Linear Regression Calculator
Enter up to eight x and y pairs, then an x value to predict from. You get the least-squares line y = mx + b, r squared, the standard error of the estimate and the predicted y.
Updated
Report a problem
Formula
For n pairs with means x-bar and y-bar, let Sxx = Σ(x - x-bar)², Syy = Σ(y - y-bar)² and Sxy = Σ(x - x-bar)(y - y-bar).
slope m = Sxy / Sxxintercept b = y-bar - m × x-barr² = Sxy² / (Sxx × Syy)standard error of the estimate = sqrt( (Syy - m × Sxy) / (n - 2) )predicted y = b + m × x
Worked example
x = 1 to 8 and y = 2, 4, 5, 4, 5, 7, 8, 9. Here x-bar = 4.5, y-bar = 5.5, Sxx = 42, Sxy = 38, Syy = 46.
- slope = 38 / 42 = 0.9048
- intercept = 5.5 - 0.9048 × 4.5 = 1.4286
- equation: y = 0.9048x + 1.4286
- r² = 38² / (42 × 46) = 0.9048
- standard error of the estimate = 0.7766
- predicted y at x = 10: 0.9048 × 10 + 1.4286 = 10.476
If every x is the same, the slope is undefined and an error is shown.
Frequently asked questions
How is the regression line calculated?
The line minimises the sum of squared vertical distances to the points. The slope is Σ(x - x-bar)(y - y-bar) / Σ(x - x-bar)² and the intercept is y-bar minus slope times x-bar. For x = 1 to 8 and y = 2, 4, 5, 4, 5, 7, 8, 9 the line is y = 0.9048x + 1.4286.
What does r squared tell me?
r squared is the share of the variation in y that the straight line explains. An r squared of 0.9048 means about 90.5% of the variation in y is accounted for by x in that data set.
What is the standard error of the estimate?
It is the typical size of the gap between the actual y values and the line, sqrt(SSE / (n - 2)). In the example it is 0.7766, so predictions miss by roughly 0.78 units in y.
How do I predict y for a new x?
Put the new x in the line equation. With y = 0.9048x + 1.4286 and x = 10, y = 0.9048 x 10 + 1.4286 = 10.476. Predictions far outside your x values are unreliable.
Why does the calculator need at least 3 pairs?
Two points always fit a line exactly, so there is nothing left to estimate the error from. The standard error divides by n - 2, so n must be at least 3, and the x values must not all be the same.