Linear Regression Calculator

Least squares · R² · residual plot · prediction

Separate X and Y with comma or space. Minimum 2 data points.
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Result

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Enter data and click Calculate

Compute the regression equation, R², correlation, and make predictions.

Scatter Plot & Regression Line

Python Compiler

What Is Linear Regression?

Linear regression models the relationship between a dependent variable (Y) and independent variable (X) by fitting the best straight line through the data using the least squares method.

Regression Equation:  y = a + bx
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Slope (b)

How much Y changes for each 1-unit increase in X. Positive = upward trend, negative = downward.

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Intercept (a)

The predicted value of Y when X = 0. The starting point of the regression line.

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R² (Fit Quality)

Proportion of variance in Y explained by X. Ranges from 0 (no fit) to 1 (perfect fit).

Key Formulas

Slope:  b = Σ[(xi − x̄)(yi − ȳ)] / Σ(xi − x̄)²
Intercept:  a = ȳ − b × x̄
R²:  R² = 1 − SSres / SStot
Standard Error:  SEE = √[Σ(yi − ŷi)² / (n − 2)]

Understanding R²

R² RangeInterpretationFit Quality
0.90 – 1.0090–100% of variance explainedExcellent
0.70 – 0.8970–89% of variance explainedGood
0.50 – 0.6950–69% of variance explainedModerate
0.00 – 0.49Less than 50% explainedWeak

Tip: A high R² does not guarantee a good model. Always visualize residuals to check for patterns that indicate model violations (non-linearity, heteroscedasticity).

Assumptions of Linear Regression

Linearity

The relationship between X and Y is approximately linear. Check with a scatter plot.

Independence

Observations are independent of each other. No autocorrelation in residuals.

Homoscedasticity

Residuals have constant variance across all X values. Fan-shaped patterns indicate violation.

Normality

Residuals are approximately normally distributed. Less critical for large samples.

Frequently Asked Questions

R² is the proportion of variance in Y explained by the linear model. An R² of 0.85 means 85% of the variation in Y is captured by the regression line. Higher is better, but always check residual plots.
Linear regression assumes: linearity (X and Y have a linear relationship), independence of observations, homoscedasticity (constant residual variance), and approximately normal residuals. Inspect residual plots to verify.
Plug any X value into y = a + bx. Be cautious about extrapolating far beyond your data range, as the linear relationship may not hold outside the observed domain.
If residuals show a curve or pattern, consider polynomial regression, logarithmic transformation, or other nonlinear models. Always visualize your data with a scatter plot first.
The slope (b) tells you how much Y changes for each 1-unit increase in X. A slope of 2.5 means Y increases by 2.5 units on average when X increases by 1 unit.
SEE measures the average distance between observed Y values and predicted values from the regression line. Smaller SEE means the model makes more precise predictions.

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