Linear Regression Calculator

Fit a straight line, parabola, exponential or power curve to paired data by least squares, with the normal equations, R² and predictions.

Fit	Straight line y = a + bx
x values	1, 2, 3, 4, 5, 6, 7, 8
y values	2.1, 3.9, 6.2, 7.8, 10.1, 12.2, 13.8, 16.1
Predict y at x = (optional)	10

Fit

x values

y values

Predict y at x = (optional)

Line of best fit

y = 1.997619x + 0.035714

The line explains 99.8839% of the variation in y (R² = 0.99884). At x = 10, the predicted y is 20.011905.

Exact fraction. Checked exactly.

Method: Least squares (normal equations)
Worked out with: AllCalcHub exact fractions
Check: Every step that can be checked was checked; the coefficients satisfy every normal equation exactly

Slope: 1.99761905
Intercept: 0.03571429
R²: 0.998839
Correlation r: 0.999419
Slope p-value: 4.889e-10

Data and fitted line

Point at the curvePoint at the chart, or Tab to it and use ← →

datafit

About the Linear Regression Calculator

Linear regression finds the line y = a + bx that makes the sum of squared vertical distances to the points as small as possible. R² is the share of the variation in y the line explains, from 0 (none) to 1 (all).

Curve fitting by least squares uses the normal equations: for a line, Σy = na + bΣx and Σxy = aΣx + bΣx²; for a parabola add c and one more equation. The steps show the sums and solve the equations exactly. Exponential and power curves are fitted by taking logarithms: ln y = ln a + bx, or ln y = ln a + b ln x.

The formula

b = Σ(x − x̄)(y − ȳ) ÷ Σ(x − x̄)²; a = ȳ − b·x̄

R²: (Sxy)² ÷ (Sxx·Syy)

Worked example

Example: Line of best fit = y = 1.99x + 1.05

Inputs: Fit: Straight line y = a + bx; x values: 1, 2, 3, 4, 5; y values: 3.1, 4.9, 7.2, 8.8, 11.1; Predict y at x = (optional): 8.
Formula: b = Σ(x − x̄)(y − ȳ) ÷ Σ(x − x̄)²; a = ȳ − b·x̄
The sums: n = 5, Σx = 15, Σx² = 55, Σy = 351/10, Σxy = 626/5
Normal equation 1: Σy = an + bΣx: 351/10 = 5a + 15b
Normal equation 2: Σxy = aΣx + bΣx²: 626/5 = 15a + 55b
Solve the equations: a = 21/20, b = 199/100
Result: Line of best fit = y = 1.99x + 1.05
Slope: 1.99
Intercept: 1.05
R²: 0.997305

Things to keep in mind

Predictions far outside the range of your x values are unreliable.
Correlation isn't causation.

Questions people ask

What's a good R²?

It depends on the field: physics often expects above 0.99, social sciences can treat 0.3 as meaningful.

Can I fit a curve?

Yes: choose a parabola (normal equations in a, b, c), an exponential or a power curve. The exponential and power fits take logs first, as textbooks do, which is least squares on ln y rather than on y.