Kurtosis Calculator

Kurtosis is the fourth standardized central moment of a distribution. This calculator uses the adjusted sample formula that corrects for bias:

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

Raw kurtosis is simply excess kurtosis plus 3 (since the normal distribution has a raw kurtosis of 3).

The three categories of kurtosis:

• Mesokurtic (excess kurtosis ≈ 0): Similar tail weight to the normal distribution. Example: the normal distribution itself.
• Leptokurtic (excess kurtosis > 0): Heavier tails, more outlier-prone. Example: Student's t-distribution, Laplace distribution.
• Platykurtic (excess kurtosis < 0): Lighter tails, fewer outliers. Example: uniform distribution, beta distributions with certain parameters.

A common misconception is that kurtosis measures "peakedness." Research by Westfall (2014) clarified that kurtosis is fundamentally about the tails, not the peak. A distribution can have high kurtosis with either a sharp peak or a flat peak, as long as the tails are heavy.

Kurtosis is essential in risk management and finance. Financial return distributions typically exhibit positive excess kurtosis ("fat tails"), meaning extreme events (market crashes, sudden spikes) occur more frequently than a normal distribution would predict. Ignoring kurtosis in risk models was a contributing factor to the underestimation of risk before the 2008 financial crisis.

The formula requires a minimum of 4 data points because the denominator includes (n-3). The adjustment factors correct for small-sample bias in the fourth moment estimator. For large n, the adjustment converges to the population kurtosis formula.

In conjunction with skewness, kurtosis provides a comprehensive picture of distribution shape. Together, they form the basis of normality tests such as the Jarque-Bera test, which tests whether data follows a normal distribution by checking if skewness and excess kurtosis are both approximately zero.

Excess kurtosis = 0 means normal-like tails. Positive values indicate heavier tails (more outlier risk). Negative values indicate lighter tails. The raw kurtosis adds 3 to the excess kurtosis for compatibility with some textbook conventions.