Skewness (Third Central Moment)

• is a measure of symmetry or asymmetry of the probability distribution of a real-valued random variable about its mean
• value can be positive, negative, 0, or undefined
  • skewness 0 indicates data is symmetric (e.g. uniform distribution, normal distribution, etc)
  • negative skew indicates data are skewed left
  • positive skew indicates data are skewed right

Fisher-Pearson Coefficient of Skewness

For a random variable 𝑋:

• population skewness = 𝐄[(𝑋- 𝜇)³]

where:

• 𝜇 - population mean

For univariate dataset {𝑋₁, …, 𝑋_𝑛}:

• sample skewness = 𝑔 = 𝐄[(𝑋_𝑖- 𝑋̅)³] = [𝛴_{1≤𝑖≤𝑛}(𝑋_𝑖- 𝑋̅)³] / [𝑠³𝑛]

where:

• 𝑋̅ - sample mean
• 𝑠 - sample standard deviation

Adjusted Fisher-Pearson Coefficient of Skewness

• 𝐺 = [√[𝑛(𝑛 - 1)] / (𝑛 - 2)] · 𝑔

Pearson 2 Skewness Coefficient

• 𝑆_𝑘2= 3 (𝑋̅ - 𝑀̅) / 𝑠

where:

• 𝑀̅ - sample median