Regression equation
Calculate the least-squares line in the form y = intercept + slope x, similar to Excel SLOPE and INTERCEPT output.
Free statistics calculator
Linear Regression Calculator: Slope, Intercept, R-Squared, and Prediction
Paste paired x-y data and calculate the least-squares regression line, slope, intercept, correlation, R-squared, residual error, slope p-value, confidence interval, prediction, and residuals. Use it to check Excel LINEST, SLOPE, INTERCEPT, RSQ, SPSS, R, Stata, or statistics homework output.
- Paired data Enter x and y values line by line or as two columns.
- Regression equation Get slope, intercept, and the fitted line.
- Model fit Review r, R-squared, SSE, and residual standard error.
- Prediction Enter an x value to estimate the corresponding y value.
Calculate a Linear Regression Model
Enter paired values with x first and y second. The calculator fits a simple linear regression model using ordinary least squares. Each row must keep the x value matched with the correct y value from the same observation.
Put one pair on each line, such as 1, 2.1 or 1 2.1.
The first number is x and the second number is y.
Results
Simple linear regression
Regression Equation
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Count
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Slope
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Intercept
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Correlation r
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R-Squared
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SSE
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Residual Std. Error
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Slope SE
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Slope P-Value
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95% CI for Slope
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Prediction
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Slope inference uses the usual simple linear regression assumptions. Always check the scatterplot, residuals, outliers, and assignment requirements before writing a conclusion.
Scatterplot and Fitted Line
| X | Y | Predicted Y | Residual |
|---|---|---|---|
| Calculate a model to view residuals. | |||
What This Calculator Includes
Simple linear regression estimates a straight-line relationship between one predictor variable and one response variable.
Model fit
Review correlation, R-squared, residual sum of squares, residual standard error, and fitted values.
Slope inference
Get the standard error, t statistic, p-value, and 95% confidence interval for the slope.
Prediction
Enter an x value and calculate the predicted y from the fitted regression line.
Linear Regression Formulas
Ordinary least squares chooses the line that minimizes the sum of squared residuals between observed y values and predicted y values.
| Measure | Formula idea | Meaning |
|---|---|---|
| Slope | Sxy / Sxx |
Expected change in y for a one-unit increase in x. |
| Intercept | y-bar - slope * x-bar |
Predicted y value when x is zero. |
| R-squared | 1 - SSE / SST |
Proportion of y variation explained by the linear model. |
| Residual | observed y - predicted y |
How far each point is from the fitted line. |
How to Calculate Linear Regression Step by Step
A strong regression answer includes the equation, model fit, residual checks, and a written interpretation of the slope and prediction.
- Enter paired x-y values and keep each observation matched.
- Check the scatterplot to see whether a straight-line model is reasonable.
- Calculate the least-squares slope and intercept.
- Use the equation y-hat = intercept + slope * x to calculate fitted values and predictions.
- Review residuals, SSE, residual standard error, correlation, and R-squared.
- Use the slope p-value and confidence interval when your assignment asks for inference.
- Write the slope interpretation in the units and context of the original problem.
Linear Regression Examples
These examples match the default paired data in the calculator and are useful for checking Excel, SPSS, R, Stata, or hand-calculation output.
| Output | Result | Meaning |
|---|---|---|
| Regression equation | y-hat = 1.1964 + 0.8286x | For each 1-unit increase in x, predicted y increases by about 0.8286. |
| Model fit | R-squared = 0.9938 | About 99.38% of y variation is explained by the linear model. |
| Slope inference | p approximately 7.57e-8; 95% CI = [0.7630, 0.8941] | The slope is positive and clearly different from zero for this example data. |
| Prediction | At x = 9, y-hat = 8.6536 | The fitted line predicts about 8.6536 when x equals 9. |
Common Mistakes to Avoid
- Treating a strong R-squared as proof that x causes y.
- Using linear regression when the scatterplot clearly curves or has separate groups.
- Ignoring outliers that strongly influence the slope and intercept.
- Making predictions far outside the observed x range without warning about extrapolation.
Linear Regression Calculator FAQs
What does the slope mean in linear regression?
The slope estimates the expected change in y for a one-unit increase in x, assuming a linear relationship is appropriate.
What does R-squared mean?
R-squared is the proportion of variation in y explained by the linear model with x. It does not prove causation.
Can this replace Excel, SPSS, R, or Stata output?
It can help you check calculations, but assignments may require software output, assumption checks, residual plots, and written interpretation.
Why do I need at least three data points for the slope p-value?
The slope test uses n - 2 degrees of freedom, so at least three paired observations are needed for residual variation.
Is this like Excel LINEST, SLOPE, INTERCEPT, and RSQ?
Yes. The calculator uses ordinary least squares, the same core method behind Excel-style SLOPE, INTERCEPT, RSQ, and simple LINEST output. Small differences can appear from rounding.
What is the difference between correlation and regression?
Correlation measures strength and direction of association. Regression fits an equation that predicts y from x and estimates how much y changes when x increases by one unit.
What does the slope p-value test?
The slope p-value tests whether the population slope could be zero under the usual simple linear regression assumptions.
Need Help Interpreting Regression Output?
Regression assignments often require more than slope and R-squared. You may need residual checks, p-values, confidence intervals, software output, ANOVA tables, and a written conclusion in context.
Related Statistics Help
- Statistics Calculators
- Correlation Coefficient Calculator
- P-Value Calculator
- Confidence Interval Calculator
- Regression and ANOVA Homework Help
- How to Do Linear Regression in Excel
- How to Interpret Regression Output in Excel
- Excel Statistics Homework Help
- R Statistics Homework Help
- SPSS Homework Help
- Statistics Homework Help
Use Calculators Responsibly
Use this calculator to check computations, understand regression output, and prepare tutoring questions. For graded work, follow your course rules and show the required method.