STEM Calculators

Simple Linear Regression Analysis

Statistics • Simple Linear Regression

Written by STEM Calculators Team Published Updated
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Simple Linear Regression Model

Ready
Accepted separators: comma, tab, semicolon, or spaces. Empty lines are ignored.
The file should contain two columns: x and y (a header row is ok).
Use “Auto-detect” unless your file is unusual.
If provided, the calculator will compute \( \hat{y}(x_0) \).
Vertical segments visualize \(e_i = y_i - \hat{y}_i\).
Drag the two orange handles to try “many possible lines.” The candidate SSE updates live.
Used for the residual distribution graph.
Candidate SSE: —
Regression assumptions checklist
\(E(\varepsilon)=0\) for each \(x\) Errors independent Errors normal (for each \(x\)) Constant variance
Scatter plot and regression line
Click a point to inspect its residual.
Residual distribution (histogram + normal overlay)
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Calculation steps (least squares)

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Frequently Asked Questions

What does simple linear regression analysis tell you?

It describes and models the linear relationship between one explanatory variable x and one response variable y. The fitted line provides an estimated slope and intercept that summarize how y changes as x changes.

What does the slope b1 mean in a regression line?

The slope b1 is the estimated change in the predicted response y for a one-unit increase in x. Its sign indicates whether the relationship is positive or negative.

What is R^2 in simple linear regression?

R^2 is the proportion of the total variation in y explained by the fitted linear model. Values closer to 1 indicate a stronger linear fit, while values closer to 0 indicate a weaker linear fit.

How do you test whether there is a significant linear relationship?

A common test is H0: beta1 = 0 versus an alternative that beta1 is not 0 (or is greater or less than 0). The calculator reports a test statistic and p-value to decide whether the slope differs from 0 at the chosen alpha.

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

A confidence interval estimates the mean response at a given x0, while a prediction interval estimates a single future observation at x0. Prediction intervals are wider because they include additional individual-to-individual variability.

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