Free statistics calculator

Linear Regression Calculator for Slope, Intercept, R-Squared, and Prediction

Paste paired x-y data and calculate the least-squares regression line, slope, intercept, correlation, R-squared, regression ANOVA table, coefficient table, mean-response confidence interval, prediction interval, and residuals. Use it to check Excel LINEST, PHStat, SPSS, R, Stata, or statistics homework output.

Use Calculator Get Regression Help

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 Get y-hat plus confidence and prediction intervals.

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.

X-Y data pairs

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.

X value for interval and prediction

Confidence level

Results

Simple linear regression

Regression Equation -

Count -

Slope -

Intercept -

Correlation r -

R-Squared -

SSE -

Residual Std. Error -

Adjusted R-Squared -

Overall F -

Slope SE -

Slope P-Value -

95% CI for Slope -

Prediction -

Confidence Interval for Mean Response -

Prediction Interval for Individual Response -

Slope inference uses the usual simple linear regression assumptions. Always check the scatterplot, residuals, outliers, and assignment requirements before writing a conclusion.

Regression ANOVA Table

Source	SS	df	MS	F	Significance F
Calculate a model to view regression ANOVA output.

Coefficient Table

Term	Coefficient	Standard Error	t Stat	P-value	Lower	Upper
Calculate a model to view coefficient output.

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.

Regression equation

Calculate the least-squares line in the form y = intercept + slope x, similar to Excel SLOPE and INTERCEPT output.

Model fit

Review correlation, R-squared, residual sum of squares, residual standard error, and fitted values.

Slope inference

Get PHStat-style coefficient output with standard errors, t statistics, p-values, and confidence intervals.

Prediction intervals

Enter an x value and calculate y-hat, mean-response confidence interval, and individual prediction interval.

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.
Regression ANOVA	`MSR / MSE`	Tests whether the simple regression model explains a significant amount of y variation.
Mean-response confidence interval	`y-hat +/- t * SE(mean)`	Interval estimate for the average y value at a chosen x value.
Prediction interval	`y-hat +/- t * SE(prediction)`	Interval estimate for one new individual y value at a chosen x value.

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, R-squared, adjusted R-squared, and the ANOVA table.
Use coefficient p-values and confidence intervals when your assignment asks for inference.
Use the mean-response confidence interval for an average y and the prediction interval for one new y.
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.
Regression ANOVA	F = 957.3439; significance F approximately 7.57e-8	The simple linear model explains a statistically significant amount of variation in y.
Prediction	At x = 9, y-hat = 8.6536; mean CI = [8.3227, 8.9845]; prediction interval = [8.1152, 9.1919]	The fitted line predicts about 8.6536 when x equals 9, with separate intervals for mean and individual response.

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.

What is the difference between a confidence interval and a prediction interval in regression?

A confidence interval estimates the mean response at a chosen x value. A prediction interval estimates where one new individual y value may fall, so it is usually wider.

Is this calculator for multiple regression?

This page is for simple linear regression with one x variable and one y variable. Multiple regression needs separate columns for several predictors and should be handled with a separate multiple regression calculator.

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.

Submit Your Regression Question Ask an Online Statistics Tutor

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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.

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