Kurtosis – Meaning & Types (Mesokurtic, Leptokurtic, Platykurtic)

Kurtosis helps us understand the peakedness or flatness of a distribution compared to a normal curve. While skewness tells us direction, kurtosis tells us shape — especially whether the distribution has heavy tails, light tails, or normal tails.

Meaning of Kurtosis

Kurtosis measures the degree of concentration of values around the mean. In simpler words:

• Is the graph tall and sharp?
• Or flat and wide?
• Or normal bell-shaped?

A distribution with many extreme values has heavy tails, leading to high kurtosis.

Types of Kurtosis

Kurtosis is generally classified into three categories, depending on the shape of the curve.

1. Mesokurtic (Normal Peak)

• Represents a normal distribution.
• Neither too peaked nor too flat.
• Often used as the reference curve.
• Value of kurtosis (β₂) ≈ 3.

2. Leptokurtic (High Peak)

• Curve is more peaked than the normal distribution.
• Indicates clustered values near the mean.
• Heavy tails: more extreme values.
• Kurtosis (β₂) > 3.

3. Platykurtic (Flat Peak)

• Curve is flatter than the normal distribution.
• Values are more spread out.
• Light tails: fewer extreme values.
• Kurtosis (β₂) < 3.

Visual Summary

         Leptokurtic
             /\
            /  \
           /    \
 Normal   /      \
 Mesokurtic        \
         Platykurtic (flatter)

Formula & Components

Karl Pearson’s measure of kurtosis uses the fourth moment:

β₂ = μ₄ / μ₂²

Where:

• μ₄ = fourth central moment
• μ₂ = second central moment (variance)

Interpretation using excess kurtosis:

Excess Kurtosis = β₂ - 3

• = 0 → mesokurtic

• 0 → leptokurtic

• < 0 → platykurtic

Steps to Calculate Kurtosis

1. Compute the mean.
2. Compute deviations (X - mean).
3. Compute second moment (μ₂) and fourth moment (μ₄).
4. Apply the formula β₂ = μ₄ / μ₂².
5. Compare with 3 to identify type.

Solved Example (Ungrouped Data)

Data: 2, 4, 6, 8, 10

Step 1: Mean

Mean = (2+4+6+8+10) / 5 = 6

Step 2: Deviations

Step 3: Compute μ₂ and μ₄

μ₂ = (16 + 4 + 0 + 4 + 16) / 5 = 8
μ₄ = (256 + 16 + 0 + 16 + 256) / 5 = 108.8

Step 4: Apply formula

β₂ = 108.8 / (8²)
    = 108.8 / 64
    = 1.70

Step 5: Interpretation

β₂ = 1.70 < 3 → Platykurtic distribution.

Practical Interpretation

Important Notes

• Kurtosis measures shape, not skewness.
• Leptokurtic → high peak = many similar values + heavy tails.
• Platykurtic → flat peak = values widely spread.
• Mesokurtic → reference normal distribution.
• Uses fourth moment, making it sensitive to extreme deviations.

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