Jarque-Bera normality test calculator definition and formula

The Jarque-Bera (JB) normality test calculator determines whether a statistical sample follows a normal Gaussian distribution by combining sample skewness and kurtosis, per the Carlos Jarque & Anil Bera 1980 formulation at Australian National University Canberra published in Economics Letters vol. 6 no. 3 pp. 255-259 "Efficient tests for normality, homoscedasticity and serial independence of regression residuals". The closed-form formula: JB = (n/6) · [S² + (1/4) · (K - 3)²], where n = sample size (number of observations), S = sample skewness (third standardized moment S = m₃/m₂^1.5), K = sample kurtosis (fourth standardized moment K = m₄/m₂² with K=3 for Gaussian normal, K-3 = excess kurtosis), m_k = k-th central moment (1/n)·Σ(xᵢ-x̄)^k. The JB statistic under null hypothesis H0 of normality asymptotically follows the chi-square distribution with 2 degrees of freedom χ²(2) (Bowman-Shenton 1975 Biometrika 62 moments derivation). The χ²(2) critical thresholds: α=10% → 4.605; α=5% → 5.991 (standard inference); α=1% → 9.210; α=0.1% → 13.816.

The p-value chi-square 2 dof exact formula: p = exp(-JB/2) (χ²(2) density upper-tail integral). Decision: if JB > threshold → reject H0 (non-normal, heavy tails or significant asymmetry); if JB ≤ threshold → do not reject (compatible with normality). The 4 JB application categories:

• GARCH/ARMA econometric model residual validation (Bollerslev 1986 fundamental for volatility clustering of financial returns)
• Italian financial returns testing BTP FTSE MIB (stock returns are leptokurtic K>3 with fat tails → JB rejects normality strongly, motivating Student-t or GED distributions)
• OLS linear regression assumption validation (residuals ε~N(0,σ²) Gauss-Markov hypothesis, JB on OLS residuals)
• ISTAT income distribution testing (typically lognormal with right Pareto tail, JB rejects normality). The test physics: under H0 normal S→0 and K→3 in probability (law of large numbers on moments), so JB→0; under deviations from normality (asymmetry S≠0 or heavy tails K>3 leptokurtic or thin tails K<3 platykurtic) the two quadratic terms in JB explode linearly with n, ensuring test consistency (power→1 as n→∞). The Italian applications:

• Bank of Italy 1893 economic research Via Nazionale Rome normality test on 10Y BTP returns, BTP-Bund spread, HICP Eurostat inflation
• ISTAT Rome 1926 national statistics household income distributions EU-SILC, population census, demography
• Bocconi University Milan 1902 Statistics Department Mario Forni econometrics + quantitative finance FTSE MIB returns
• Bruno de Finetti Trieste 1931 subjective probability foundations of Italian Bayesian inference
• University of Bologna 1088 oldest Cataldo Agostinelli mathematical statistics
• SISSA Trieste 1978 + ICTP 1964 Statistical Methods Higher Studies
• Politecnico Milano 1863 + Politecnico Torino 1859 statistical engineering quality control SPC
• Catholic University Sacred Heart Milan 1921 econometric distributions
• INFN AI Atlas/CMS Higgs 2012-2024 Carlo Rubbia 1984 Nobel Monte Carlo distributional tests
• ICRA Sapienza Rome 1303 + INDAM mathematics
• Cineca Marconi100/Leonardo HPC bootstrap Monte Carlo simulations of JB
• Consob 1974 IVASS 2012 Solvency II returns regulation VaR backtesting
• Eutekne taxation + Italian asset management INPS pension funds. PNRR Mission 4 Component 2 €6.9B AI/data science research

Jarque-Bera Italian application table BTP FTSE MIB ISTAT

(Source: Jarque & Bera 1980 Economics Letters 6(3) pp. 255-259 + Bowman-Shenton 1975 Biometrika 62 + D'Agostino-Pearson 1973 + Bank of Italy 1893 + ISTAT 1926 + Borsa Italiana).

Italian Jarque-Bera calculation examples

• Perfect normal distribution (control): n=100, S=0, K=3 (exact Gaussian). JB = (100/6) · [0² + (1/4)·(3-3)²] = (100/6) · [0 + 0] = 0. p-value = exp(-0/2) = 1.000. Decision: do not reject H0 obvious (perfectly normal sample)
• Moderately non-normal distribution: n=100, S=0.5 (positive asymmetry), K=4 (mild leptokurtic). JB = (100/6) · [0.5² + (1/4)·(4-3)²] = 16.667 · [0.25 + 0.25] = 16.667 · 0.5 = 8.33. p-value = exp(-8.33/2) = exp(-4.167) = 0.0155. JB > 5.991 (α=5%) → reject H0 at 5% but not at 1% (8.33 < 9.21)
• FTSE MIB daily returns 5 years Borsa Italiana 1808: n=1,250, S=-0.40 (negative skew = heavy left tails crash), K=6.5 (strong leptokurtosis fat tails). JB = (1,250/6) · [0.16 + (1/4)·12.25] = 208.33 · [0.16 + 3.0625] = 208.33 · 3.2225 = 675.0. p-value ≈ 10⁻¹⁴⁷ ≈ 0. Strong rejection H0 normality → motivates GARCH(1,1) Bollerslev 1986 for VaR/Solvency II IVASS
• BTP 10Y decennial returns Bank of Italy 1893: n=1,250 (5 years daily), S=0.30, K=5.20. JB = (1,250/6) · [0.09 + (1/4)·4.84] = 208.33 · [0.09 + 1.21] = 208.33 · 1.30 = 270.8. p-value ≈ 0
• IT HICP monthly inflation ISTAT-Eurostat 240 months 2005-2024: n=240, S=0.20, K=3.30. JB = (240/6) · [0.04 + (1/4)·0.09] = 40 · 0.0625 = 2.5. p-value = exp(-1.25) = 0.287. JB < 5.991 → do not reject H0 (monthly inflation compatible with normality)
• ISTAT EU-SILC household income distribution 20,000 obs: n=20,000, S=1.80 (strong right Pareto skew high incomes), K=8.5. JB = (20,000/6) · [3.24 + (1/4)·30.25] = 3,333.3 · [3.24 + 7.5625] = 3,333.3 · 10.8025 = 36,008. Reject H0 at all levels → motivates log transformation or lognormal model
• GARCH(1,1) FTSE residuals Bocconi Mario Forni econometrics: n=1,250, S=0.05 (post-fit standardized), K=3.40. JB = (1,250/6) · [0.0025 + (1/4)·0.16] = 208.33 · [0.0025 + 0.04] = 208.33 · 0.0425 = 8.85. JB > 5.991 → reject at 5% (residuals still slightly leptokurtic, motivates GARCH-t Student)
• Higgs Monte Carlo simulation INFN Atlas/CMS: n=10,000, S=0.02, K=2.98. JB = (10,000/6) · [0.0004 + (1/4)·0.0004] = 1,666.67 · 0.0005 = 0.83. Do not reject (MC well-calibrated Gaussian)

Italian statistics econometrics research Jarque-Bera

The Italian statistics and econometrics research centuries-old roots:

• University of Bologna 1088 oldest in the world, Cataldo Agostinelli mathematical statistics + Cesare Marchetti probability calculus
• Bocconi University Milan 1902 Statistics Department Mario Forni econometrics + Carlo Favero quantitative finance BAFFI Carefin + Francesco Corielli stochastic finance + Igor Pruenster Bayesian nonparametric
• Bruno de Finetti Trieste 1931 subjective probability founder of Italian Bayesian school, "La prévision: ses lois logiques, ses sources subjectives" 1937 Annales Poincaré
• ISTAT Rome 1926 Corrado Gini concentration coefficient 1912 + Luigi Bodio national statistics + decennial population censuses + EU-SILC household income + Labour Force RCFL quarterly
• Bank of Italy 1893 economic research Via Nazionale Rome 91 General Directorate Studies Economics Statistics head Sergio Nicoletti Altimari + econometricians Filippo Altissimo Stefano Siviero Italy macro analysis unemployment rate GDP HICP deflator BTP yields
• Catholic University Sacred Heart Milan 1921 Statistical Sciences Department Mario Faliva Maria Grazia Zoia macroeconometrics + Gianluca Cubadda
• SISSA Trieste 1978 + ICTP Abdus Salam 1964 Statistical Methods Higher Studies Bayesian inference + machine learning
• Politecnico Milano 1863 + Politecnico Torino 1859 statistical engineering quality control SPC EWMA CUSUM Shewhart
• Sapienza Rome 1303 Faculty of Statistics Demography Maurizio Vichi cluster analysis + ICRA Remo Ruffini astrostatistics
• University of Padua 1222 Statistics Department Adriano Paggiaro biostatistics
• University of Florence 1321 Department of Statistics Bruno de Finetti school
• INDAM Rome mathematics 1939 National Institute Higher Mathematics Francesco Severi GNAMPA probability analysis group
• Cineca Casalecchio Bologna 1969 Marconi100 30 PFlops + Leonardo pre-exascale 250 PFlops 2023 5th TOP500 worldwide Monte Carlo bootstrap JB power analysis 10⁹ samples
• INFN AI Atlas/CMS Higgs 2012-2024 Carlo Rubbia 1984 Nobel CERN W and Z bosons discovery 1983 + INFN Monte Carlo distributional tests 5σ Higgs 125 GeV
• Consob 1974 National Companies Stock Commission returns regulation VaR backtesting + IVASS 2012 Solvency II insurance RCA 2026 €420 + Covip 1993 pension funds
• Eutekne taxation Italian asset management Anima Holding Mediolanum Generali fund returns backtesting. Italian R&D tax bonuses 2025: R&D Tax Credit art. 1 c. 200 L. 160/2019 up to 25%; Patent Box L. 178/2020 50%; PNRR Mission 4 Component 2 AI/data science €6.9B; Decreto Sostegni-bis L. 106/2021 and Budget Law 2026 = L. 199/2025 extended

Updated July 2026