Linear Regression Calculator

Least-squares line y = a + bx with standard errors, t and p for the slope, r², F, residual standard error and prediction at any x.

The Linear Regression Calculator fits an ordinary least-squares line to (x, y) pairs: slope b = Sxy/Sxx, intercept a = ȳ − b·x̄. It reports standard errors of both coefficients, the t-test and p-value for the slope, r², the F statistic, the residual standard error and a predicted y for a chosen x.

Free, runs in your browser. Nothing you enter is uploaded.

How to use it

Paste X and Y.
Read the fitted equation and statistics.
Enter an x to predict y.

What you enter

Paired X and Y values (n ≥ 3)

What you get

Slope and intercept with SE
t and p for slope
r², F, residual SE
Predicted y

Formula

b = Σ(x − x̄)(y − ȳ) / Σ(x − x̄)²   |   a = ȳ − b·x̄   |   SE(b) = s_res / √Sxx

Least squares chooses the line minimising the sum of squared vertical residuals. r² is the fraction of variance in y explained by x.

Worked example

Input: X 1…10, Y 2.1 3.9 6.2 7.8 10.1 12.2 13.8 16.1 18.0 20.3

Result: ŷ = 0.0133 + 2.0230·x, r² = 0.9988, p < 0.0001

Each unit of x raises y by about 2.02; the model explains 99.9 % of the variance.

Questions

What does r² mean?

The proportion of variation in y accounted for by the linear relationship with x; 0.85 means 85 % explained.

Is the slope significant?

The p-value tests whether the slope differs from zero. Below 0.05 the linear relationship is statistically significant.

Limitations

Simple (one predictor) regression only.
Assumes linear relationship, independent residuals with constant variance.

References

Altman DG, Gardner MJ (1988). Calculating confidence intervals for regression and correlation. BMJ 296:1238-1242.

More Statistics

All free tools · By Dr. Zubair Khalid, DVM, MS, PhD.