Least Squares Regression
Paste data points to fit a least-squares regression line y = ax + b with slope a and intercept b. See the sums (Σx, Σy, Σx², Σxy), Sxx and Sxy, the correlation r, R², a scatter plot with the fitted line, residuals and predictions from x — with the working shown.
Comma, tab or space separated. e.g. 1, 3 or 1 3
Try an example
Enter data to fit a regression line.
Done
Least Squares Regression
How to use
- 1
Paste the data
Enter one x, y pair per line, separated by commas, tabs or spaces.
- 2
Check the line and metrics
See the regression line y = ax + b, slope a, intercept b, correlation r, R² and the scatter plot.
- 3
Predict and check
Enter an x to get the predicted ŷ, and use the residual table to look for outliers.
Features
- Fit a least-squares line y = ax + b from pasted data
- See the sums (Σx, Σy, Σx², Σxy) and the working for Sxx, Sxy and Syy
- Get the correlation r and coefficient of determination R² with a strength guide
- View a scatter plot with the fitted line and a table of residuals (y − ŷ)
- Predict ŷ from an x value, with an extrapolation warning outside the data range
- Runs entirely in your browser; no sign-up and nothing uploaded
Use cases
Calibration curves in lab reports
Find the slope, intercept and R² of a calibration line and inspect residuals for outliers.
Spot the trend in data
Use the scatter plot and fitted line to see the relationship between two variables.
Check homework and reports
Verify a hand-computed slope, intercept and correlation against the working shown.
Details
Least squares finds the line that minimises the sum of squared residuals. Writing the regression line as y = ax + b, the slope a and intercept b follow from the sum of squared deviations Sxx and the sum of products Sxy: a = Sxy / Sxx and b = ȳ − a·x̄ (where x̄ and ȳ are the means).
Sxx = Σx² − (Σx)²/n and Sxy = Σxy − (Σx·Σy)/n let you compute everything from sums instead of subtracting the mean point by point. This tool shows the sums (Σx, Σy, Σx², Σxy), Sxx, Sxy, Syy and the coefficients with the working.
The correlation coefficient r = Sxy / √(Sxx·Syy) ranges from −1 to 1 and measures the strength of the linear relationship. The coefficient of determination R² = r² ranges from 0 to 1 and indicates how much of the variation in y the line explains.
Many statistics textbooks write the line as y = a + bx (a: intercept, b: slope), which swaps the letters used here. State clearly which letter is the slope and which is the intercept in your report. By convention x is the explanatory variable and y is the response variable.
FAQ
What is the least squares method?
It fits the line that minimises the sum of squared vertical differences (residuals) between the data points and the line. Squaring prevents positive and negative differences from cancelling out.
Why are the slope and intercept letters different from my textbook?
This tool follows the site convention y = ax + b (a: slope, b: intercept). Some statistics textbooks write y = a + bx (a: intercept, b: slope), which swaps the letters. State clearly which letter is the slope in your answer.
What is the difference between r and R²?
r measures the strength and direction of the linear relationship (−1 to 1), while R² = r² measures how much of the variation in y the line explains (0 to 1). If r = 0.7, then R² = 0.49, so the line explains about 49% of the variation.
How should I handle outliers?
Least squares is strongly affected by outliers. Check the residual table for points that deviate strongly, and consider whether a measurement error justifies excluding them. Do not remove inconvenient points without a reason.
Can I use it for curved relationships?
No, this tool fits straight lines only. For curved data a straight line gives a low r and R² and can mislead. Some curved relationships can be linearised with a log or other transformation.
How reliable are the predictions?
Predictions inside the range of the data (interpolation) are relatively stable, but predictions outside it (extrapolation) are less reliable; the tool warns you. Correlation also does not prove causation.
Related guide
GuideHow to Find a Regression Line with Least Squares
Learn the least squares method step by step: build the sum table, compute Sxx and Sxy, find the slope a and intercept b, and read r and R² correctly. Includes worked examples, residuals, extrapolation and outlier cautions, and a free browser calculator for checking by hand.
GuideMean, Median and Standard Deviation Guide
Calculate mean, median and standard deviation with a five-value worked table. Compare N versus N−1, interpret outliers, and check calculator results against Excel.
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All processing happens in your browser. Your files are never uploaded.
Verified: Cross-checked against independent calculations (perfect line y=2x+1 with r=1, five scattered points giving a=0.6, b=2.2 and r=0.7746, negative correlation y=−2x+12 with r=−1, a five-point calibration giving a=0.99, b=0.08 and R²=0.9948, prediction at x=10 giving 21) plus error handling are covered by browser tests
Did you know?
Squaring the correlation coefficient r gives the coefficient of determination R²: if r = 0.7 then R² = 0.49, so the line explains about 49% of the variation in y. A strong correlation and accurate predictions are not the same thing.