Coefficient of Determination Calculator

Calculate R-squared (R²) to measure how well a regression model fits the data

About the Coefficient of Determination Calculator

The Coefficient of Determination, commonly known as R-squared (R²), is a fundamental statistical metric used to evaluate the goodness-of-fit of a regression model. It quantifies the proportion of the variance in the dependent variable that is predictable from the independent variable(s). By comparing the error of the model against the error of a simple mean-based baseline, researchers and data analysts can determine how much 'explanatory power' their statistical model actually possesses.

This tool is widely utilized by economists, data scientists, and engineers to validate predictive models. Whether you are analyzing stock market trends, interpreting laboratory results, or forecasting sales growth, R-squared provides a standardized score to help you decide if your model is reliable or if it requires further refinement. It serves as a bridge between raw data analysis and actionable insights, offering a clear percentage-based interpretation of a model's performance.

Formula

R² = 1 - (SSR / SST)

In this formula, SSR stands for the Sum of Squared Residuals (also known as the Error Sum of Squares), which measures the discrepancy between the actual data points and the values predicted by the model. SST stands for the Total Sum of Squares, which measures the total variance in the dependent variable by calculating the squared differences between each observation and the mean of the data set.

The resulting R-squared value typically ranges from 0 to 1. An R-squared of 1 indicates that the model explains all the variability of the response data around its mean, while a value of 0 indicates the model explains none of the variability.

Worked examples

Example 1: A researcher is testing a new linear model where the Total Sum of Squares (SST) is calculated as 500 and the Sum of Squared Residuals (SSR) is 75.

1. Identify the Total Sum of Squares: SST = 500
2. Identify the Sum of Squared Residuals: SSR = 75
3. Divide SSR by SST: 75 / 500 = 0.15
4. Subtract the result from 1: 1 - 0.15 = 0.85

Result: R² = 0.85. This means that 85% of the variance in the dependent variable is explained by the model.

Example 2: A marketing analyst finds that a campaign model has an SST of 1200 and an SSR of 720.

1. SST = 1200
2. SSR = 720
3. Divide SSR by SST: 720 / 1200 = 0.60
4. Subtract from 1: 1 - 0.60 = 0.40

Result: R² = 0.40. Only 40% of the variance is captured, suggesting a relatively weak fit for this specific model.

Common use cases

An e-commerce analyst uses R-squared to see how much of the variation in monthly sales can be explained by changes in advertising spend.
A medical researcher calculates R² to determine the strength of the relationship between dosage levels and patient recovery times.
Real estate appraisers use the coefficient to evaluate how accurately a pricing model predicts home values based on square footage.
Engineers apply R-squared to test the reliability of a sensor calibration curve against known physical standards.

Pitfalls and limitations

R-squared does not indicate whether the independent variables are a cause of the changes in the dependent variable.
Adding more independent variables to a model will never decrease R-squared, even if the variables are completely irrelevant.
Low R-squared values are not always bad; they may simply indicate that the process is inherently noisy but still statistically significant.
R-squared is sensitive to outliers, which can disproportionately inflate or deflate the resulting fit.

Frequently asked questions

difference between r and r squared in statistics

While both measure the strength of a relationship, the correlation coefficient (r) indicates direction and linear strength, while R-squared (r^2) quantifies the proportion of variance explained. R-squared is essentially the square of the correlation coefficient in a simple linear regression.

what is a good r-squared value for regression

A 'good' R-squared value is entirely dependent on the field of study; in social sciences, a value of 0.50 might be considered high, whereas in physical laboratory experiments, researchers often expect values above 0.90.

can r squared be negative in regression analysis

A negative R-squared can occur when using a non-linear model or when a linear model is forced through a specific intercept that does not fit the data trend at all. It indicates that the chosen model fits the data worse than a simple horizontal line representing the mean of the dependent variable.

when to use adjusted r-squared vs r-squared

Adjusted R-squared is used to account for the number of predictors in a model, preventing the score from increasing simply by adding irrelevant variables. Standard R-squared will always stay the same or increase as more variables are added, even if they add no real value.

is a high r squared always good

A high R-squared does not guarantee that the model is correct because it does not account for omitted variable bias or non-linear patterns. You must always check a residual plot to ensure the model assumptions are met despite a high coefficient.