Measure of Distribution Shape - (Skewness & Kurtosis)

Measures of Distribution Shape describe how data are distributed around the centre — whether they are symmetrical, skewed, or have heavy or light tails.

They help you understand the pattern of the data distribution beyond just the mean and standard deviation.

There are two main shape measures:

1. Skewness – describes the direction of asymmetry.
2. Kurtosis – describes the tailedness or peakedness.

Skewness

Skewness measures the asymmetry of a distribution around its mean.

• A symmetrical distribution (like a normal curve) has skewness ≈ 0.
• A positively skewed distribution has a longer right tail (mean > median).
• A negatively skewed distribution has a longer left tail (mean < median).

Interpretation:

• Symmetrical
• • Value: ≈ 0
  • Tail direction: None
  • Relationship: mean ≈ median ≈ mode
• Positively Skewed (Right)
• • Value: ≈ > 0
  • Tail direction: Right tail longer
  • Relationship: mean > median > mode
• Negatively Skewed (Left)
• • Value: ≈ < 0
  • Tail direction: Left tail longer
  • Relationship: mean < median < mode

Kurtosis

Kurtosis measures the tailedness of a distribution — how peaked or flat it is compared to a normal distribution.

It indicates whether data have:

• Heavy tails (more outliers)
• Light tails (fewer outliers)

Types of Kurtosis (using excess kurtosis):

• Mesokurtic
• • Value: 0
  • Shape: Normal shape
  • Interpretation: Moderate tails
• Leptokurtic
• • Value: > 0
  • Shape: Sharper peak, heavy tails
  • Interpretation: More outliers
• Platykurtic
• • Value: < 0
  • Shape: Flatter peak, light tails
  • Interpretation: Fewer outliers

How Skewness and Kurtosis Work Together

Visual example

In Machine Learning / Data Science

• Skewness helps detect when features are not symmetric — you might need transformations like
• • log(x), sqrt(x), or Box-Cox to make them more normal.
• Kurtosis helps check for outliers or extreme values — high kurtosis means heavy tails.
• Many algorithms (e.g., linear regression, PCA) assume normally distributed features, so understanding shape is essential.

Python Example

Output:

Interpretation:

• Skewness > 0 → distribution is right-skewed (outlier on the right).
• Kurtosis > 0 → distribution is leptokurtic (heavy tails, peaked).