Formula, in Risk & Portfolio Construction
Skewness Formula
Skewness is the third standardized moment of a return distribution: the average cubed deviation from the mean, divided by the cube of the standard deviation. It measures asymmetry. Negative skew means a long left tail (occasional large losses), which most strategies exhibit and which deflates risk-adjusted scores that assume symmetry.
4 variables. Published May 26, 2026. By AI Fin Hub Research.
Skew = ( (1/n) x sum_i (R_i - mean_R)^3 ) / sigma^3
where sigma = population standard deviation (divisor n) Variables
R_i
Period return
Individual return observation in the sample.
mean_R
Mean return
Average of the returns, the center from which deviations are measured.
(R_i - mean_R)^3
Cubed deviation
Deviation from the mean raised to the third power. Cubing preserves sign, so large positive deviations and large negative deviations push the result in opposite directions; the imbalance between them is the skew.
sigma^3
Cube of standard deviation
Standardizes the third moment so skewness is unitless and comparable across assets. Using the population standard deviation (divisor n) matches the population skewness definition.
Step by step
- 1
Compute the mean and the population standard deviation of the returns.
Returns -10%, +2%, +3%, +5% have mean 0% and standard deviation computed below.
- 2
For each return, cube its deviation from the mean and average the cubes.
Deviations -0.10, 0.02, 0.03, 0.05; cubes -0.001, 0.000008, 0.000027, 0.000125; mean = -0.00021000.
- 3
Compute the cube of the population standard deviation.
Variance = mean of squared deviations = (0.01 + 0.0004 + 0.0009 + 0.0025)/4 = 0.00345; sigma = 0.05874; sigma^3 = 0.0002027.
- 4
Divide the mean cubed deviation by sigma cubed.
-0.00021000 / 0.0002027 = -1.036.
Worked example
Detecting left-tail risk in a four-return sample
Returns
-10%, +2%, +3%, +5%
Mean
0%
Deviations equal the returns (mean is 0). Cubes: (-0.10)^3 = -0.001, (0.02)^3 = 0.000008, (0.03)^3 = 0.000027, (0.05)^3 = 0.000125. Sum = -0.00084; mean = -0.00021. Variance = (0.01 + 0.0004 + 0.0009 + 0.0025)/4 = 0.01380/4 = 0.00345; sigma = 0.058737; sigma^3 = 0.00020266. Skewness = -0.00021 / 0.00020266 = -1.036.
Skewness of about -1.04, a clear negative skew. The single -10% return drives the long left tail: most outcomes are small positives, but the rare loss is large. Strategies with this profile (selling options, carry trades) look attractive on mean and Sharpe yet hide concentrated downside, which is exactly why the deflated Sharpe ratio penalizes negative skew.
Common variations
Run the numbers next
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Paste a returns CSV. Sharpe, Sortino, Calmar, Omega, alpha, beta, tracking error, information ratio, max drawdown, and tail moments — plus.
- Calculators Sharpe Ratio Calculator
Paste daily returns; get Sharpe, Sortino, Calmar, and Omega side-by-side with a recommendation on which ratio fits your distribution.
Sources
- Biometry: The Principles and Practice of Statistics in Biological Research , Robert R. Sokal and F. James Rohlf, W. H. Freeman (1995)
- The Statistics of Sharpe Ratios , Andrew W. Lo, Financial Analysts Journal (2002)
Keep the topic connected
- Formula Kurtosis Formula
The kurtosis formula: the fourth standardized moment of returns. Measures tail heaviness, with excess kurtosis relative to the normal, and an example.
- Formula Deflated Sharpe Ratio Formula
The deflated Sharpe ratio formula: the probability a strategy's Sharpe is real after correcting for the number of trials, return skew, kurtosis, and sample length.
- Formula Downside Deviation Formula
The downside deviation formula: root mean square of returns below a target, over total observations. The risk term inside the Sortino ratio.
- Learn Volatility
Volatility as the standard deviation of returns: realized vs implied, the annualization gotcha, and why volatility-of-volatility matters.