Calculator Inputs
Example Data Table
| Observation | X | Y |
|---|---|---|
| 1 | 1.0 | 2.1 |
| 2 | 2.0 | 2.8 |
| 3 | 3.0 | 3.6 |
| 4 | 4.0 | 4.5 |
| 5 | 5.0 | 5.0 |
| 6 | 6.0 | 6.2 |
| 7 | 7.0 | 6.8 |
| 8 | 8.0 | 7.9 |
This sample dataset shows a positive linear trend. You can paste these values into the calculator to test the fitted line and interval widths.
Formula Used
Simple linear regression: ŷ = b₀ + b₁x
Slope: b₁ = Sxy / Sxx
Intercept: b₀ = ȳ − b₁x̄
Residual standard error: s = √(SSE / (n − 2))
Leverage at x₀: h(x₀) = 1/n + (x₀ − x̄)² / Sxx
Confidence band for mean response: ŷ(x₀) ± t × s × √h(x₀)
Prediction band for future observation: ŷ(x₀) ± t × s × √(1 + h(x₀))
Here, n is the number of observations, x̄ and ȳ are sample means, SSE is the residual sum of squares, and t is the two-sided critical value from the Student t distribution.
How to Use This Calculator
- Enter matching X and Y values as lists.
- Choose the confidence level for the interval width.
- Select whether you want confidence bands, prediction bands, or both.
- Pick automatic evaluation points or provide custom X locations.
- Choose your preferred decimal precision.
- Press the calculate button to show the results above the form.
- Review diagnostics, charted bands, and the full interval table.
- Use the CSV or PDF buttons to export the calculated output.
Frequently Asked Questions
1. What does a confidence band show?
A confidence band shows the uncertainty around the estimated mean regression line. It tells you where the average response is likely to fall at each X value.
2. What is the difference between a confidence band and a prediction band?
A confidence band estimates uncertainty for the mean fitted trend. A prediction band is wider because it includes both model uncertainty and the natural spread of individual observations.
3. Why do the bands widen near the edges?
Bands often widen away from the center because leverage increases when X values move farther from the sample mean. That makes interval estimates less certain at the extremes.
4. Can I use this tool for nonlinear relationships?
This version is designed for simple linear regression. If your data curves strongly, transform the variables or use a nonlinear model before interpreting the bands.
5. How many data points should I enter?
You need at least three paired observations. More observations usually improve stability, reduce noise impact, and make the regression line and interval estimates more reliable.
6. Does a higher confidence level change the line?
No. The fitted regression line stays the same. A higher confidence level only widens the interval bands because the critical value becomes larger.
7. What if my X values are repeated?
Repeated X values are acceptable if the overall X list still varies. The calculator only fails when every X value is identical, because a slope cannot be estimated then.
8. What exports are included?
The calculator includes CSV and PDF download buttons for the band table. These exports make it easier to document analysis, share results, or archive interval outputs.