Skewness - Definition

Definition

The skewness of a random variable X is the third standardized moment, denoted γ₁ and defined as

$\gamma_1 = \operatorname{E}\Big = \frac{\mu_3}{\sigma^3} = \frac{\operatorname{E}\big}{\ \ \ ( \operatorname{E}\big )^{3/2}} = \frac{\kappa_3}{\kappa_2^{3/2}}\,$

where μ₃ is the third moment about the mean μ, σ is the standard deviation, and E is the expectation operator. The last equality expresses skewness in terms of the ratio of the third cumulant κ₃ and the 1.5th power of the second cumulant κ₂. This is analogous to the definition of kurtosis as the fourth cumulant normalized by the square of the second cumulant.

The skewness is also sometimes denoted Skew.

The formula expressing skewness in terms of the non-central moment E can be expressed by expanding the previous formula,

$\begin{align} \gamma_1 &= \operatorname{E}\bigg \\ & = \frac{\operatorname{E} - 3\mu\operatorname E + 3\mu^2\operatorname E - \mu^3}{\sigma^3}\\ &= \frac{\operatorname{E} - 3\mu(\operatorname E -\mu\operatorname E) - \mu^3}{\sigma^3}\\ &= \frac{\operatorname{E} - 3\mu\sigma^2 - \mu^3}{\sigma^3}\ . \end{align}$

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