Question 1

What is the Jarque-Bera test?

Accepted Answer

The Jarque-Bera (JB) test checks whether sample skewness and excess kurtosis jointly match a normal distribution. Under H₀ (data is normal): skewness = 0 and excess kurtosis = 0. The JB statistic combines both into a single chi-square test with 2 degrees of freedom. A large JB (small p-value) indicates the data deviates significantly from normality in its shape.

Question 2

What does the p-value tell me?

Accepted Answer

p < 0.05: Reject H₀ — significant evidence that the data is NOT normally distributed (at α = 0.05). p ≥ 0.05: Fail to reject H₀ — no significant evidence of non-normality (data is consistent with normal, but this does not prove normality). Note: With very large samples (n > 1000), even tiny departures from normality produce p < 0.05 because the test has high power. Practical significance matters as much as statistical significance.

Question 3

What are skewness and excess kurtosis?

Accepted Answer

Skewness measures asymmetry: positive skew = right tail is longer (data bunched on left), negative skew = left tail is longer. For a normal distribution, skewness = 0. Excess kurtosis measures tail heaviness relative to normal: positive = heavier tails (leptokurtic), negative = lighter tails (platykurtic). For normal, excess kurtosis = 0. Guidelines: |skewness| < 0.5 is fairly symmetric, |excess kurtosis| < 1 is roughly normal-shaped.

Question 4

Should I always test for normality before t-tests?

Accepted Answer

Not necessarily. The t-test is fairly robust to modest normality violations, especially with n ≥ 30 (CLT). Testing normality is most important for: small samples (n < 30), one-sided tests, or when you want to decide between parametric and non-parametric methods. A practical approach: plot a histogram and Q-Q plot first. If the JB test rejects normality but n ≥ 30, a t-test is usually still fine. If n < 15 with strong skew, prefer the Wilcoxon test.

Question 5

What are limitations of the Jarque-Bera test?

Accepted Answer

JB has low power for small samples (n < 30) — it may miss non-normality. For large samples (n > 500), it is hypersensitive and often rejects normality for trivial deviations. The test only detects departures in skewness and kurtosis, not bimodality or other shapes. Alternative normality tests include: Shapiro-Wilk (best for small samples, up to n = 5000), Anderson-Darling, and Kolmogorov-Smirnov.

Question 6

What should I do if my data fails the normality test?

Accepted Answer

Options: (1) Apply a transformation — log(x) for right-skewed data, square root, or Box-Cox. Re-test the transformed data. (2) Use a non-parametric test: Wilcoxon signed-rank test instead of paired t-test, Mann-Whitney U instead of two-sample t-test, Kruskal-Wallis instead of one-way ANOVA. (3) If n ≥ 30, proceed with parametric tests — the CLT makes them robust. (4) Use bootstrapping for any test.

Test you want to run	Normality critical?	Non-parametric alternative
One-sample t-test	Yes (n < 30)	Wilcoxon signed-rank (vs constant)
Paired t-test	Yes for differences (n < 30)	Wilcoxon signed-rank test
Two-sample t-test	Yes (n < 30 per group)	Mann-Whitney U test
One-way ANOVA	Yes (n < 30 per group)	Kruskal-Wallis test
Pearson correlation	Yes (for inference)	Spearman rank correlation
Linear regression (residuals)	For prediction intervals	Robust regression

Measure	Normal value	Problematic range	What it means
Skewness	0	\|S\| > 1	Strong asymmetry — long tail on one side
Excess kurtosis	0	\|K\| > 2	Very heavy or very light tails vs normal

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