Regression Coefficient Calculator

Ordinary least squares chooses the slope and intercept that minimise the sum of squared vertical residuals. Differentiating that sum gives the two normal equations, whose solution is b1 = Sxy divided by Sxx and b0 = mean of Y minus b1 times mean of X, so the fitted line always passes through the centroid of the data. The slope carries the units of the problem: if X is drug concentration in mg/L and Y is cell viability in percent, a slope of -2.90 means each additional mg/L costs 2.9 percentage points of viability. Precision of the slope is se(b1) = s divided by the square root of Sxx, where s is the residual standard deviation with n-2 degrees of freedom, so spreading the X values further apart tightens the estimate as effectively as adding cases. Each coefficient is tested with t = estimate over its standard error on n-2 degrees of freedom, and in simple regression the overall F statistic is exactly the square of the slope t. R squared reports the fraction of total variation removed by the fit, while adjusted R squared subtracts a penalty for degrees of freedom used and is the fairer figure when comparing models. The usual assumptions are linearity, independent observations, constant residual variance and approximately normal residuals; violations bias the standard errors more than the slope itself. Residual plots and leverage diagnostics such as Cook's distance are the standard checks. Predictions outside the observed range of X are extrapolation and carry no guarantee that the linear form still holds.

The least squares criterion was first published by Adrien-Marie Legendre in 1805, in an appendix to Nouvelles Methodes pour la Determination des Orbites des Cometes, as a practical rule for reconciling more astronomical observations than unknowns. Carl Friedrich Gauss presented it in Theoria Motus in 1809 together with a probabilistic justification based on the normal error law, and claimed to have used the method since 1795, which triggered a long priority dispute. Laplace supplied a large-sample rationale shortly afterwards. The word regression is later and comes from biology. Francis Galton, studying the heights of parents and offspring, published Regression towards Mediocrity in Hereditary Stature in 1886 and found that tall parents had tall but less extreme children. George Udny Yule recast Galton's idea in least squares terms in his 1897 paper on the theory of correlation for the Journal of the Royal Statistical Society and extended it to several predictors, an approach Karl Pearson developed further around 1898 to 1903. Ronald Fisher supplied the inferential machinery in the 1920s, giving the correct degrees of freedom, the t tests for individual coefficients and the analysis of variance decomposition of the sum of squares. Once electronic computing arrived in the 1960s, multiple regression became routine in dose-response work, epidemiology and laboratory calibration.