Entered returns · location, spread and shape

Return Distribution Analyzer

Summarize ordered equal-frequency returns with NIST population moments and an explicit N plus one percentile convention, including quartiles, skewness and kurtosis.

Runs in your browserPopulation moments disclosedNo normality claim

Enter observed equal-frequency returns

Use signed percentage returns from one consistent sampling and net-or-gross convention. Order is retained as entered, while percentile ranks use an ascending copy.

Entered

Enter percentage numbers without percent signs. Separate with spaces, commas, semicolons or new lines. Maximum 500.

Distribution boundary: This page describes the entered sample only. Skewness and kurtosis do not establish a probability model, predict tail losses or validate a strategy.

Entered return distribution arithmetic

Entered Strategy Diagnostics 1.0.0.

Derived
No return distribution calculated yetEnter at least four non-identical returns, or load the audited example.

How the return distribution is summarized

Skewness = (Σ(Y[i] − Ȳ)³ ÷ N) ÷ σ³
Kurtosis = (Σ(Y[i] − Ȳ)⁴ ÷ N) ÷ σ⁴
Percentile position = p(N + 1), linearly interpolated

Version 1.0.0 follows the NIST unadjusted Fisher-Pearson skewness and original kurtosis definitions. Their standard deviation uses N in the denominator; excess kurtosis is also shown as original kurtosis minus three.

Percentiles use the NIST p times N plus one rank with linear interpolation between adjacent ordered observations. Positions beyond the available endpoints are clipped to the entered minimum or maximum.

Assumptions and limits

  • Enter 4 to 500 signed percentage returns under one consistent sampling convention.
  • A constant series is rejected because standardized shape moments divide by zero spread.
  • Different software may use adjusted skewness, excess-only kurtosis or another percentile definition.
  • Extreme observations can materially change the mean, standard deviation, skewness and kurtosis.
  • No normality, tail-risk forecast, expected shortfall, strategy grade or recommendation is produced.

Worked example from the audited fixture

For the audited 30-return fixture, the mean is 0.110000%, the median is 0.150000%, the minimum is −1.200000% and the maximum is 1.500000%.

P10 = −0.890000%; Q1 = −0.600000%; Q3 = 0.825000%; P90 = 1.190000%
IQR = 1.425000%; population standard deviation = 0.770000%
Skewness = 0.097500; kurtosis = 1.822869; excess kurtosis = −1.177131

How to interpret the result

The percentiles use the disclosed p(N + 1) interpolation, while skewness and kurtosis use population moments with N in the denominator. The small positive skew and below-three kurtosis describe only these 30 values; they do not prove a distribution or predict tail losses.

Frequently asked questions

  • Enter 4 to 500 non-identical signed percentage returns under one consistent sampling and net-or-gross convention.
  • Version 1.0.0 uses the NIST unadjusted Fisher-Pearson population-moment formula with N-divisor standard deviation.
  • The fourth population central moment is divided by population standard deviation to the fourth power.
  • The original NIST convention has a normal reference of three; excess kurtosis subtracts three and therefore has a normal reference of zero.
  • The page uses the NIST p times N plus one rank, linear interpolation and disclosed clipping at the entered endpoints.
  • Standardized shape moments divide by zero spread, so the calculation is rejected.
  • Programs may use adjusted skewness, excess-only kurtosis or another percentile rank and interpolation convention.
  • No. It describes the entered sample and does not prove normality, forecast losses, validate a strategy or make a recommendation.

Sources and methodology

Return Volatility CalculatorCompare the existing sample-deviation and annualization contract.Sharpe Ratio CalculatorNormalize entered excess returns by sample deviation separately.Sortino Ratio CalculatorCompare entered target downside under its explicit denominator convention.

Compare the trading records behind your sample

Keep one broker account, return convention, fee treatment and sampling rule across the entered observations before comparing arithmetic.

XM

Review available account statements, instruments and trading terms.

Check XM terms

FBS

Compare account-history exports and cost conventions for your region.

Check FBS terms

FXOpen

Confirm account statements and trading conditions behind the entered sample.

Check FXOpen terms

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Disclaimer: The results from this tool are estimates for educational and informational purposes only and may differ from your broker's figures. This is not financial or investment advice. Trading forex and CFDs carries a high level of risk and can result in the loss of all your capital. Always verify calculations with your broker and trade within your risk tolerance.