Calculator Inputs
Enter model summary values to estimate regression interval bounds.
Example Data Table
These sample scenarios show how interval width changes with different standard errors, leverage values, residual variation, and confidence levels.
| Scenario | β̂ | SE(β̂) | df | Confidence | ŷ(x₀) | s | h(x₀) | Coefficient CI | Mean CI | Prediction Interval |
|---|---|---|---|---|---|---|---|---|---|---|
| Retail demand model | 1.42 | 0.18 | 48 | 95% | 72.40 | 4.60 | 0.07 | 1.06 to 1.78 | 69.95 to 74.85 | 62.83 to 81.97 |
| Churn risk model | -0.63 | 0.14 | 58 | 90% | 31.80 | 2.90 | 0.11 | -0.86 to -0.40 | 30.19 to 33.41 | 26.69 to 36.91 |
| Revenue lift model | 3.05 | 0.52 | 36 | 99% | 110.20 | 8.40 | 0.05 | 1.64 to 4.46 | 105.09 to 115.31 | 86.79 to 133.61 |
Formula Used
This calculator reports three related interval estimates from regression summary statistics.
1) Coefficient confidence interval
2) Mean response confidence interval at x₀
3) Prediction interval for a future observation at x₀
Here, β̂ is the estimated coefficient, SE(β̂) is its standard error, s is the residual standard error, h(x₀) is leverage at the chosen input point, and t* is the critical value from the t distribution using the selected confidence level and degrees of freedom.
Prediction intervals are always wider than mean response intervals because they include both model uncertainty and future observation noise.
How to Use This Calculator
- Enter the estimated coefficient and its standard error from your regression summary.
- Add residual degrees of freedom from the fitted model output.
- Choose a confidence level such as 90%, 95%, or 99%.
- Provide the fitted value at x₀ if you want mean and prediction intervals.
- Enter residual standard error and leverage at x₀ for interval band calculations.
- Select displayed decimals, then submit the form.
- Review the interval table, summary cards, and Plotly graph.
- Use the export buttons to save the calculated results as CSV or PDF.
Frequently Asked Questions
1) What does this calculator estimate?
It estimates a coefficient confidence interval, a mean response confidence interval, and a prediction interval for a future observation at the selected input point.
2) Why is the prediction interval wider?
A prediction interval includes uncertainty in the fitted mean plus the random variation of a single future outcome, so it must be wider.
3) What is leverage in regression?
Leverage measures how unusual the selected predictor location is relative to the data cloud. Higher leverage typically produces wider interval bands.
4) Which degrees of freedom should I enter?
Use the regression model’s residual degrees of freedom. In many cases, this equals sample size minus the number of fitted parameters.
5) Can I use 90% or 99% confidence levels?
Yes. Lower confidence gives narrower intervals, while higher confidence gives wider intervals because the critical t value increases.
6) Does this work for multiple regression?
Yes, if you already know the coefficient estimate, standard error, fitted value, residual standard error, leverage, and degrees of freedom from the model output.
7) What happens if leverage is zero?
The mean response interval collapses to the fitted value because the leverage contribution is zero. The prediction interval still remains positive.
8) Why might my interval still be very wide?
Large standard errors, small samples, high leverage, high residual variation, or very high confidence levels all increase interval width.