Skew (Skewness)
Skewness describes the asymmetry of a data distribution, indicating whether the tail is longer on the left or right side.
Definition
Skew describes whether a data distribution is lopsided: a right skew has a long tail on the right, a left skew has a long tail on the left, and a symmetric distribution has no skew. Positive (right) skew has mean > median > mode, while negative (left) skew has mean < median < mode. Formally, skewness is the standardized third central moment: $\gamma_1 = \mu_3/\sigma^3$, where $\mu_3 = E[(X-\mu)^3]$; the sample skewness estimator is $g_1 = m_3/m_2^{3/2}$, where $m_k$ is the $k$-th sample central moment.
Example
Home prices are right-skewed: most homes are moderately priced, but a few very expensive mansions create a long tail on the right. Annual incomes are positively skewed too: most people earn moderate incomes, but a small number of high earners create a long right tail, pulling the mean above the median (which is why the Census reports median household income). For an exponential distribution with rate $\lambda$, skewness $= 2$ (always positively skewed); for a log-normal distribution $LN(\mu, \sigma^2)$, skewness $= (e^{\sigma^2}+2)\sqrt{e^{\sigma^2}-1}$, increasing with $\sigma$, which explains why log-transforming right-skewed data often achieves approximate normality.
Key Insight
When a distribution is skewed, the mean gets pulled toward the tail (where the unusual values are) while the median stays closer to the bulk of data, so the median is more reliable for skewed data; a useful memory trick is that skew is named for the direction of the tail, not the bulk (a right-skewed distribution has its bulk on the left and tail on the right). The Jarque-Bera test for normality uses both skewness $g_1$ and excess kurtosis $g_2$: $JB = n(g_1^2/6 + g_2^2/24)$, which follows a chi-squared distribution with $2$ degrees of freedom under normality, providing a formal test for departure from the normal distribution.