Linear Regression Calculator

How it works

The fitted model is ŷ = a + bX. With centered sums Sxx = Σ(X−X̄)² and Sxy = Σ(X−X̄)(Y−Ȳ), the slope is b = Sxy/Sxx and the intercept is a = Ȳ−bX̄. Accepted values first become IEEE 754 binary64 browser numbers. Each represented value is then decoded as an integer times a power of two; sums, products and regression moments are accumulated exactly with bounded integers. Slope, intercept, each fitted value, each residual and the residual sum of squares are formed from those exact moments before conversion back to browser numbers for reporting. This handles both adjacent X values with an unrepresentable midpoint and mixed-scale cancellation. Reported values are rounded browser numbers, so subtracting two rounded CSV columns can differ from the separately computed residual. This is exact accumulation of the accepted binary64 values, not exact arithmetic on every entered decimal digit. Entered decimals that cannot round-trip through the browser number’s decimal representation are rejected before fitting, so silently lost input digits do not create a false constant series. The guard does not make binary arithmetic exact; rescale the original measurements if large values conceal small differences.

A residual is observed Y minus fitted Y. Sum of squared residuals is SSE. The displayed RMSE is √(SSE/n), while residual standard deviation is √(SSE/(n−2)); those are different denominators. The latter is undefined for two pairs. R² is 1−SSE/Syy for nonconstant Y, and Pearson r is the normalized centered cross-product. Constant Y makes r and R² undefined rather than automatically proving a perfect model. Constant X prevents slope estimation and is rejected. The fit requires at least two complete pairs.

The line minimizes vertical squared residuals in the supplied units. This page does not fit a zero-intercept, weighted, robust, polynomial or multiple-predictor model. It does not infer error distributions, calculate p-values or confidence/prediction intervals, correct for measurement error, or prove that a linear model is suitable. Inspect residuals and the actual problem before interpreting fit quality. A high R² can coexist with a misleading model, unusual leverage points or a relation that will not hold for new observations. An X outside the observed range receives an extrapolation warning.

In pair mode, put X first and Y second, one pair per line, separated by a comma or whitespace. Blank lines are ignored; partial pairs and extra columns are rejected. In CSV mode, provide unique nonblank column headers, choose comma, semicolon or tab, read the columns, and select two different columns. The explicit blank-pair policy either rejects an incomplete record or omits the whole pair with a visible count. Other non-numeric selected cells always receive a record-specific error. The tool never independently filters X and Y and accidentally pairs values from different records.

Limits are 1 MiB of valid UTF-8 text, 10,000 pasted pairs, or 10,000 CSV records including the header under the shared 200-column/200,000-cell CSV guard. Each numeric token has at most 200 characters and must be a decimal or e-notation value. Nonzero magnitudes range from 1e−100 to 1e100. Inputs that overflow or underflow are rejected. A calculation outside the finite numerical range is also rejected. Local files are read into this browser tab; no account or dataset upload is needed.

The scatter preview samples at most 500 observations in original order and clips the displayed line to its plot. It is a preview, not an assurance that every unusual point is visible. The expandable table shows the first 20 rows. CSV and JSON contain all fitted rows, original record numbers and residuals; JSON additionally contains the method, computed summary, selected columns, blank policy, any current prediction and a JSON-safe exact-moment audit model. Displayed values round to 12 significant digits while exports retain computed precision. Editing observations or settings clears old results and estimates; changing the prediction X clears its old answer. Data are kept only in tab memory, not a URL or persistent history. Optional analytics receives fixed action labels and input mode, never values, column names or filenames.

Common use cases

• Check a simple calibration line and inspect every observed-versus-fitted residual.
• Compare two numeric CSV columns while retaining their original row pairing.
• Create a transparent classroom regression example with a plot and exported calculation.
• Estimate a fitted value inside an observed range while keeping extrapolation clearly labeled.

Frequently asked questions

Standards and limitations

• NIST: least squares and the linear model
• NIST: regression model validation
• NIST Statistical Reference Datasets: linear least squares