Linear Regression Calculator

How it works

Linear regression finds the one straight line that comes closest to a set of points, where "closest" means the sum of the squared vertical distances from the points to the line is as small as possible. That line is ŷ = a + bx: b is the slope (how much y changes for each unit of x) and a is the intercept (the value of y where x = 0).

The slope comes from how x and y vary together, Σ(x − x̄)(y − ȳ), divided by how x varies on its own, Σ(x − x̄)². The intercept is then fixed by the fact that the line always passes through the point of means (x̄, ȳ).

r² says what fraction of the spread in y the line accounts for: 1 is a perfect fit, 0 means x tells you nothing about y. The standard error of the estimate is the typical size of a residual (a point's vertical distance from the line), in the units of y, so it tells you how far a prediction is likely to be off.

Formula

x̄ = Σx ÷ n          ȳ = Σy ÷ n
Sxx = Σ(x − x̄)²     Syy = Σ(y − ȳ)²     Sxy = Σ(x − x̄)(y − ȳ)
b  = Sxy ÷ Sxx                     (slope)
a  = ȳ − b·x̄                       (intercept)
r² = Sxy² ÷ (Sxx · Syy)            r = ±√r², with the sign of b
SE = √( Σ(y − ŷ)² ÷ (n − 2) )      (standard error of the estimate)
ŷ  = a + b·x                       (prediction)

Example

Eight weeks of practice (x = 1 to 8) and test scores y = 52, 57, 61, 68, 70, 75, 80, 85. The means are x̄ = 4.5 and ȳ = 68.5; Sxy = 195 and Sxx = 42, so the slope is 195 ÷ 42 = 4.6429 points per week and the intercept is 68.5 − 4.6429 × 4.5 = 47.6071. The line is ŷ = 47.6071 + 4.6429x.

Syy = 910, so r² = 195² ÷ (42 × 910) = 0.9949 and r = 0.9974: the line explains 99.5% of the variation in scores. The standard error of the estimate is 0.8797 points, and the predicted score after 10 weeks is 47.6071 + 4.6429 × 10 = 94.0357.

Assumptions and limitations

• This is ordinary least squares: x is taken as exact and only the vertical distance from each point to the line is minimised. If both variables carry measurement error of similar size, a different fit is appropriate.
• A high r² means the line fits these points well. It does not mean x causes y, and it says nothing about what happens outside the range of x you entered.
• The standard error divides by n − 2, so at least three points are needed.
• The x and y lists are paired by position: the third y belongs to the third x. A missing value shifts every later pair, so check the two lists are the same length and in the same order.

Frequently asked questions