Kurtosis

Kurtosis is a statistical measure that describes the tailedness of a probability distribution—specifically, the propensity of a distribution to produce extreme values or outliers relative to a normal distribution (Pearson K. 1905, p.170)^[1]. The normal distribution has kurtosis of 3 (or excess kurtosis of 0). Distributions with fatter tails produce more extreme observations; distributions with thinner tails produce fewer. When analyzing financial returns, the difference matters enormously—a model assuming normal tails will catastrophically underestimate the probability of market crashes.

The term derives from the Greek "kyrtos" meaning curved or bulging. Karl Pearson introduced the measure in 1905, though understanding of what kurtosis actually measures evolved over decades. A common misconception equates kurtosis with "peakedness"—how pointed or flat a distribution appears at its center. This is wrong. Kurtosis measures tail behavior, and tails determine outlier frequency.

Mathematical definition

Kurtosis is defined using the fourth moment:

For a random variable X with mean μ and standard deviation σ:

Kurtosis = E[(X - μ)⁴] / σ⁴

This is the fourth standardized moment. Values are typically reported as either kurtosis (raw) or excess kurtosis (kurtosis minus 3, so that normal distributions have excess kurtosis of 0)^[2].

Sample estimation

For a sample of n observations, sample kurtosis is calculated as:

g₂ = [n(n+1) / ((n-1)(n-2)(n-3))] × Σ[(xᵢ - x̄)⁴ / s⁴] - 3(n-1)² / ((n-2)(n-3))

This formula provides an unbiased estimator under normal assumptions. Software packages differ in their default formulas—users should verify which calculation is applied.

Three types of distributions

Distributions are classified by kurtosis:

Mesokurtic

Normal distribution benchmark. Mesokurtic distributions have kurtosis of 3 (excess kurtosis of 0). The normal distribution is the prototypical mesokurtic distribution.

Moderate tails. Neither unusually prone to outliers nor unusually unlikely to produce them.

Leptokurtic

Heavy tails. Leptokurtic distributions have kurtosis greater than 3 (positive excess kurtosis). They produce more extreme values than normal distributions^[3].

Outlier prone. The probability of observations far from the mean is higher than normal. Financial returns typically exhibit leptokurtosis—market crashes and rallies occur more frequently than normal models predict.

Examples. Student's t-distribution (especially with low degrees of freedom), Laplace distribution, logistic distribution. Stock market returns are typically leptokurtic.

Etymology. "Lepto" from Greek meaning slender—originally a reference to the peak, though this terminology is misleading since kurtosis doesn't measure peak shape.

Platykurtic

Light tails. Platykurtic distributions have kurtosis less than 3 (negative excess kurtosis). They produce fewer extreme values than normal distributions.

Outlier resistant. Observations tend to cluster closer to the mean with fewer extreme departures.

Examples. Uniform distribution (discrete and continuous), raised cosine distribution. The Bernoulli distribution with p = 0.5 has the minimum possible excess kurtosis of -2^[4].

Etymology. "Platy" from Greek meaning flat or broad.

Common misconceptions

Kurtosis is frequently misunderstood:

Peak misconception

Wrong interpretation. Many sources describe kurtosis as measuring "peakedness"—how tall and pointed versus flat and spread a distribution appears at its center. This is incorrect.

Correct interpretation. Kurtosis measures tail behavior. A distribution can be perfectly flat-topped yet have high kurtosis, or infinitely peaked yet have low kurtosis. The shape of the center is largely irrelevant.

The confusion origin. Early statisticians conflated peak shape with tail behavior. Distributions that happen to have both peaked centers and fat tails (like Student's t) reinforced the misconception^[5].

What kurtosis actually measures

Tail weight. High kurtosis means more probability mass in the tails relative to the shoulders (the region between the center and tails). It indicates higher probability of extreme observations.

Outlier frequency. Leptokurtic distributions generate more outliers than mesokurtic distributions. Risk models must account for this.

Applications

Kurtosis informs various analyses:

Finance and risk management

Fat tails in returns. Financial asset returns typically exhibit excess kurtosis of 1-10 or more. The normal distribution dramatically underestimates crash probability.

Value at Risk. VaR models assuming normal distributions understate risk. Fat-tailed distributions yield more conservative risk estimates.

Option pricing. The Black-Scholes model assumes normal returns. Observed fat tails create pricing discrepancies, particularly for out-of-the-money options^[6].

Leverage effects. Crashes are more severe than rallies, creating asymmetry that kurtosis alone doesn't capture (skewness is needed for this).

Quality control

Process monitoring. Changes in kurtosis may indicate process instability even when mean and variance remain stable. Increasing kurtosis suggests growing outlier frequency.

Acceptance sampling. Understanding distribution shape helps determine appropriate sampling strategies.

Data analysis

Normality assessment. Significant departure from kurtosis of 3 suggests non-normal data, affecting the validity of parametric statistical tests.

Outlier detection. High kurtosis warns of outlier-prone data requiring robust statistical methods.

Distribution selection. Kurtosis helps identify appropriate probability distributions for modeling data^[7].

Interpretation guidelines

Rules of thumb for interpreting kurtosis:

Excess kurtosis near 0. Distribution is approximately mesokurtic. Normal-based methods may be appropriate.

Excess kurtosis > 1. Noticeably heavier tails than normal. Outlier-prone data. Consider robust methods.

Excess kurtosis > 3. Very heavy tails. Normal assumptions strongly violated. Fat-tailed distributions needed.

Excess kurtosis < -1. Unusually light tails. Data tightly clustered with few outliers.

Relationship to other moments

Kurtosis is one of four commonly used moments:

Mean (first moment). Central location.

Variance (second moment). Spread around the mean.

Skewness (third moment). Asymmetry—whether tails extend further in one direction^[8].

Kurtosis (fourth moment). Tail weight—propensity for extreme values.

Higher moments exist but are rarely used in practice due to increasing estimation difficulty and interpretation complexity.

Limitations

Kurtosis has constraints:

Sample sensitivity. Small samples produce unreliable kurtosis estimates. A single extreme observation dramatically affects sample kurtosis.

Not fully descriptive. Different distributions can have identical kurtosis yet different shapes. Kurtosis doesn't uniquely identify distributions.

Combines tail effects. Kurtosis reflects both tails jointly. A distribution with one fat tail and one thin tail might have moderate kurtosis, obscuring asymmetry.

Kurtosis — recommended articles

Descriptive statistics — Statistical measures — Risk management — Probability distribution

References

• Pearson K. (1905), Das Fehlergesetz und seine Verallgemeinerungen durch Fechner und Pearson, Biometrika, Vol. 4.
• DeCarlo L.T. (1997), On the Meaning and Use of Kurtosis, Psychological Methods, Vol. 2, No. 3.
• Westfall P.H. (2014), Kurtosis as Peakedness, The American Statistician, Vol. 68, No. 3.
• Mandelbrot B., Hudson R.L. (2004), The Misbehavior of Markets, Basic Books.

Footnotes

1. ↑ Pearson K. (1905), Das Fehlergesetz, p.170
2. ↑ DeCarlo L.T. (1997), On the Meaning and Use of Kurtosis, pp.292-307
3. ↑ Westfall P.H. (2014), Kurtosis as Peakedness, pp.191-195
4. ↑ DeCarlo L.T. (1997), On the Meaning and Use of Kurtosis, pp.297-302
5. ↑ Westfall P.H. (2014), Kurtosis as Peakedness, pp.191-193
6. ↑ Mandelbrot B., Hudson R.L. (2004), Misbehavior of Markets, pp.112-134
7. ↑ DeCarlo L.T. (1997), On the Meaning and Use of Kurtosis, pp.303-307
8. ↑ Pearson K. (1905), Das Fehlergesetz, pp.171-178

Author: Sławomir Wawak