression line, SST, SSE, SSR, and \(r^2\).
Use the visualization controls to show/hide:
- Total errors to the mean line (\(y_i - \bar{y}\)) for SST
- Residuals to the regression line (\(y_i - \hat{y}_i\)) for SSE
Use Copy summary or Download results CSV to export the computed values.

Note: If all \(y\)-values are identical, then \(\text{SST} = 0\) and \(r^2\) is not defined (there is no variation in \(y\) to explain).

Frequently Asked Questions

What is the coefficient of determination R^2?

R^2 is the proportion of the total variation in the response variable y that is explained by the regression model. In simple linear regression it summarizes how well the line fits the data.

How is R^2 computed from SST and SSE?

A common formula is R^2 = 1 - SSE/SST, where SSE is the sum of squared residuals and SST is the total sum of squares about the mean of y. The calculator computes these from your data and the fitted line.

Does a high R^2 mean the regression model is correct?

A high R^2 indicates the model explains a large fraction of the variability in y, but it does not guarantee causation or that the linear form is appropriate. Always check the scatter plot and residual behavior.

Can R^2 be low even if the slope is statistically significant?

Yes, a slope can be statistically different from 0 while R^2 remains small, especially with large sample sizes or when the relationship is weak but consistent. R^2 measures explained variation, not statistical significance.

When is R^2 not meaningful?

R^2 can be misleading when the model form is wrong, when outliers dominate the fit, or when the data are non-linear. It is also not comparable across different response variables or different study designs without context.

Coefficient of Determination

Coefficient of Determination (r²)

Key sums of squares

Fit summary

Computation table

Calculation steps

Rate this calculator

Frequently Asked Questions

Coefficient of Determination (r²)

Key sums of squares

Fit summary

Computation table

Calculation steps

Rate this calculator

Frequently Asked Questions

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