# Linear Regression Equation Calculator

Free linear regression calculator. Enter paired X-Y data to compute the regression equation, slope, intercept, R-squared, and predict Y for any X value.

## What this calculates

Compute the best-fit linear regression equation from paired X and Y data using the least squares method. This calculator gives you the slope, intercept, R-squared value, and lets you predict Y for any given X.

## Inputs

- **Number of Data Points** — min 3, max 10 — Number of paired observations (3-10).
- **Predict Y for X =** — Enter an X value to predict the corresponding Y.
- **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

- **Slope (b1)** — Change in Y for each unit increase in X.
- **Intercept (b0)** — Predicted Y when X is zero.
- **Equation** — formatted as text — The linear regression equation.
- **R-Squared** — Proportion of variance in Y explained by X.
- **Predicted Y** — Predicted Y value for the given X.

## Details

Linear regression fits a straight line y = b0 + b1*x through a set of data points by minimizing the sum of squared residuals (the vertical distances between the observed and predicted Y values). The slope b1 tells you how much Y changes for each unit increase in X, and the intercept b0 is the predicted value of Y when X is zero.

The least squares formulas are: b1 = (nSigma(xy) - Sigma(x)Sigma(y)) / (nSigma(x^2) - (Sigma(x))^2) and b0 = y-bar - b1x-bar. These formulas minimize the total squared error, which is the standard optimality criterion for linear models.

The R-squared value tells you how well the line fits the data. An R-squared of 1.0 means the line passes through every point perfectly. In practice, R-squared above 0.7 indicates a good fit for many applications. Use the equation to predict Y for new X values, but be cautious about extrapolating far beyond the range of your data.

## Frequently Asked Questions

**Q: What does the slope mean in a regression equation?**

A: The slope (b1) represents the average change in Y for each one-unit increase in X. A positive slope means Y increases as X increases; a negative slope means Y decreases as X increases.

**Q: Can I use linear regression for prediction?**

A: Yes, within the range of your data (interpolation). Predicting beyond the observed range (extrapolation) is risky because the linear relationship may not hold outside the data. Always check R-squared to ensure the model fits well before relying on predictions.

**Q: What if the relationship is not linear?**

A: If a scatter plot of your data shows a curve rather than a straight line, linear regression will give poor results. Consider polynomial regression, logarithmic transformations, or other non-linear models. A low R-squared is often a sign that the relationship is not linear.

**Q: How is this different from the correlation calculator?**

A: The correlation calculator measures the strength and direction of the linear relationship (r). The regression calculator provides the actual equation (y = b0 + b1*x) that lets you make predictions. Regression gives you the line; correlation tells you how well the data fits it.

---

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