Skew (Skewness)
Statistics & ProbabilitySkewness 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. A symmetric distribution has no skew.
Example
Home prices are right-skewed: most homes are moderately priced, but a few very expensive mansions create a long tail on the right side of the distribution.
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. Median is more reliable for skewed data.
Definition
Skewness measures the asymmetry of a distribution. Positive (right) skew: the right tail is longer; mean > median > mode. Negative (left) skew: the left tail is longer; mean < median < mode. A symmetric distribution has skewness = 0.
Example
Annual incomes are positively skewed: most people earn moderate incomes, but a small number of high earners create a long right tail. Mean income is pulled above the median by these high earners. The Census reports median household income for this reason.
Key Insight
An easy memory trick: the skew is named for the direction of the tail, not the bulk of the data. A right-skewed distribution has its bulk on the left and tail on the right.
Definition
The skewness of a distribution is the standardized third central moment: $\gamma_1 = \mu_3/\sigma^3$, where $\mu_3 = E[(X-\mu)^3]$. Positive skewness indicates a longer right tail; negative indicates a longer left tail. The sample skewness estimator is $g_1 = m_3/m_2^{3/2}$, where $m_k$ is the $k$-th sample central moment.
Example
For an exponential distribution with rate $\lambda$, skewness $= 2$ (always positively skewed). For a log-normal distribution $LN(\mu, \sigma^2)$, skewness $= (e^{\sigma^2}+2)\sqrt{e^{\sigma^2}-1}$, which increases with $\sigma$. This explains why log-transforming right-skewed data often achieves approximate normality.
Key Insight
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.