Kurtosis Calculator

Sample Kurtosis

K = [n(n+1)/((n-1)(n-2)(n-3))] × Σ[(x-μ)⁴/σ⁴]

• Fourth standardized moment
• Measures tail weight
• Sample size adjustment
• Unbiased estimator
• Distribution shape indicator
• Outlier sensitivity measure
• Peak sharpness analysis

Excess Kurtosis

K_excess = K - 3

• Normal distribution reference
• Zero for normal distribution
• Relative tail weight
• Comparative measure
• Distribution classification
• Financial risk indicator
• Market analysis tool

Population Kurtosis

γ₂ = E[(X-μ)⁴]/σ⁴

• Theoretical measure
• No bias correction
• Large sample applications
• Distribution modeling
• Theoretical properties
• Asymptotic behavior

Leptokurtic (K > 3)

• Heavy tails
• High, sharp peak
• Financial returns
• Risk assessment
• Extreme value analysis
• Market volatility
• Outlier presence

Mesokurtic (K = 3)

• Normal distribution
• Reference distribution
• Balanced tails
• Moderate peak
• Natural phenomena
• Statistical baseline
• Theoretical model

Platykurtic (K < 3)

• Light tails
• Flat peak
• Uniform-like
• Bounded data
• Controlled processes
• Quality measures
• Discrete distributions

Financial Analysis

• Risk measurement
• Portfolio optimization
• Market efficiency
• Asset allocation
• Trading strategies
• Value at Risk (VaR)
• Option pricing

Quality Control

• Process capability
• Specification limits
• Manufacturing control
• Product uniformity
• Defect analysis
• System stability
• Performance metrics

Research Methods

• Normality testing
• Distribution fitting
• Outlier detection
• Model validation
• Experimental design
• Data transformation
• Statistical power