What Is a Kurtosis Calculator?

A kurtosis calculator measures the tailedness of a distribution — how prone the data is to producing extreme outliers. Pre-filled with 13 values that cluster around the center, the calculator computes population and sample kurtosis, excess kurtosis, classifies the distribution (leptokurtic/mesokurtic/platykurtic), and shows a distribution histogram.

Kurtosis vs Excess Kurtosis

Kurtosis (raw, 4th standardized moment): κ = E[(X−μ)⁴]/σ⁴. A normal distribution has κ = 3. Excess kurtosis = κ − 3, zeroed at the normal distribution. Most software (R, Python, Excel KURT) reports excess kurtosis. Positive excess means heavier tails than normal; negative means lighter tails.

Leptokurtic, Mesokurtic, Platykurtic

Leptokurtic (excess > 0): heavy tails, sharp peak. More probability in the tails than normal — extreme events are more likely. Examples: stock returns, income, insurance claims. Mesokurtic (excess ≈ 0): similar to normal. Platykurtic (excess < 0): light tails, flat top. Bounded data with few extremes. Example: uniform distribution (excess = −1.2).

Kurtosis in Risk Management

Financial returns are leptokurtic: the S&P 500 has excess kurtosis around 2–4 on daily returns. This means market crashes and extreme gains are far more frequent than a normal model predicts. Standard VaR calculations using normal assumptions underestimate risk. Fat-tailed models (Student's t-distribution, extreme value theory) provide more accurate tail risk estimates.

The Jarque-Bera Normality Test

JB = (n/6) × [S² + (K²/4)], where S = skewness, K = excess kurtosis. Under H₀ (normality): JB ~ χ²(2). Large JB → reject normality. This test efficiently combines both shape measures. A dataset with skewness = 0.5 and excess kurtosis = 1.5 with n=100 gives JB = 100/6 × (0.25 + 0.5625) ≈ 13.5, rejecting normality at the 1% level (critical value ≈ 9.21).