OneTaskBy Tim King · Updated
Linear Regression Calculator
Least Squares Regression Equation, Slope, Intercept and R-Squared From x and y
OneTask is free, with no sign-up, and built by one person.
If this tool saved you time, a coffee helps keep new ones coming.
How the Linear Regression Calculator works
This linear regression calculator fits the straight line that best describes how y changes with x. Paste your paired values and it returns the equation, the slope, the intercept, R², the p-value for the slope and a prediction, plus the fitted value and residual for every point.
Simple linear regression draws the one straight line through a scatter of points that keeps the squared vertical misses as small as possible, which is why it is called least squares. The slope tells you how much y changes for each extra unit of x; the intercept is the predicted y at x = 0.
How the linear regression calculator works
The formula for the slope is b = Sxy ÷ Sxx, where Sxy = Σ(x − x̄)(y − ȳ) and Sxx = Σ(x − x̄)². The intercept is a = ȳ − b × x̄, giving ŷ = a + bx. R² is the share of the variation in y the line explains, and the slope's t statistic tests whether the true slope could be zero.
First example. The default data, hours revised against exam mark for 22 students, gives a slope of about 4 marks per hour and an R² close to 0.99.
Second example. Points (1, 2), (2, 4) and (3, 6) sit exactly on y = 2x, so R² = 1 and every residual is zero.
How many points do you need? At least three for a p-value, but ten or more for a trend you can trust.
Fitted values and residuals for each data point
| x | y | Predicted ŷ | Residual y − ŷ |
|---|---|---|---|
| 1 | 38 | 40.0772 | -2.0772 |
| 2 | 42 | 44.1371 | -2.1371 |
| 2 | 45 | 44.1371 | 0.8629 |
| 3 | 47 | 48.1969 | -1.1969 |
| 3 | 50 | 48.1969 | 1.8031 |
| 4 | 52 | 52.2568 | -0.2568 |
| 4 | 55 | 52.2568 | 2.7432 |
| 5 | 54 | 56.3166 | -2.3166 |
| 5 | 58 | 56.3166 | 1.6834 |
| 6 | 60 | 60.3764 | -0.3764 |
| 6 | 63 | 60.3764 | 2.6236 |
| 7 | 62 | 64.4363 | -2.4363 |
| 7 | 66 | 64.4363 | 1.5637 |
| 8 | 68 | 68.4961 | -0.4961 |
| 8 | 71 | 68.4961 | 2.5039 |
| 9 | 70 | 72.556 | -2.556 |
| 9 | 74 | 72.556 | 1.444 |
| 10 | 76 | 76.6158 | -0.6158 |
| 10 | 78 | 76.6158 | 1.3842 |
| 11 | 80 | 80.6757 | -0.6757 |
| 12 | 83 | 84.7355 | -1.7355 |
| 12 | 85 | 84.7355 | 0.2645 |
Tips
- Order matters. The nth x value pairs with the nth y value, so keep both lists in the same order.
- Do not extrapolate far. A prediction well outside the range of your x values is a guess, not an estimate.
- Check the residuals. A curved pattern in them means a straight line is the wrong model.
FAQ
What does R² tell me?
R² is the proportion of the variation in y explained by the straight line, from 0 to 1. An R² of 0.8 means the line accounts for 80% of the spread in y and the rest is scatter around it. A high R² does not prove that x causes y, only that they move together.
What is a residual?
A residual is the actual y minus the y the line predicts for that x. Positive residuals sit above the line and negative ones below. They always sum to zero for a least-squares line, and a large one flags a point worth checking for a typing error or a genuine outlier.
What is the difference between regression and correlation?
Correlation measures how tightly two variables move together on a scale from −1 to 1. Regression goes further and gives an equation for predicting one from the other. In simple regression R² is just the correlation coefficient squared, so the two always agree on strength.
See also: all Statistics tools, Arithmetic, Number Theory, Algebra and Geometry, Trigonometry and Calculus.
Related tools
Correlation Coefficient Calculator
Pearson r, Spearman rho and significance from paired data.
Slope Calculator
Slope, y = mx + b, angle and grade from two points.
Average Calculator
Mean, median, mode and range of any list of numbers.
Standard Deviation Calculator
Sample and population standard deviation and variance of a list.
Binomial Probability Calculator
Exactly, at most and at least k successes in n trials.