Regression Results
Enter data and select Calculate Regression.
Formula Used
Simple linear regression estimates a straight line from paired observations. The fitted value uses an intercept and slope. Residuals measure vertical prediction errors.
ŷ = b₀ + b₁x b₁ = Σ[(xᵢ − x̄)(yᵢ − ȳ)] / Σ[(xᵢ − x̄)²] b₀ = ȳ − b₁x̄ R² = 1 − SSE / SST
Confidence bands describe uncertainty around the fitted mean response. Prediction bands include additional uncertainty for one new observation. They are therefore wider than confidence bands.
How to Use
- Enter paired X and Y values manually.
- Select a regression model and confidence level.
- Choose plot, diagnostic, and transformation options.
- Add future X values for predictions.
- Select Calculate Regression to generate results.
- Review plots, tables, warnings, and model comparisons.
- Export the completed analysis when needed.
Example Data
| Study hours | Exam score |
|---|---|
| 1 | 52 |
| 2 | 55 |
| 3 | 61 |
| 4 | 66 |
| 5 | 72 |
| 6 | 78 |
| 7 | 83 |
| 8 | 88 |
Frequently Asked Questions
What does the slope represent?
The slope estimates the average Y change for one X unit. Positive slopes rise, while negative slopes fall. Its scale depends on both variable units.
What does R² mean?
R² describes the proportion of observed Y variation explained by the model. Higher values indicate stronger fitted agreement. It does not prove causation or correctness.
Why are prediction bands wider?
Prediction bands include fitted-mean uncertainty and individual observation noise. Confidence bands include only mean uncertainty. New observations therefore need wider intervals.
Can I use negative values?
Linear and polynomial models accept negative values. Logarithmic and power models require positive X values. Exponential models require positive Y values.
What is an influential point?
An influential point noticeably changes fitted coefficients when included. Cook’s distance combines residual size and leverage. Large values deserve careful investigation.
Should detected outliers be removed?
Outliers should not be removed automatically. First check measurement quality and context. Compare models with and without them cautiously.
What is heteroscedasticity?
Heteroscedasticity means residual spread changes across fitted values. It can weaken standard inference. The residual plot helps reveal changing variation.
When should polynomial regression be used?
Use polynomial regression when curvature is scientifically plausible. Avoid excessive degrees that memorize noise. Compare validation evidence and residual patterns.
What is extrapolation?
Extrapolation predicts beyond the observed X range. Relationships may change outside measured data. Such predictions require stronger caution and context.