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Key Takeaways
- Skewness shows which side of the mean the large outcomes come from: negative for the left tail, positive for the right.
- Negative skewness means the largest deviations sit below the mean, the shape of a short out-of-the-money option’s returns, so size the position for the left tail.
- Positive skewness means the largest deviations sit above the mean, the shape of a long out-of-the-money option’s payoff.
- The standard estimate cubes the deviation from the mean, which makes rare extreme observations dominate the number.
- Read skewness with standard deviation, tail quantiles and the trade’s payoff, and compare the underlying’s historical skew with the volatility skew its options imply.
What is skewness?
Skewness is a measure of the asymmetry of a distribution, calculated as the standardized third moment: the average cubed deviation from the mean divided by the standard deviation cubed. Negative skewness means negative cubed deviations dominate, as left-tail outliers do; positive skewness means the positive contributions dominate. For returns, the sign shows which side of the mean the large outcomes mainly come from.
The statistic is unit-free. Adding a constant leaves it unchanged. Multiplying by a positive constant leaves it unchanged, while multiplying by a negative constant reverses its sign.
How is skewness calculated?
For n observations, define m2 as the average squared deviation with divisor n and m3 as the average cubed deviation with the same divisor. Fisher-Pearson skewness is g1 = m3 / m2^(3/2).
The adjusted coefficient is G1 = √[n(n − 1)] / (n − 2) × g1 for n > 2 and positive variance. Keep the adjustment and the divisor-n variance together, because mixing conventions gives the wrong number.
Formula
m_k = (1 / n) × Σ(R_i − Rbar)^k; Fisher-Pearson g1 = m3 / m2^(3/2). Adjusted G1 = √[n(n − 1)] / (n − 2) × g1, for n > 2 and m2 > 0.- Rᵢ is each observed return, R̄ is the sample mean return, and N is the number of observations.
- The unadjusted Fisher-Pearson formula uses variance m2 with divisor n, and the adjusted coefficient multiplies by the sample-size factor; plug in a divisor n − 1 standard deviation and the result comes out wrong.
- Cubing the deviations preserves sign, so left-tail outliers push the statistic negative and right-tail outliers push it positive.
Skewness example: 146 return observations
Use the chart’s 146 observations at the displayed return values. Returns −12%, −10%, −8%, −6%, −4%, −2%, 0%, 2%, 4% and 6% have counts 1, 2, 3, 5, 9, 16, 30, 42, 28 and 10.
- Mean return: 0.7%; median and mode: 2%
- Variance using divisor 146: 12.53 percentage-points squared; standard deviation: 3.54 percentage points
- Unadjusted Fisher-Pearson skewness g1: −1.03
- Adjusted Fisher-Pearson skewness G1: −1.04
The 146 returns skew negative: rare losses as deep as −12% sit far below the mean, while the gains cluster near it. In this sample the typical period returns 2% and the mean sits at 0.7%, so the left tail is the risk to size for.
The ten displayed return values have counts 1, 2, 3, 5, 9, 16, 30, 42, 28 and 10, totaling 146 observations. Treat each displayed value as the exact return for its count.
The mean is 0.7%, the median and mode are 2%, and the divisor-n standard deviation is 3.54 percentage points. Unadjusted skewness is −1.03; adjusted skewness is −1.04.
The losses do the work. The −12% to −6% returns sit far enough below the mean that their cubes outweigh all the gains combined.
Skewness vs win rate, mean-median order and symmetry
Calculate the win rate and the mean-median order from the returns themselves. Shifting every return upward changes the fraction of positive returns without changing skewness at all.
A symmetric distribution with a finite third moment has zero skewness, and a lopsided distribution also reads zero when its cubed deviations cancel. Values −2, −1, 1 and 3 with probabilities 20%, 25%, 50% and 5% have mean zero and third moment zero, yet the distribution is asymmetric.
For returns with tails so fat that the third moment is infinite, the sample coefficient still prints a number, and it jumps with every new extreme, so read it as a description of the sample in hand.
Skewness vs volatility and volatility skew
Read skewness with standard deviation, empirical tail quantiles and the payoff distribution. A sign-reversed return series has unchanged dispersion and reversed skewness.
Historical return skewness summarizes past returns, while volatility skew shows how option prices charge for the tails across strikes today. A position has its own skew, and the Options Profit Calculator draws it: the payoff across price, date and volatility shows which tail holds the large gains and which holds the large losses.
Limitations of skewness
- Recheck the estimate when an outlier enters or leaves the window. Cubing the deviations lets a few extreme observations dominate, so one new crash day moves the number sharply.
- Compare like with like: g1 with g1, or G1 with G1, on samples of similar length.
- Measure expected return and win rate directly. The sign tells you which side the large deviations fall on, and the mean, median and win rate come from the returns themselves.
- Plot the returns before you call them symmetric. A lopsided distribution reads zero when its cubed deviations cancel, and fat-tailed returns with an infinite third moment have no population skewness at all.
- Price the tail from the option chain. Historical skewness covers the past sample, and option-implied volatility skew shows what the market charges for tail risk today.
Bottom line
Skewness shows whether a return distribution’s large deviations sit below or above the mean, which tells you which tail to size for. Use it with standard deviation, tail quantiles and the actual payoff, compare it across the same estimator and sample size, and check the option chain’s volatility skew for what the market charges for that tail.
Clear answers
Frequently Asked Questions
Sources and Further Reading
- NIST/SEMATECH e-Handbook of Statistical Methods: Measures of Skewness and Kurtosis
- Federal Reserve: Stock Market Cross-Sectional Skewness and Business Cycle Fluctuations
- CFA Institute: How Sharp Is the Sharpe Ratio? An Analysis of Global Stock Indices
How entries are sourced, calculated and updated: editorial methodology.