Calculate Kurtosis
Online calculator to calculate the kurtosis (peakedness) of a data distribution
Kurtosis Calculator
Kurtosis
Kurtosis is a measure of peakedness of a distribution. It describes the height of the peak and the heaviness of the tails.
Kurtosis Visualization
Kurtosis describes the shape of the distribution.
High kurtosis: Pointed peak, heavy tails (more outliers).
━ Platykurtic (flat) ━ Mesokurtic (normal) ━ Leptokurtic (peaked)
What is Kurtosis?
Kurtosis is an important measure in descriptive statistics for describing distribution shape:
- Definition: Measure of peakedness (peak height) and tail heaviness
- Also Called: Excess or fourth standardized moment
- Moment: Based on the fourth standardized moment
- Property: Describes concentration around mean and outlier risk
- Application: Risk management, model selection, financial analysis
- Interpretation: Comparison with normal distribution (κ=0)
Types of Kurtosis
Depending on the value, three types of kurtosis are distinguished:
Platykurtic
κ < 0: Flatter distribution than normal distribution. Fewer values in peak and tails.
Example: Uniform distribution, certain Beta distributions
Mesokurtic
κ = 0: Peakedness like normal distribution. Reference value for comparisons.
Example: Normal distribution (Gaussian distribution)
Leptokurtic
κ > 0: Sharper distribution than normal distribution. More values in peak and tails (outliers!).
Example: t-distribution, Laplace distribution, financial returns
Applications of Kurtosis
Kurtosis is used in many fields:
Finance
- Risk management (tail risk, extreme losses)
- Analyze return distribution
- Value-at-Risk (VaR) modeling
- Portfolio optimization (higher moments)
Data Analysis
- Describe distribution shape
- Normality testing (κ should ≈ 0)
- Outlier detection (high κ indicates many outliers)
- Model selection (e.g., t-distribution when κ > 0)
Formulas for Calculating Kurtosis
Sample Kurtosis (Excess, g₂)
Biased estimator - the -3 makes normal distribution equal to 0
Population Kurtosis (G₂) - Corrected
Unbiased estimator - for inferential statistics
Alternative: Without Excess Correction
Without -3, normal distribution has κ = 3
Relationship to Moments
m₄ = fourth central moment
Symbol Explanations
| \(g_2\) | Sample kurtosis (excess) |
| \(G_2\) | Population kurtosis |
| \(x_i\) | Individual data value |
| \(\overline{x}\) | Arithmetic mean |
| \(s\) | Standard deviation |
| \(n\) | Number of values |
Example Calculations for Kurtosis
Example 1: Nearly Normal-Distributed Data
Calculate: Kurtosis of the data
1. Mean & Standard Deviation
Standard deviation:
s ≈ 2.28
2. Z-Values to Fourth Power
| z₁⁴ = ((2-5.2)/2.28)⁴ ≈ 3.97 |
| z₂⁴ = ((5-5.2)/2.28)⁴ ≈ 0.001 |
| z₃⁴ = ((8-5.2)/2.28)⁴ ≈ 3.14 |
| z₄⁴ = ((7-5.2)/2.28)⁴ ≈ 0.98 |
| z₅⁴ = ((4-5.2)/2.28)⁴ ≈ 0.12 |
3. Calculate Kurtosis
Sum: 8.22
\[g_2 = \frac{8.22}{5} - 3\] \[\approx \color{blue}{-1.36}\]
Platykurtic!
Flatter than normal distribution
Example 2: Leptokurtic Distribution (With Outliers)
Data with extreme values at the tails
Statistics
| Mean: | 5.33 |
| Median: | 5.5 |
| Standard Deviation: | 5.46 |
| Number of Extreme Values: | 2 (15, -5) |
Most values lie at 5-6, but two extreme outliers
Kurtosis Calculation
After calculation:
g₂ ≈ 2.54
Highly leptokurtic!
The outliers (15 and -5) increase kurtosis greatly. This shows heavy tails - typical for financial returns!
Key Insight
Positive Kurtosis (g₂ = 2.54): Indicates that data contains more extreme values (outliers) than a normal distribution. This is important for risk management: high kurtosis means higher risk of extreme losses or gains. In finance, such distributions are often modeled with t-distributions or other fat-tail distributions.
Example 3: Platykurtic Distribution (Uniform)
Uniformly distributed data
Statistics
| Mean: | 5.5 |
| Median: | 5.5 |
| Standard Deviation: | 2.87 |
All values uniformly distributed, no concentration
Kurtosis Calculation
After calculation:
g₂ ≈ -1.22
Platykurtic!
Uniform distributions typically have negative kurtosis. Values are uniformly distributed without concentration in center or at tails.
Mathematical Foundations of Kurtosis
Kurtosis is the fourth standardized moment and describes the tail-heaviness (heaviness of tails) and peakedness (peak height) of a distribution.
Excess Kurtosis vs. Kurtosis
Two Definitions:
- Kurtosis (κ): κ = E[(X-μ)⁴]/σ⁴, normal distribution has κ = 3
- Excess Kurtosis (g₂): g₂ = κ - 3, normal distribution has g₂ = 0
Advantage of Excess: The -3 correction makes normal distribution the reference point (0). Positive values show more concentration and heavier tails than normal distribution.
Kurtosis Interpretation
g₂ < 0
Platykurtic
Flat peak
Light tails
Fewer outliers
g₂ ≈ 0
Mesokurtic
Like normal distribution
Reference value
Normal outlier rate
g₂ > 0
Leptokurtic
High peak
Heavy tails
More outliers
Kurtosis in Finance
Why is kurtosis important for finance?
- Fat Tails: Financial returns often have g₂ > 0, i.e., more extreme events than normal distribution
- Black Swan Events: High kurtosis indicates higher risk of extreme losses
- VaR Modeling: Standard models (normal distribution) underestimate risk with high kurtosis
- Portfolio Optimization: Higher moments (skewness, kurtosis) important for realistic risk management
- t-Distribution: Often used for financial returns due to heavier tails (positive kurtosis)
Practical Considerations
When to Analyze Kurtosis?
- Risk Management: Tail risk, extreme events
- Normality Testing: κ should ≈ 0
- Model Selection: t-distribution when g₂ > 0
- Outlier Detection: High κ indicates outliers
- Distribution Comparison: Different data sources
Be Careful With
- Small Samples: Kurtosis very unstable for n < 100
- Outliers: Fourth power greatly amplifies influence
- Multimodal Distributions: Interpretation difficult
- Skewed Distributions: Consider skewness first
- Non-Numeric Data: Kurtosis not meaningful
Kurtosis of Various Distributions
| Distribution | Excess Kurtosis (g₂) | Interpretation |
|---|---|---|
| Normal Distribution | 0 | Reference value, mesokurtic |
| Uniform Distribution | -1.2 | Platykurtic, flat |
| t-Distribution (df=5) | 6 | Highly leptokurtic, heavy tails |
| Laplace Distribution | 3 | Leptokurtic, peaked |
| Exponential Distribution | 6 | Highly leptokurtic, very skewed |
| Logistic Distribution | 1.2 | Slightly leptokurtic |
Summary
Kurtosis is an indispensable tool for describing the shape of a distribution, particularly for peak height and tail-heaviness. As the fourth standardized moment, it is very sensitive to outliers. Excess kurtosis (g₂ = κ - 3) makes normal distribution the reference point: g₂ < 0 (platykurtic, flat), g₂ = 0 (mesokurtic, normal), g₂ > 0 (leptokurtic, peaked with heavy tails). In finance, positive kurtosis is particularly important as it indicates higher risk of extreme events (fat tails, black swans). Standard models like normal distribution underestimate this risk. Interpretation should always be in context with other statistics (mean, standard deviation, skewness) and visual representations. For small samples (n < 100), kurtosis is very unstable.
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