# R-Squared (Coefficient of Determination) Calculator

Free R-squared calculator. Enter paired data to compute R², adjusted R², and correlation r to measure how well a linear regression fits your data.

## What this calculates

Calculate R-squared from paired X and Y data to determine what proportion of the variance in your dependent variable is explained by the linear regression model. This tool also provides the adjusted R² and the Pearson correlation coefficient.

## Inputs

- **Number of Data Points** — min 3, max 10 — Number of paired observations (3-10).
- **X1** — X value for observation 1.
- **X2** — X value for observation 2.
- **X3** — X value for observation 3.
- **X4** — X value for observation 4.
- **X5** — X value for observation 5.
- **X6** — X value for observation 6.
- **X7** — X value for observation 7.
- **X8** — X value for observation 8.
- **X9** — X value for observation 9.
- **X10** — X value for observation 10.
- **Y1** — Y value for observation 1.
- **Y2** — Y value for observation 2.
- **Y3** — Y value for observation 3.
- **Y4** — Y value for observation 4.
- **Y5** — Y value for observation 5.
- **Y6** — Y value for observation 6.
- **Y7** — Y value for observation 7.
- **Y8** — Y value for observation 8.
- **Y9** — Y value for observation 9.
- **Y10** — Y value for observation 10.

## Outputs

- **R²** — Coefficient of determination (0 to 1).
- **Adjusted R²** — R² adjusted for the number of predictors.
- **Correlation (r)** — Pearson correlation coefficient.
- **Interpretation** — formatted as text — Strength of the model fit.

## Details

The coefficient of determination (R²) is one of the most widely reported statistics in regression analysis. It ranges from 0 to 1, where 0 means the model explains none of the variability in the response variable and 1 means it explains all of it. For example, an R² of 0.85 means 85% of the variance in Y is accounted for by the linear relationship with X.

R² is calculated as R² = 1 - (SS_res / SS_tot), where SS_res is the sum of squared residuals (the differences between observed and predicted values) and SS_tot is the total sum of squares (the differences between observed values and the mean). This approach works directly from the data without needing to compute the correlation coefficient first.

The adjusted R² corrects for the number of predictors in the model. In simple linear regression with one predictor, the adjustment is minor, but in multiple regression it prevents the false inflation of R² that occurs when adding irrelevant variables. A decreasing adjusted R² when a new variable is added is a signal that the variable does not improve the model.

## Frequently Asked Questions

**Q: What does R-squared tell you?**

A: R² tells you the proportion of variance in the dependent variable (Y) that is predictable from the independent variable (X). An R² of 0.75 means 75% of the variability in Y is explained by the linear relationship with X.

**Q: Can R-squared be negative?**

A: In standard linear regression with an intercept, R² is always between 0 and 1. However, adjusted R² can be negative if the model is a very poor fit, indicating the model is worse than simply using the mean of Y as a prediction.

**Q: What is the difference between R² and adjusted R²?**

A: R² always increases (or stays the same) when you add more predictors, even useless ones. Adjusted R² penalizes for adding predictors that do not improve the model, giving a more honest assessment of fit quality in multiple regression.

**Q: What is a good R-squared value?**

A: It depends on the field. In physics or engineering, R² above 0.95 is common. In social sciences or biology, R² of 0.5-0.7 may be considered strong. There is no universal threshold; context matters.

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Source: https://vastcalc.com/calculators/statistics/r-squared
Category: Statistics
Last updated: 2026-04-21