Kurtosis — Definition, Formula & Examples

Kurtosis is a statistical measure that describes how heavy or light the tails of a distribution are compared to a normal distribution. Higher kurtosis means more data in the extreme tails (more outliers), while lower kurtosis means thinner tails (fewer outliers).

Kurtosis is the standardized fourth central moment of a probability distribution, defined as $\kappa_4 = \frac{\mu_4}{\sigma^4}$ , where $\mu_4$ is the fourth central moment and $\sigma$ is the standard deviation. Excess kurtosis subtracts 3 (the kurtosis of a normal distribution) so that a normal distribution has excess kurtosis of zero.

Key Formula

\text{Excess Kurtosis} = \frac{n(n+1)}{(n-1)(n-2)(n-3)}\sum_{i=1}^{n}\left(\frac{x_i - \bar{x}}{s}\right)^4 - \frac{3(n-1)^2}{(n-2)(n-3)}

Where:

$n$ = Sample size
$x_i$ = Each individual data value
$\bar{x}$ = Sample mean
$s$ = Sample standard deviation

How It Works

Kurtosis quantifies the shape of a distribution's tails, not its peakedness (a common misconception). A distribution with excess kurtosis greater than 0 is called leptokurtic — it has heavier tails than a normal distribution, meaning extreme values are more likely. A distribution with excess kurtosis less than 0 is called platykurtic — it has lighter tails. A normal distribution has excess kurtosis exactly 0 and is called mesokurtic. In practice, you compute kurtosis from sample data and compare the result to 0 to judge whether your data has unusually many or few outliers relative to a normal distribution.

Worked Example

Problem: A dataset has 5 values: 2, 4, 4, 4, 8. Compute the population kurtosis and excess kurtosis.

Step 1: Find the mean.

\bar{x} = \frac{2+4+4+4+8}{5} = \frac{22}{5} = 4.4

Step 2: Compute the deviations raised to the 2nd and 4th powers, then find the second and fourth central moments.

\mu_2 = \frac{(-2.4)^2 + (-0.4)^2 + (-0.4)^2 + (-0.4)^2 + (3.6)^2}{5} = \frac{5.76+0.16+0.16+0.16+12.96}{5} = 3.84

Step 3: Compute the fourth central moment.

\mu_4 = \frac{(-2.4)^4 + (-0.4)^4 + (-0.4)^4 + (-0.4)^4 + (3.6)^4}{5} = \frac{33.1776+0.0256+0.0256+0.0256+167.9616}{5} = 40.2432

Step 4: Divide to get population kurtosis, then subtract 3 for excess kurtosis.

\kappa_4 = \frac{40.2432}{3.84^2} = \frac{40.2432}{14.7456} \approx 2.73

Answer: The population kurtosis is approximately 2.73, and the excess kurtosis is

2.73 - 3 = -0.27

. Since excess kurtosis is slightly negative, this distribution is mildly platykurtic (lighter tails than a normal distribution).

Why It Matters

Kurtosis is critical in finance for assessing tail risk — a portfolio with high kurtosis faces more frequent extreme gains or losses than a normal model predicts. It also appears in quality control and hypothesis testing when checking whether data follow a normal distribution (normality tests like Jarque-Bera use both skewness and kurtosis).

Common Mistakes

Mistake: Interpreting kurtosis as a measure of how peaked or flat a distribution is.

Correction: Kurtosis measures tail heaviness, not peak height. Distributions with the same kurtosis can have very different peak shapes. Focus on what kurtosis tells you about outlier frequency.