Skewness and kurtosis — what do they measure?

medium

Answer

$\mathrm{Skewness}\; = \;E[((X\; - \;{\mu}) / {\sigma})^{3}]$
$Positive: long right tail (log-normal, income).$
$Negative: long left tail.$
$\mathrm{Kurtosis}\; = \;E[((X\; - \;{\mu}) / {\sigma})^{4}]$
$Normal has kurtosis 3 (excess kurtosis 0).$
$Higher (leptokurtic): heavier tails, more outliers (t, Laplace, financial returns).$
$Lower (platykurtic): lighter tails than normal.$
$Both give quick diagnostic beyond mean/variance.$

Check yourself — multiple choice

Only variance
Skew: asymmetry (3rd standardized moment); kurtosis: tailedness (4th); normal has kurtosis 3
Same as variance
Not useful

Skew (3rd moment): asymmetry; Kurtosis (4th): tailedness; normal ref = 3.

#moments#descriptive

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