Linear regression calculator, the line and the working.

A linear regression calculator finds the straight line that best describes how y changes with x. It gives the line as an equation, y = mx + b, where m is the slope and b is the intercept. This one uses ordinary least squares, the standard method: the line that makes the sum of the squared vertical gaps between your points and the line as small as possible. It also reports r, R squared and the standard error, so you can tell how well a straight line fits your data.

How to use the linear regression calculator

1. Put your numbers in the Data box, one pair on each line: the x value, then the y value, like 3 61. A tab, comma, semicolon or space between them all work, so you can paste straight from Excel, Google Sheets or Numbers.
2. Or click Open a CSV file, or drop a file on the box. If the first row holds names, they label the chart axes.
3. Read the cards: the equation, slope, intercept, correlation r, R squared and standard error. They update as you type.
4. To predict, type an x under y for this x or a y under x for this y.
5. Scroll down for the working table and the residuals. Use Download CSV for a file with each point, its value on the line and its residual, Save chart PNG for the picture, or Copy link to share the data and settings.

The formulas

First find the mean of x and the mean of y. Then add up two sums over all points: Sxx is the sum of (x − mean x) squared, and Sxy is the sum of (x − mean x) times (y − mean y). The slope is m = Sxy ÷ Sxx. The intercept is b = mean y − m × mean x. The correlation is r = Sxy ÷ √(Sxx × Syy), where Syy is the sum of (y − mean y) squared, and R squared is r times r. The standard error of the estimate is s = √(sum of squared residuals ÷ (n − 2)).

A worked example you can check by hand

Click the 5 points example: x is 1, 2, 3, 4, 5 and y is 2, 4, 5, 4, 5.

The means are 3 and 4. The slope is 6 ÷ 10 = 0.6. The intercept is 4 − 0.6 × 3 = 2.2, so the line is y = 0.6x + 2.2. The correlation is 6 ÷ √(10 × 6) = 0.7746, and R squared is 0.6. The line gives 2.8, 3.4, 4.0, 4.6 and 5.2, so the residuals are −0.8, 0.6, 1.0, −0.6 and −0.2. Their squares add up to 2.4, and with n − 2 = 3 the standard error is √(2.4 ÷ 3) = 0.8944. To predict y at x = 6, work out 0.6 × 6 + 2.2 = 5.8. To find x when y is 4.6, work out (4.6 − 2.2) ÷ 0.6 = 4.

Reading r, R squared and the p value

• r runs from −1 to 1. Near 1, y rises with x along a straight line. Near −1, y falls as x rises. Near 0, there is no straight-line link.
• R squared is the share of the changes in y that the line explains. An R squared of 0.6 means 60 percent.
• Standard error is the typical size of a residual, in the units of y.
• p value of the slope tests whether the true slope could be zero. With only five points, the example above has a p value of 0.124, so five points are too few to be sure.

Things to watch

• A high R squared does not show that x causes y.
• One far-off point can tilt the whole line. Look at the chart and the residuals.
• A prediction outside your x values is a guess. The page tells you when you do that.
• If all x values are equal the line is vertical and has no slope. If all y values are equal the slope is 0 and r is undefined.
• This is a straight line only. For a curve, try the scatter plot maker, which can fit a curved trend line.

For the spread of a single list of numbers, use the standard deviation calculator or the average calculator.