Least-squares regression prediction calculator

It returns the slope, intercept, prediction, and readable equation in one deterministic result. Use it to check statistics exercises, turn a small calibration table into an estimate, or automate repeatable linear forecasts without a spreadsheet. The calculation runs entirely from the values supplied and does not use a network service, random sampling, or changing external data.

Fit the least-squares line from paired observations

Enter the observed predictor values in x and the corresponding response values in y, keeping each pair at the same array position. At least two pairs are required, both arrays must have equal length, and the x values must contain some variation. The calculator first computes the mean of each series. It then measures how x varies around its mean and how the centered x and y values vary together. Dividing the centered cross-product sum by the centered square sum for x gives the fitted slope. Subtracting the slope times mean x from mean y gives the intercept. These are the ordinary least-squares coefficients for the line written as y equals slope times x plus intercept. This line minimizes the sum of squared vertical differences between the observed responses and their fitted values. Every supplied value must be a finite number, so missing entries, numeric strings, infinities, and malformed arrays are rejected instead of being silently ignored or coerced into a misleading model. The calculation accepts up to ten thousand observation pairs per request.

Predict the response at a new input value

Supply the new predictor value in predict_x. After fitting the coefficients from the observed pairs, the calculator substitutes this new value into the same equation and returns the result as prediction. The output also includes n, slope, intercept, predict_x, the printable equation, and the ordinary_least_squares method label, making it straightforward to record both the estimate and the model that produced it. A prediction inside the observed x range is interpolation and is usually easier to defend because nearby data support the line. A prediction outside that range is extrapolation. The arithmetic is still valid, but the relationship may bend, level off, or change beyond the measurements, so interpret an extrapolated value carefully. The result is a fitted response, not a guarantee about a future observation, and this focused calculator does not produce confidence or prediction intervals. Identical input always produces identical output because the implementation uses a closed-form calculation with no network calls, randomness, timestamps, or learned model. Successful API requests use the published base price of $0.002 per item.

Check assumptions and interpret the coefficients

The slope describes the fitted change in y for a one-unit increase in x. A positive slope indicates an increasing fitted relationship, while a negative slope indicates a decreasing one. The intercept is the fitted y value when x equals zero, although it may have little practical meaning when zero is far outside the observed range. Before relying on the prediction, plot the points when possible and look for curvature, unusual observations, separated clusters, or changing vertical spread. A straight-line summary can hide all of those patterns. Also consider whether observations are meaningfully independent and whether x is measured accurately enough for ordinary least squares to suit the task. The calculator deliberately rejects a dataset whose x values are all identical because its slope denominator is zero and no unique finite least-squares slope exists. It also rejects unequal array lengths rather than guessing which observations belong together. Use the returned equation as a transparent computational summary, retain the original units when explaining slope and prediction, and avoid presenting extra decimal places as evidence of greater real-world certainty than the measurements support.