Linear Regression Calculator – Line of Best Fit (y = mx + b)

How it works

Simple linear regression finds the straight line that best describes how Y changes with X. This calculator uses ordinary least squares: it chooses intercept b₀ and slope b₁ so that the sum of squared vertical residuals is as small as possible. Enter X and Y as two lists of equal length, or paste two columns (one pair per line). Optionally predict ŷ for any x — including values outside the observed range. Results include the equation, R², the standard error of estimate, a residual table, and an inline scatter plot of the points with the fitted line.

Required Inputs & Typical Use Cases

• X and Y data

  Separate comma/space-separated lists, or pasted columns of x y pairs. Need at least two pairs with non-constant X.

• Predict y for x (optional)

  Any finite x; ŷ = b₀ + b₁x is evaluated at that point and marked on the graph.

Use this tool for homework line-of-best-fit problems, quick trend checks on small datasets, residual inspection before a report, or predicting a next value from a simple linear trend. Pair it with correlation when you care about association strength, or with a t-test when you need formal inference on means.

The Formula

b₁ = Sxy / Sxx
b₀ = ȳ − b₁ x̄
ŷ = b₀ + b₁ x
R² = 1 − SSE / SST
SEE = √(SSE / (n − 2))   (n > 2)
where Sxx = Σ(xᵢ − x̄)², Sxy = Σ(xᵢ − x̄)(yᵢ − ȳ),
SSE = Σ(yᵢ − ŷᵢ)², SST = Σ(yᵢ − ȳ)²

• b₁

  Slope (change in ŷ per unit x)

• b₀

  Y-intercept (ŷ when x = 0)

• R²

  Coefficient of determination

• SEE

  Standard error of the estimate

• residual

  yᵢ − ŷᵢ for each observation

Worked Example

1. Default sample: X = 1…5, Y = 2, 4, 5, 4, 5

   x̄ = 3, ȳ = 4, Sxx = 10, Sxy = 6 → slope b₁ = 0.6, intercept b₀ = 2.2, so ŷ = 2.2 + 0.6x. R² = 0.6. Predicting at x = 6 gives ŷ = 2.2 + 0.6×6 = 5.8.

Tips, Assumptions & Limitations

• X and Y must have the same number of values; mismatched lengths are a common paste error.
• If all X values are identical, Sxx = 0 and no unique regression line exists.
• R² close to 1 means a tight linear fit — not that X causes Y.
• Extrapolating far beyond your X range can be misleading even when the in-sample fit looks excellent.
• Inspect residuals in Show steps: large patterned residuals suggest a nonlinear relationship or outliers.
• SEE is undefined for n = 2 (no leftover degrees of freedom); you still get slope, intercept, and R².