Understanding Skewness and Kurtosis

#ai #discuss

When analyzing datasets, it’s not enough to know measures of central tendency (mean, median, mode) and variability (variance, standard deviation).

Skewness: The Measure of Asymmetry

Definition: Skewness measures the degree and direction of asymmetry in a distribution around its mean.
Formula:

Skewness=(n1∑i=1n(xi−xˉ)2)3/2n1∑i=1n(xi−xˉ)3

Real-Life Example:

Income distribution → Often positively skewed because most people earn average wages, but a small number of high earners stretch the tail to the right.
Exam scores → If most students score high but a few fail badly, the distribution is negatively skewed.

Kurtosis: The Measure of Tailedness

Definition: Kurtosis measures the heaviness of tails in a distribution compared to a normal distribution.
Formula:

Kurtosis=(n1∑i=1n(xi−xˉ)2)2n1∑i=1n(xi−xˉ)4

A normal distribution has kurtosis ≈ 3 (called mesokurtic).
To make interpretation easier, analysts often use excess kurtosis = kurtosis – 3.

Real-Life Example:

Stock returns → Usually leptokurtic (heavy-tailed). This means extreme ups and downs occur more frequently than in a normal curve.
Heights of people → Typically close to mesokurtic, since extreme deviations are rare.
Uniform distribution → Often platykurtic (light-tailed), with fewer outliers.

Key Difference

Skewness → Tells us about the direction of data spread (left, right, or symmetric).
Kurtosis → Tells us about the intensity of tails (normal, heavy, or light).