Calculator Input
Example Data Table
This sample shows paired values for a simple linear regression test.
| Observation | X | Y |
|---|---|---|
| 1 | 1.0 | 2.1 |
| 2 | 2.0 | 3.9 |
| 3 | 3.0 | 5.8 |
| 4 | 4.0 | 8.2 |
| 5 | 5.0 | 10.1 |
| 6 | 6.0 | 11.7 |
| 7 | 7.0 | 13.9 |
| 8 | 8.0 | 16.2 |
Formula Used
1. Slope: b₁ = Sxy / Sxx
2. Intercept: b₀ = ȳ − b₁x̄
3. Sample standard deviation: SD = √[Σ(value − mean)² / (n − 1)]
4. Residual sum of squares: SSE = Σ(y − ŷ)²
5. Mean square error: MSE = SSE / (n − 2)
6. Residual standard deviation: s = √MSE
7. Standard error of the slope: SE(b₁) = √(MSE / Sxx)
8. Correlation: r = Sxy / √(Sxx × Syy)
9. Coefficient of determination: R² = r²
The standard error of the slope measures slope precision. Standard deviation measures spread in the original values. They answer different questions.
How to Use This Calculator
- Enter the X dataset in the first field.
- Enter the matching Y dataset in the second field.
- Use commas, spaces, or line breaks between values.
- Set chart labels and decimal places if needed.
- Click Calculate Now to run the regression.
- Review the slope, SE of slope, and standard deviations.
- Inspect fitted values, residuals, and the two charts.
- Download the output as CSV or PDF.
FAQs
1. What does the standard error of the slope measure?
It measures how precisely the regression slope is estimated from sample data. A smaller value means the slope estimate is more stable across repeated samples.
2. How is standard deviation different here?
Standard deviation measures spread in the raw X or Y values. It does not directly measure slope precision. It describes variability, not estimation uncertainty.
3. Why can SE of the slope be small?
It becomes smaller when residual noise is lower, sample size is larger, or X values are more spread out. Those conditions improve slope precision.
4. Why do I need paired X and Y values?
The calculator runs simple linear regression. Each X value must match one Y value. Unpaired entries break the fitted relationship and invalidate the slope estimate.
5. What is residual standard deviation?
Residual standard deviation summarizes the typical prediction error around the regression line. It shows how far observed Y values usually fall from fitted Y values.
6. What does R² add to the analysis?
R² shows the proportion of Y variation explained by the line. Higher values indicate a tighter linear fit, though they do not replace uncertainty measures.
7. Can I use this for nonlinear data?
This page is built for straight-line relationships. Nonlinear patterns may produce misleading slopes, residuals, and uncertainty values. Use a nonlinear model for curved data.
8. When should I compare SE of slope and SD together?
Compare them when you want both estimation precision and raw data spread. SE describes confidence in the slope. SD describes how widely the original values vary.