Kurtosis

Descriptive Statistics

Where Do Extreme Events Hide?

Kurtosis measures the heaviness of a distribution's tails — and reveals whether extreme events are rare surprises or regular visitors.

Understanding kurtosis helps you:

Assess tail risk — know if crashes and outliers are more frequent than a normal model predicts
Evaluate models — detect when a normal assumption dangerously underestimates extreme events
Classify distributions — distinguish mesokurtic, leptokurtic, and platykurtic shapes
Test normality — combine with skewness in the Jarque-Bera test for a rigorous check

In finance and science, the tails are where the action is. Kurtosis tells you how thick they are.

What is Kurtosis?

Definition

Kurtosis measures the tailedness of a distribution — how much probability mass is in the tails relative to a normal distribution. Excess kurtosis subtracts 3 (the normal distribution's kurtosis) so that the normal baseline is zero.

import numpy as np
from scipy import stats

np.random.seed(42)
n = 5000
normal  = np.random.normal(0, 1, n)
t3      = np.random.standard_t(df=3, size=n)    # fat tails
uniform = np.random.uniform(-3, 3, n)            # thin tails

for name, data in [("Normal", normal), ("t(df=3) Fat-tail", t3), ("Uniform Thin-tail", uniform)]:
    ek = stats.kurtosis(data, fisher=True)   # Fisher's: normal=0
    print(f"{name:<22}: Excess Kurtosis = {ek:+.4f}")

Type	Excess Kurtosis	Shape	Example
Mesokurtic	= 0	Normal-like	Normal distribution
Leptokurtic	greater than 0	Fat tails, sharp peak	t-distribution, daily returns
Platykurtic	less than 0	Thin tails, flat peak	Uniform distribution

Fat Tails in Finance

# Simulate daily stock returns — compare crash frequency
normal_rets = np.random.normal(0, 0.01, 2520)
fat_rets    = np.random.standard_t(5, 2520) * 0.01

threshold = 0.03  # 3% daily move
print(f"Normal model >3% moves: {np.sum(np.abs(normal_rets)>threshold)}/2520")
print(f"Fat-tail model >3% moves: {np.sum(np.abs(fat_rets)>threshold)}/2520")
print("Fat tails vastly increase extreme event frequency!")

Jarque-Bera Test for Normality

# Combines skewness and kurtosis into a single normality test
for name, data in [("Normal", normal), ("t(df=3)", t3)]:
    jb_stat, p = stats.jarque_bera(data)
    reject = "YES — not normal" if p < 0.05 else "NO — cannot reject normality"
    print(f"{name}: JB={jb_stat:.2f}, p={p:.6f} -> Reject normality? {reject}")

Kurtosis in Machine Learning

ML Application	Kurtosis Usage	Why
Outlier detection	High kurtosis = heavy tails	More outliers expected
Feature engineering	Low kurtosis → transform	Normality helps linear models
Risk management	Fat tails = extreme events	Financial ML models
Data quality	Unexpected kurtosis = data issues	Sanity check pipeline

import numpy as np
from scipy.stats import kurtosis

np.random.seed(42)

# Kurtosis and outlier detection
normal_data = np.random.normal(0, 1, 1000)
heavy_tailed = np.random.standard_t(df=3, size=1000)

print(f"Normal: kurtosis = {kurtosis(normal_data):.3f} (mesokurtic)")
print(f"t(3):   kurtosis = {kurtosis(heavy_tailed):.3f} (leptokurtic, more outliers)")

# Count extreme values
extreme_normal = np.sum(np.abs(normal_data) > 3) / len(normal_data)
extreme_heavy = np.sum(np.abs(heavy_tailed) > 3) / len(heavy_tailed)
print(f"\nValues beyond ±3σ:")
print(f"Normal: {extreme_normal:.3%}")
print(f"t(3):   {extreme_heavy:.3%} (more extremes due to heavy tails)")

Key Takeaways

Excess kurtosis = 0 for normal; greater than 0 = fat tails; less than 0 = thin tails

Leptokurtic distributions produce more extreme outliers than normal theory predicts

Financial returns are almost always leptokurtic — risk models must account for this

Jarque-Bera test formally tests normality using both skewness and kurtosis

"Kurtosis is the probability of the improbable. Ignore it, and the tails will bite you."

Kurtosis — Fat Tails and Extreme Events