Linear Regression Calculator

In short Least-squares regression fits the straight line that minimises the sum of the squared VERTICAL distances from the points to the line. For the eight pairs loaded above it gives y = 0.9167x + 1.5, with r² = 0.9318. The slope is the practical result: every one-unit rise in x goes with a change of 0.9167 in y. The page also reports the standard error of the slope, its confidence interval, and the significance test — which is the same t-test the correlation page reports for r, because a zero slope and a zero correlation are the same hypothesis. Three things are easy to get wrong. Regressing y on x and x on y give DIFFERENT lines, not the same line rearranged, because only vertical distances are minimised; the two slopes multiply to r². The intercept is the fitted value at x = 0, and if zero is outside the range of your data it is an extrapolation the data does not evidence, however precise it looks. And there are two prediction intervals, not one: the interval for the MEAN y at a given x is much narrower than the interval for a SINGLE new observation there, and the second is almost always the one people actually want. Finally, a high r² means the line fits, not that the model is right. Plot the residuals — a clear pattern in them is the data telling you a straight line was the wrong shape.