Calculator Input
Use commas, spaces, or new lines. You can test residuals from paired actual and predicted values, or enter residuals directly.
Example Data Table
This example uses paired observations. Residuals equal Actual minus Predicted.
| # | Actual | Predicted | Residual |
|---|---|---|---|
| 1 | 52 | 50 | 2 |
| 2 | 49 | 51 | -2 |
| 3 | 57 | 56 | 1 |
| 4 | 63 | 62 | 1 |
| 5 | 60 | 61 | -1 |
| 6 | 66 | 64 | 2 |
| 7 | 70 | 69 | 1 |
| 8 | 73 | 74 | -1 |
| 9 | 75 | 76 | -1 |
| 10 | 81 | 79 | 2 |
Formula Used
Residual: ei = yi - ŷi
Standardized Residual: zi = (ei - ē) / s
Skewness: moment-based shape measure of asymmetry using the residual mean and second and third central moments.
Kurtosis: moment-based tail measure using the fourth central moment. Excess kurtosis equals kurtosis minus three.
Jarque-Bera Statistic: JB = (n/6)g₁² + (n/24)g₂², where g₁ is skewness and g₂ is excess kurtosis.
p-Value: this tool uses the chi-square distribution with two degrees of freedom. For two degrees, p = exp(-JB/2).
Decision Rule: reject normality when the p-value is smaller than the chosen significance level α.
How to Use This Calculator
- Select Actual and Predicted Values if you want the page to compute residuals automatically.
- Select Residual Values Only if you already have residuals from another model.
- Enter values with commas, spaces, or line breaks.
- Choose the significance level, histogram bins, flag threshold, and decimal precision.
- Press Calculate Residual Normality.
- Review the decision banner, summary metrics, and residual detail table.
- Use the histogram, QQ plot, and standardized residual chart to inspect shape, tails, and unusual observations.
- Download the results as CSV or PDF for reporting, notes, or coursework.
FAQs
1. What does this calculator test?
It checks whether model residuals look normally distributed using Jarque-Bera statistics, skewness, kurtosis, standardized residuals, a histogram, and a QQ plot.
2. Should I enter actual and predicted values or residuals?
Use paired values when the model output is still in raw form. Use direct residual entry when another system already calculated residuals for you.
3. What does a small p-value mean?
A small p-value means the residual shape differs enough from normality that the Jarque-Bera test rejects the normal assumption at your selected alpha level.
4. Is one normality test always enough?
No. A good review also checks residual plots, leverage, domain logic, model form, and whether your downstream inference is sensitive to nonnormal errors.
5. Why are skewness and kurtosis included?
Skewness measures asymmetry. Kurtosis measures tail weight and peak shape. Jarque-Bera combines both to evaluate how far residuals drift from normal behavior.
6. Can outliers change the result?
Yes. A few extreme residuals can strongly affect skewness, kurtosis, the histogram, and the QQ plot, which may cause normality rejection.
7. What sample size is reasonable?
The calculator works with small samples, but normality judgments become more stable as the number of residuals grows and plots become more informative.
8. Why use the QQ plot with the histogram?
The histogram shows overall shape, while the QQ plot shows where residuals depart from theoretical normal quantiles, especially in the tails.