How to use this calculator
- Enter or choose x values (predictor).
- Enter or choose y values (response, same order).
- Enter or choose confidence level.
- The slope (b₁) updates as you type. Use Copy result or Share to keep it.
Method
b₁ = Sxy ÷ Sxx; b₀ = ȳ − b₁x̄; s² = SSE ÷ (n − 2); SE(b₁) = √(s² ÷ Sxx); t = b₁ ÷ SE; CI = b₁ ± t*·SE
- Sxy
- Σ(x − x̄)(y − ȳ)
- Sxx
- Σ(x − x̄)²
- SSE
- sum of squared residuals
- t*
- two-sided t critical value with n − 2 df
Worked example
- Sxy = 6 and Sxx = 10, so b₁ = 0.6; b₀ = 4 − 0.6 × 3 = 2.2.
- Residuals −0.8, 0.6, 1, −0.6, −0.2 give SSE = 2.4 and s² = 2.4 ÷ 3 = 0.8.
- SE = √(0.8 ÷ 10) = 0.2828; t = 2.121 with 3 df, two-sided p = 0.124; 95% CI = 0.6 ± 3.182 × 0.2828.
Each extra unit of x adds 0.6 to y on average, but with five points the 95% interval (−0.30 to 1.50) still includes zero.
What the result means
The slope is in units of y per unit of x. A confidence interval that excludes 0 corresponds to a two-sided p-value below 1 − confidence. A significant slope shows association in this range of x; do not extrapolate far beyond the data.
Assumptions and limits
- Ordinary least squares: the relation is linear, residuals are independent with constant variance and roughly normal.
- x is measured without important error.
- The test is two-sided, H₀: slope = 0.
- Outliers and high-leverage points can dominate the fit; check a residual plot.
- One predictor only; multiple regression needs a matrix method.
Questions
How is the slope related to the correlation?
b₁ = r × (s_y ÷ s_x). The slope keeps the units; r is unit-free.
Why use n − 2 degrees of freedom?
Two parameters (slope and intercept) are estimated from the data before measuring the scatter around the line.