Simple Linear Regression Calculator

When to use

• Use simple linear regression to estimate a straight-line relationship between one independent variable x and one dependent variable y.
• Use it when a scatterplot and residual pattern support an approximately linear relationship over the observed x range.
• Use multiple, nonlinear, weighted, time-series, or mixed models when the design requires more than one straight-line predictor.

How to use

1. Enter matched x,y pairs with the independent variable first and dependent variable second, or select two aligned CSV columns.
2. Calculate the least-squares slope and intercept only after confirming variation in x.
3. Inspect the scatterplot, fitted line, residuals, residual standard error, r, R squared, slope interval, and p-value.
4. Check influential outliers and avoid predictions outside the observed x range.

Inputs

Independent variable x

The predictor or explanatory value; it must vary across observations.

Dependent variable y

The response paired row-by-row with x.

Paired rows

Each x,y row must refer to the same observational unit; missing-pair handling must preserve alignment.

Alpha

The prespecified threshold for the slope test and corresponding confidence interval.

Notation

Interpretation

• The slope is the estimated change in y per one-unit increase in x; the intercept is fitted y at x=0 and may lack practical meaning when zero is outside the data range.
• R squared is the sample share of y variation represented by the fitted line, while r gives direction and linear-association strength.
• Residuals reveal departures from linearity, changing variance, and influential observations; neither correlation nor regression alone proves causation.

Effect-size guidance

• Report slope with units and a confidence interval, not only its p-value.
• Use residual standard error to describe typical vertical deviation around the line.
• Limit prediction to the observed x range unless a defensible external model supports extrapolation.

Worked examples

Assumptions

• Inputs are finite numeric values after parsing, with missing or invalid values handled according to the selected workflow.
• The relationship is reasonably linear over the observed x range.
• Residual behavior is suitable for least-squares inference.

Practical guidance

• Inspect the scatterplot before fitting and the residual pattern afterward.
• Check source records for high-leverage or influential points rather than deleting them automatically.
• Use subject knowledge and study design to distinguish prediction from causal explanation.

Common mistakes

• Extrapolating beyond observed x values.
• Treating regression as causal proof.

Limitations

• Multiple regression and logistic regression are out of scope.
• Outliers can strongly affect least squares.

Related workflow links