reference · Performance & testing
Skewness and Kurtosis: Inspect Return Shape and Extreme Values
Skewness describes asymmetry, while kurtosis measures a standardized fourth moment that is sensitive to extreme observations. Both require a stated formula and enough context to avoid reading a small sample as a reliable distribution.
Key points
- A skewness near zero does not prove normally distributed returns.
- Ordinary normal-distribution kurtosis is 3; excess kurtosis subtracts 3.
- Small samples and single extreme outcomes can dominate these estimates.
Use shape statistics alongside the data
Two return histories can have the same mean and standard deviation while differing in asymmetry and extreme outcomes. Skewness and kurtosis add information about shape, but a histogram, time plot and the actual large observations remain essential context. They are not substitutes for understanding how a strategy creates gains and losses.
NIST describes multiple skewness and kurtosis conventions. A basic moment skewness divides the mean cubed deviation by the population-moment standard deviation cubed. Ordinary kurtosis uses the fourth power instead; excess kurtosis subtracts 3. Sample-bias corrections can produce different software outputs.
Check an explicit symmetric example
Consider returns of −2%, −1%, +1% and +2%. The mean is zero. Using percentage-point units and population moments, the second moment is (4 + 1 + 1 + 4) ÷ 4 = 2.5. The third moment is (−8 − 1 + 1 + 8) ÷ 4 = 0, so moment skewness is zero.
The fourth moment is (16 + 1 + 1 + 16) ÷ 4 = 8.5. Ordinary moment kurtosis is 8.5 ÷ 2.5 squared = 1.36, and excess kurtosis is −1.64. This four-value illustration is a formula check, not evidence that a trading process has thin tails. A bias-adjusted sample function may report a different value because it answers a differently defined estimation problem.
Interpret signs without turning them into forecasts
Negative skewness can arise when the left tail contains relatively large negative outcomes; positive skewness can arise from relatively large positive outcomes. Multimodal or unusual distributions can complicate that shorthand. A strategy with many small wins and an occasional large loss deserves direct inspection of those losses rather than a judgment based only on the sign.
High kurtosis can reflect heavy tails or influential extreme observations relative to variance. Calling it simply “peakedness” loses useful information. A symmetric sample can still contain severe outcomes on both sides, so zero skewness does not imply low tail risk. Likewise, low observed kurtosis in a short quiet period cannot rule out a future jump.
Make comparisons reproducible
Document sample dates, frequency, cost treatment, formula variant and missing-data policy. Check that a program's “kurtosis” field means ordinary or excess kurtosis. Confirm whether standard deviation inside the formula uses n or n − 1 and whether a small-sample adjustment is applied.
A constant series has zero variance, making standardized higher moments undefined. Very short samples can also violate a software estimator's minimum observation count. For copied accounts, align returns and preserve extreme execution events instead of removing them as inconvenient outliers. A mistaken quote may need correction with evidence; an actual gap loss belongs in the analysis. These statistics describe observed shape and do not establish a stable future payoff distribution.
Questions and answers
Why does normal kurtosis appear as 0 in one program and 3 in another?
One may report excess kurtosis, which subtracts 3 from ordinary kurtosis. Verify the convention before comparing values.
Can a zero-skewness sample still be risky?
Yes. Symmetry can coexist with large positive and negative tails, and a short sample may miss important extreme events entirely.
Sources and further checks
Use the current source for your exact instrument, account and platform. Referencing a general specification does not establish support for every TradeCopier workflow.
- NIST: measures of skewness and kurtosis · Checked September 19, 2026
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