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Kurtosis Calculator

Calculate Kurtosis of Data Distribution with Step-by-Step Derivation

Enter data values (comma separated)

Kurtosis Formulas

Sample Kurtosis = {[n(n+1)/((n-1)(n-2)(n-3))] × Σ[(xᵢ - x̄)⁴/σ⁴]} - {[3(n-1)²/((n-2)(n-3))]}
Excess Kurtosis = Sample Kurtosis - 3

Kurtosis measures the "tailedness" of a distribution. Compare to normal distribution (kurtosis = 3, excess = 0).

Leptokurtic (excess >0): heavy tails, sharp peak. Platykurtic (excess <0): light tails, flat peak.

What is Kurtosis?

Kurtosis is a measure of the "tailedness" of the probability distribution of a real-valued random variable. It describes the shape of the distribution's tails and peak.

Mesokurtic

Excess kurtosis = 0, like normal distribution

Leptokurtic

Excess kurtosis > 0, heavy tails, sharp peak

Platykurtic

Excess kurtosis < 0, light tails, flat peak

Fourth Moment

Based on fourth power of deviations from mean

💡 Example: Uniform distribution (1,2,3,4,5,6,7,8,9,10) has negative excess kurtosis (platykurtic). Data with outliers has positive excess kurtosis (leptokurtic).

Applications

Statistics Finance Risk Management Data Science Quality Control

Frequently Asked Questions

What is kurtosis?
Kurtosis measures the "tailedness" of a probability distribution. High kurtosis means heavy tails and sharp peak; low kurtosis means light tails and flat peak.
How to interpret kurtosis?
Excess kurtosis = 0: mesokurtic (normal). Excess >0: leptokurtic (heavy tails). Excess <0: platykurtic (light tails).
What is excess kurtosis?
Excess kurtosis = kurtosis - 3. Subtracts 3 to compare to normal distribution (kurtosis = 3).
Why is kurtosis important?
Kurtosis indicates tail risk, identifies outliers, and helps select appropriate statistical models and tests.

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