Simple Linear Regression Model

Statistics • Simple Linear Regression

Written by STEM Calculators Team Published December 27, 2025 Updated February 24, 2026

Goal: paste (x, y) data, fit the least-squares line, and visualize the scatter plot with the regression line (and optional residuals).

Use x for the explanatory (independent) variable and y for the response (dependent) variable.

Data pairs (x, y)

Accepted formats per line: x,y or x y (commas, tabs, spaces). Header rows are allowed.

Load CSV

Predict at x₀ (optional)

If provided, the calculator reports ŷ(x₀) and the mean-response standard error (when n ≥ 3).

Axis ticks (2–8)

Controls how many labeled tick marks appear on each axis.

Plot options

Show residuals Animate

Residuals are vertical distances from each point to the fitted line at the same x.

Plot padding (0–20%)

Adds breathing room around the extreme data values.

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Results

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Simple Linear Regression Model

This calculator fits a simple linear regression line that predicts a response variable y from a single explanatory variable x.

How to use the calculator

Paste your data into Data pairs as two columns per row (x then y). Accepted separators: commas, tabs, or spaces. A header row is allowed.
Example: x,y
1,2.2
2,2.8
Or click Load CSV to import a .csv file containing x and y values.
(Optional) Enter a value for Predict at x₀ to compute the fitted value \(\hat{y}(x_{0})\).
Click Calculate. The scatter plot with the fitted regression line appears first, followed by results and calculation steps.

What the results mean

Fitted line: \(\hat{y} = b_{0} + b_{1}\,x\). The slope \(b_{1}\) is the change in predicted y for a 1-unit increase in x; the intercept \(b_{0}\) is the predicted y when \(x = 0\).
Residuals: \(e = y - \hat{y}\). When “Show residuals” is enabled, vertical segments visualize these errors on the plot.
SSE and s: \(\text{SSE} = \sum e^{2}\) and \(s = \sqrt{\text{SSE}/(n-2)}\) (requires \(n \ge 3\)). Smaller values indicate the points are closer to the fitted line.
\(R^{2}\): the proportion of variation in y explained by the linear model (when defined). Larger values (closer to 1) indicate a stronger linear fit.

Export and copy options

Copy cleaned CSV outputs your data as a standard two-column x,y file.
Copy summary copies key model values (n, \(b_{0}\), \(b_{1}\), and \(R^{2}\) when available).

Frequently Asked Questions

What does a simple linear regression model compute?

It finds the best-fitting straight line that predicts a response y from a single explanatory variable x using least squares. The output includes the fitted equation yhat = b0 + b1 x and measures of fit.

How are the slope and intercept found in least-squares regression?

The slope b1 is computed from how x and y vary together relative to how x varies, and the intercept b0 is then set so the line passes through the point (xbar, ybar). This ensures the fitted line matches the sample means.

What are residuals in a regression plot?

A residual is the vertical difference between an observed y value and the fitted value at the same x: e = y - yhat. Showing residuals helps visualize how far points are from the regression line.

What does R^2 mean in a simple linear regression model?

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

How do I use this calculator to predict y at a specific x0?

Enter your (x, y) data, then fill in the Predict at x0 field and click Calculate. The calculator will report yhat(x0) and the mean-response standard error when the sample size is sufficient.

Results

Calculation steps

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Frequently Asked Questions

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