Calculate linear regression equation, slope, y-intercept, and R-squared. Fit a line to your data points.
Calculate linear regression equation (y = mx + b) from data.
The Linear Regression Calculator is a fundamental statistical tool that finds the line of best fit through your data points using the least squares method. Whether you're analyzing sales trends, predicting scientific outcomes, or exploring relationships between variables, this calculator computes the regression equation (y = mx + b), slope, y-intercept, and R-squared coefficient to quantify how well your data fits a linear model.
Linear regression is one of the most widely used statistical techniques in data science, economics, and research. It answers the question: "What is the mathematical relationship between two variables?" By fitting a straight line through scattered data points, you can make predictions, identify trends, and understand how changes in one variable affect another. This calculator uses the ordinary least squares (OLS) method—the industry standard approach that minimizes the sum of squared residuals.
Understanding linear regression empowers you to move beyond simple correlation and into predictive analytics. While correlation tells you that two variables are related, regression gives you the equation to predict one from the other. This makes it invaluable for forecasting, trend analysis, and causal inference across virtually every quantitative field.
y = Predicted value (dependent variable)
m = Slope (change in y per unit change in x)
x = Input value (independent variable)
b = Y-intercept (value of y when x = 0)
x̄, ȳ = Mean values of x and y datasets
R² measures the proportion of variance in Y explained by your regression model:
| R² Value | Interpretation | Fit Quality | Typical Fields |
|---|---|---|---|
| 0.90 – 1.00 | 90-100% of variance explained | Excellent | Physics, engineering, controlled experiments |
| 0.70 – 0.89 | 70-89% of variance explained | Good | Biology, chemistry, business analytics |
| 0.50 – 0.69 | 50-69% of variance explained | Moderate | Economics, social sciences, marketing |
| 0.30 – 0.49 | 30-49% of variance explained | Weak | Psychology, behavioral research |
| 0.00 – 0.29 | 0-29% of variance explained | Poor | Consider non-linear models or additional variables |
For valid regression results, your data should meet these key assumptions:
❌ Extrapolating beyond your data range: If your X values range from 10-50, don't predict for X=100. Relationships may not hold outside observed ranges. Stick to interpolation within your data bounds.
❌ Using linear regression on non-linear data: If a scatter plot shows a curve, a straight line won't fit well. Consider polynomial regression, logarithmic transformation, or other non-linear models instead.
❌ Ignoring outliers: A single extreme point can dramatically shift your regression line. Always visualize data first and investigate unusual values before running regression.
❌ Confusing correlation with causation: A strong regression relationship doesn't prove X causes Y. Ice cream sales and drowning deaths are correlated (both increase in summer) but ice cream doesn't cause drowning.
❌ Using too few data points: With only 2-3 points, you can fit a line but R² will be unreliable. Aim for at least 10-20 observations for meaningful regression analysis.
| Field | Example Application | X Variable | Y Variable |
|---|---|---|---|
| Business | Sales forecasting | Advertising spend | Revenue |
| Economics | Demand modeling | Price | Quantity demanded |
| Science | Physical laws verification | Force applied | Acceleration |
| Medicine | Dosage response | Drug dosage | Patient response |
| Real Estate | Property valuation | Square footage | Sale price |
| Education | Performance prediction | Study hours | Test scores |
Sources & Methodology: This calculator implements the Ordinary Least Squares (OLS) method, the standard approach for linear regression as described in statistical references including NIST/SEMATECH e-Handbook of Statistical Methods and academic statistics textbooks. R-squared calculation uses the standard formula R² = 1 - (SSres/SStot). For advanced regression analysis including multiple regression, residual analysis, and hypothesis testing, consult statistical software such as R, Python (statsmodels), or SPSS. Calculator updated January 2026.
Linear regression is a statistical method that finds the best-fitting straight line through a set of data points by minimizing the sum of squared vertical distances (residuals) between each point and the line. The result is an equation y = mx + b, where m is the slope (rate of change) and b is the y-intercept (value when x=0). The least squares method calculates m = Σ(xi-x̄)(yi-ȳ)/Σ(xi-x̄)² and b = ȳ - m×x̄, finding the unique line that minimizes prediction errors.
R-squared (R² or coefficient of determination) measures how well your regression line fits the data, ranging from 0 to 1. It represents the percentage of variance in Y explained by X. An R² of 0.85 means 85% of the variation in your dependent variable is explained by the model. Generally: R² > 0.9 is excellent, 0.7-0.9 is good, 0.5-0.7 is moderate, and < 0.5 is weak. However, interpretation depends on your field—social sciences often accept lower R² than physical sciences.
Simple linear regression uses ONE independent variable (X) to predict the dependent variable (Y), producing the equation y = mx + b. Multiple linear regression uses TWO OR MORE independent variables, producing y = b₀ + b₁x₁ + b₂x₂ + ... + bₙxₙ. For example, predicting house price with just square footage is simple regression; predicting price with square footage, bedrooms, and location is multiple regression. This calculator performs simple linear regression—for multiple variables, specialized statistical software is needed.