Jarque–Bera Normality Test Calculator
Calculate the Jarque–Bera statistic from one entered observation count, sample skewness and raw kurtosis. The page shows the NIST large-sample chi-square reference only when n is at least 2,000, withholds unsupported smaller-sample p-values, and never turns the output into a normality verdict or trading signal.
Enter moments from one return sample
Use observation count, skewness and Pearson/raw kurtosis calculated from the same return series, sampling frequency and preprocessing convention.
Whole sample length. The chi-square p-value is shown only from n = 2,000.
Skewness calculated from the same entered sample convention.
Pearson/raw kurtosis, where the normal reference is 3—not excess kurtosis.
Entered Jarque–Bera arithmetic
Entered Return Statistical Diagnostics 1.0.0.
On smaller screens, scroll horizontally to inspect the complete audit table.
| Quantity | Value | Role in this calculation |
|---|
How the Jarque–Bera reference is calculated
For n ≥ 2,000 only: p = P(χ² with 2 df ≥ JB) = e−JB ÷ 2
Here n is the entered sample length, S is sample skewness and K is Pearson/raw kurtosis, where the normal reference is 3. The two moment departures are squared, so positive and negative skew contribute symmetrically.
NIST documents a two-degree-of-freedom chi-square comparison only for samples of at least 2,000 observations and uses 100,000 normal simulations below that boundary. Version 1.0.0 therefore reports the statistic at every supported n but withholds the asymptotic p-value when n is smaller.
A p-value, when available, is an asymptotic reference under the stated null and conventions. It is not the probability that the data are normal, the probability that a strategy is valid, or a forecast of future returns.
Worked example from the audited fixture
The published NIST example has n = 195, skewness = −0.0085 and raw kurtosis = 3.049.
- S² = 0.00007225 and (K − 3)² ÷ 4 = 0.00060025.
- JB = 195 ÷ 6 × (0.00007225 + 0.00060025) = 0.02185625, displayed as 0.0219 at four decimals. Because n is below 2,000, this calculator withholds a chi-square p-value instead of reproducing NIST’s separate simulation.
Reproduce it: select “Load audited example” above. The governed engine retains full precision and rounds only the visible interface.
How to interpret the result
- The statistic is zero only when entered skewness is zero and raw kurtosis is exactly 3.
- A larger statistic means the entered moment departures contribute more after sample-size scaling; it does not identify the economic cause.
- The separate skewness and kurtosis components reveal which entered moment contributes to the total.
- Do not read “p-value withheld” as evidence for or against normality; it marks a methodology boundary.
- Even an available large-sample reference does not test stationarity, serial dependence, tail stability, data selection or future performance.
Assumptions and limits
- The calculator accepts summary statistics and cannot verify how skewness, kurtosis or sample size were estimated.
- Different software can use bias-adjusted skewness, excess kurtosis or another sample-moment convention.
- Raw kurtosis must use the normal reference 3; entering excess kurtosis would change the statistic.
- Autocorrelation, volatility clustering and structural breaks can make a normal-reference diagnostic incomplete.
- Large samples can make small moment departures produce large statistics without establishing economic importance.
- No normal/not-normal label, significance decision, strategy grade, forecast, signal or recommendation is generated.
Which return diagnostic answers which question?
Distribution shape, one-lag autocorrelation, joint residual autocorrelation and multi-period variance are related but not interchangeable. The comparison below preserves each tool’s evidence requirement and prevents one statistic from being presented as a universal strategy test.
| Diagnostic | Evidence entered | Question answered | Main boundary |
|---|---|---|---|
| Return Distribution Analyzer | Entered return series | Location, spread, percentiles, skewness and raw kurtosis | Descriptive moments only. |
| Jarque–Bera | Entered n, skewness and raw kurtosis | Joint normal-reference moment statistic | Small-sample p-value withheld; no normality verdict. |
| Return Autocorrelation | Entered return series plus one lag | One sample autocorrelation coefficient | No joint multi-lag reference. |
| Ljung–Box | Ordered residuals, maximum lag and fitted orders | Joint asymptotic portmanteau reference | No residual-independence or adequacy verdict. |
| Observed Variance Ratio | Entered returns plus horizon q | Overlapping multi-period variance divided by scaled one-period variance | No corrected z test, p-value or random-walk verdict. |
Frequently asked questions
- It combines entered squared skewness and squared raw-kurtosis departure from 3, then scales that sum by the entered sample length divided by six.
- Enter Pearson or raw kurtosis, where a normal distribution has reference value 3. Do not enter excess kurtosis, whose normal reference is zero.
- NIST uses a 100,000-draw normal simulation below that boundary. Version 1.0.0 does not simulate, so it withholds rather than invents the smaller-sample reference.
- At n of at least 2,000, the statistic is compared with chi-square having two degrees of freedom; its survival probability is exp minus JB divided by two.
- No. Observation count, skewness and raw kurtosis must describe the same return series, frequency, preprocessing and estimator convention.
- No single statistic proves a distribution. The asymptotic reference is conditional on assumptions and does not diagnose dependence, structural breaks or economic causes.
- No. Failure to show a large moment departure under one reference is not proof that the full distribution is normal or stable.
- No. It creates no normality verdict, verified edge, strategy grade, forecast, signal, position instruction or recommendation.
Sources and methodology
- NIST — Jarque–Bera Test — Published formula, sample-size boundary, simulation treatment and worked example.
Version 1.0.0 was locked only after the governing formulas and boundaries were checked in the cited primary or standards sources. Independent fixtures recompute the displayed statistics and reference probabilities separately from the browser adapter.
Continue the entered-return review
Verify the return and residual evidence
Reconcile the exact statement period, sampling frequency, timezone, realized results, spread, commission, financing, conversion and fitted-model preprocessing before deriving inputs. These browser calculations cannot certify that an entered sample is complete, stationary or representative.
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