Least Squares Regression Calculator - Fit, Audit, Predict

A least squares regression calculator fits a straight line through paired x and y values by choosing the slope and intercept that make the sum of the squared vertical distances between the data points and the line as small as possible. Use it to summarize a paired dataset, audit how well a line explains the spread in y, and estimate a y value for an x you have not measured yet.

• • Audit a hand-fit line: Paste the same x and y values you plotted, then read the slope, intercept, R-squared, and the standard error of the estimate.
• • Compare a trend before and after a change: Run the fit on the pre-change and post-change data side by side to compare slopes, R-squared, and sums of squares.
• • Forecast a single new value: Enter a predict-x to read the matching predicted y, then check the standard error of the estimate to size the prediction.
• • Teach or check a textbook example: Verify worked homework problems with a transparent fit, including sums of squares and the F-statistic.

The method is the ordinary least squares solution to y = a + b * x. The same closed-form normal-equations approach appears in introductory statistics, econometrics, and many time-series and machine-learning baselines.

The calculator is built for two pasted lists. It does not weight individual observations or handle more than one predictor at a time. For weighted fits, robust regression, or multiple regression with several predictors, use a statistics package.

If you only need the slope, intercept, and a confidence or prediction interval around a new x, the Linear Regression Calculator is a tighter tool for that specific workflow.

The least squares regression calculator parses the two lists, builds the normal-equation sums, solves for the slope and intercept, then reports the full sums-of-squares decomposition and standard errors so the fit is auditable.

• xValues and yValues: Two pasted lists of equal length, with at least 3 finite numbers each, used to fit one straight line.
• Slope b: The fitted change in y per one-unit change in x.
• Intercept a: The fitted y value when x equals 0, derived from the means so the line passes through the centroid (xbar, ybar).
• SSE, SSR, SST: SSE is the sum of squared residuals around the fitted line, SSR is the explained variation, and SST is the total variation around the mean of y.
• R-squared and adjusted R-squared: R-squared equals 1 minus SSE over SST. Adjusted R-squared applies a small-sample penalty using n - 2 in the denominator.
• Standard errors and F-statistic: The standard error of the estimate summarizes the typical residual size. The F-statistic compares the regression mean square to the residual mean square.

The same formulas power most spreadsheet trendlines and chart-platform best-fit overlays. The calculator keeps the intermediate sums visible so the user can verify how the slope and intercept were reached.

The output is dimensionless: slope, intercept, R-squared, sums of squares, and the F-statistic are plain numbers because the units come from the pasted data.

According to NIST/SEMATECH e-Handbook of Statistical Methods, the least squares estimates for the slope and intercept of a simple linear regression are given by formulas that use the sums of x, y, x^2, and xy.

The SST term uses the same y minus mean-of-y residuals that the Mean Median Mode Range Calculator summarizes, so reading the two outputs together helps you keep the variation metric consistent across reports.

These four ideas separate a least squares regression output from a black-box trendline and help you decide whether the fit deserves to be trusted.

Most chart platforms hide all four of these numbers behind a single R-squared label. The calculator surfaces them so the user can answer two questions: how much variation the line explains, and how large the typical residual is.

For a prediction problem, the standard error of the estimate is the most useful number because it puts a concrete band around a forecast. For an explanatory problem, the F-statistic and adjusted R-squared are the headline.

The Pearson r reported here is the same correlation coefficient that the Pearson Correlation Calculator returns, so the two outputs should match exactly when you paste the same paired data into both.

Paste two parallel lists, optionally set a predict-x, then read the slope, intercept, R-squared, and prediction from the result panel.

1. 1 Paste the X values: Enter the independent variable values, separated by commas, spaces, semicolons, or line breaks.
2. 2 Paste the Y values: Enter the dependent variable values in the same order, with the same separators accepted.
3. 3 Set an optional predict-x: Leave the predict-x field blank to skip the prediction, or enter any finite number to read the matching predicted y.
4. 4 Read the slope, intercept, and equation: Confirm the fitted equation in y = a + b * x form before interpreting R-squared, so the slope direction matches the data.
5. 5 Check the sums of squares: Use SST, SSR, and SSE to confirm SSR plus SSE equals SST, which proves the decomposition is consistent.

Suppose X = 1, 2, 3, 4, 5 and Y = 2, 4, 5, 4, 5. The fitted line is y = 2.2 + 0.6 * x, R-squared is 0.6, and the predicted y at x = 6 is 5.8. The standard error of the estimate near 0.89 is reasonable compared with the spread of y, so the prediction is usable for planning.

If the predict-x falls between two observed x values and the underlying shape is closer to a curve than a line, the Bilinear Interpolation Calculator is a smoother alternative for the same new-x problem.

An ordinary least squares fit is a useful baseline whenever a paired dataset has a roughly linear shape, and a transparent calculator version makes the answer easier to defend.

• • Auditable fit: Show the sums of squares and the standard errors next to the slope and intercept, so anyone can verify how the line was reached.
• • Consistent units: Keep the units of x and y whatever the pasted data already uses, so the slope and intercept are reported in the same units as the inputs.
• • Forecast a single new value: Read a predicted y for any new x in one step, without retyping the slope and intercept into a separate formula.
• • Fit-quality summary: Use R-squared, adjusted R-squared, and the F-statistic together to judge whether a linear model is enough.
• • Worked-example teaching: Verify textbook worked examples by comparing the calculator output with the published slope, intercept, and R-squared.

The benefit is that the least squares line is a shared language. Most adjacent statistics and finance tools expect a slope, an intercept, an R-squared, and a standard error, and this calculator surfaces all of them in one pass.

For more advanced workflows, treat the output as a baseline. Once the simple line is fit, the same dataset can move to weighted least squares, multiple regression, or a non-linear curve without losing the simple model as a reference.

Once the F-statistic is in hand, the P-Value Calculator helps you turn the overall model-fit F into a p-value so you can quote a probability statement alongside the slope and intercept.

The slope, intercept, R-squared, and standard errors all change shape when the data, the units, or the chosen predict-x change.

• • The calculator fits one straight line and does not perform weighted least squares, robust regression, or multiple regression with several predictors.
• • R-squared does not prove causation. A high R-squared only means the linear pattern is strong, not that x causes the change in y.
• • The standard errors assume the residuals are approximately independent, equal in variance, and roughly normal.

Use the calculator as a baseline audit, not as a final answer. If the residual pattern is systematic, the next step is a different model, not a tighter confidence band.

For finance, science, or operations datasets, document the data source and cleaning steps so the reader can judge whether the linear assumption is reasonable for that sample.

According to Wikipedia Ordinary Least Squares, the ordinary least squares estimator chooses the regression coefficients that minimize the sum of the squared residuals between the observed and predicted values of the dependent variable.

According to Wikipedia Coefficient of Determination, R-squared for a simple linear regression equals 1 minus the residual sum of squares divided by the total sum of squares around the mean of y.