Definition:Skewness/Coefficient

Definition

Let $X$ be a random variable with mean $\mu$ and standard deviation $\sigma$.

The coefficient of skewness of $X$ is the coefficient:

\(\ds \gamma_1\)

\(=\)

\(\ds \dfrac {\mu_3} { {\mu_2}^{3/2} }\)

\(\ds \)

\(=\)

\(\ds \expect {\paren {\dfrac {X - \mu} \sigma}^3}\)

where:

$\mu_i$ denotes the $i$th central moment of $X$

$\mu$ denotes the expectation of $X$, that is, its first central moment

$\sigma$ denotes the standard deviation of $X$, that is, the square root of its second central moment.

Notation

Various notations can be found to denote the coefficient of skewness.

$\mathsf{Pr} \infty \mathsf{fWiki}$ uses the notation $\gamma_1$, which originates from Karl Pearson.

Other notations that may be encountered are:

$c_1$

$\alpha_3$

$\sqrt {\beta_1}$

depending on author.

Also see

• Results about skewness can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): coefficient of skewness
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): skewness
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): coefficient of skewness
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): skewness