Kurtosis measures the peakedness or flatness of a frequency distribution relative to a normal distribution.
## Core concept
Kurtosis describes the shape of the tail and concentration of values around the mean in a distribution, independent of skewness. While variance and standard deviation measure spread, kurtosis focuses on how the distribution peaks or flattens.
- Mesokurtic: Distribution with kurtosis similar to a normal distribution (baseline reference).
- Leptokurtic: Sharp peak, heavy tails, high concentration around mean (kurtosis > 0).
- Platykurtic: Flat top, light tails, values more uniformly distributed (kurtosis < 0).
Kurtosis is calculated as the fourth central moment divided by the fourth power of standard deviation.
## Formula / rule
Kurtosis (β₂):
$$\beta_2 = \frac{m_4}{(m_2)^2}$$
Where: - m₄ = Fourth central moment = $\frac{\Sigma f(X - \bar{X})^4}{N}$ - m₂ = Second central moment = Variance (σ²) - (m₂)² = (Variance)²
Excess Kurtosis (γ₂):
$$\gamma_2 = \beta_2 - 3$$
- For normal distribution: β₂ = 3, so excess kurtosis = 0
- This adjustment makes comparison to normal distribution intuitive
Step-by-step calculation:
- Calculate mean ($\bar{X}$)
- Find deviations: (X − $\bar{X}$)
- Raise deviations to the fourth power and multiply by frequency
- Sum all fourth powers and divide by N → m₄
- Calculate variance (σ²) = m₂
- Apply formula: β₂ = m₄ ÷ (σ²)²
- For excess kurtosis: γ₂ = β₂ − 3
## Common exam applications
- Interpreting distribution shape: Exam asks to classify a given distribution using calculated kurtosis value.
- Comparing two datasets: Determine which distribution has heavier tails or sharper peak.
- Complete statistical analysis: After calculating skewness, standard deviation, and quartiles, kurtosis completes the shape description.
- Financial/risk analysis: Understanding tail risk—leptokurtic distributions indicate extreme values more likely (practical relevance for economics-based questions).
## Worked example
Data: 10, 15, 20, 25, 30 (frequencies: 2, 3, 4, 3, 2)
- Mean ($\bar{X}$) = 20
- Variance (m₂) = 25; thus σ² = 25
| X | f | (X − 20) | (X − 20)⁴ | f(X − 20)⁴ | |---|---|---------|----------|-----------| | 10 | 2 | −10 | 10,000 | 20,000 | | 15 | 3 | −5 | 625 | 1,875 | | 20 | 4 | 0 | 0 | 0 | | 25 | 3 | 5 | 625 | 1,875 | | 30 | 2 | 10 | 10,000 | 20,000 |
m₄ = (20,000 + 1,875 + 0 + 1,875 + 20,000) ÷ 14 = 43,750 ÷ 14 ≈ 3,125
β₂ = 3,125 ÷ (25)² = 3,125 ÷ 625 = 5
γ₂ = 5 − 3 = 2 (Excess kurtosis is positive → leptokurtic, sharp peak)
## Common mistakes
- Confusing skewness with kurtosis: Skewness measures asymmetry (left/right); kurtosis measures peakedness (up/down shape).
- Using variance instead of variance squared: m₄ must be divided by (σ²)², not σ².
- Ignoring excess kurtosis adjustment: Always subtract 3 from β₂ for meaningful interpretation relative to normal distribution.
- Misinterpreting excess kurtosis sign: Positive = sharper/heavier tails; negative = flatter distribution.