Mathwords logoMathwords

Excess Kurtosis — Definition, Formula & Examples

Excess kurtosis is a measure of how much the tails of a distribution differ from those of a normal distribution. A value of zero means the tails match a normal distribution, positive values indicate heavier tails, and negative values indicate lighter tails.

Excess kurtosis is defined as the fourth standardized central moment of a distribution minus 3, where the subtraction of 3 normalizes the measure so that a normal distribution has an excess kurtosis of zero. It quantifies the propensity of a distribution to produce outliers relative to a Gaussian baseline.

Key Formula

Excess Kurtosis=1n∑i=1n(xi−xˉ)4s4−3\text{Excess Kurtosis} = \frac{\dfrac{1}{n}\displaystyle\sum_{i=1}^{n}(x_i - \bar{x})^4}{s^4} - 3
Where:
  • nn = Number of data points
  • xix_i = Each individual data value
  • xˉ\bar{x} = Sample mean
  • ss = Sample standard deviation

How It Works

Compute the mean and standard deviation of your data, then find the average of each data point's deviation from the mean raised to the fourth power, scaled by the standard deviation to the fourth power. Subtract 3 from this ratio to get excess kurtosis. A positive result (leptokurtic) means heavier tails and a sharper peak than normal, signaling more extreme values. A negative result (platykurtic) means lighter tails and a flatter peak. A value near zero (mesokurtic) suggests tail behavior similar to a normal distribution.

Worked Example

Problem: Find the excess kurtosis of the dataset: 2, 4, 4, 4, 5, 5, 5, 5, 7, 9.
Step 1: Compute the mean.
xˉ=2+4+4+4+5+5+5+5+7+910=5010=5\bar{x} = \frac{2+4+4+4+5+5+5+5+7+9}{10} = \frac{50}{10} = 5
Step 2: Compute the sum of squared deviations and find the variance (using the population formula for this illustration).
s2=9+1+1+1+0+0+0+0+4+1610=3210=3.2s^2 = \frac{9+1+1+1+0+0+0+0+4+16}{10} = \frac{32}{10} = 3.2
Step 3: Compute the sum of fourth-power deviations.
∑(xi−xˉ)4=81+1+1+1+0+0+0+0+16+256=356\sum(x_i-\bar{x})^4 = 81+1+1+1+0+0+0+0+16+256 = 356
Step 4: Compute the fourth moment ratio and subtract 3.
Excess Kurtosis=356/10(3.2)2−3=35.610.24−3≈3.477−3=0.477\text{Excess Kurtosis} = \frac{356/10}{(3.2)^2} - 3 = \frac{35.6}{10.24} - 3 \approx 3.477 - 3 = 0.477
Answer: The excess kurtosis is approximately 0.477, indicating slightly heavier tails than a normal distribution (mildly leptokurtic).

Why It Matters

In finance, excess kurtosis helps analysts gauge the likelihood of extreme returns—high kurtosis warns of fat tails and greater risk of market crashes. In quality control and data science, checking kurtosis helps determine whether standard normal-based methods (like control charts or confidence intervals) are appropriate for a given dataset.

Common Mistakes

Mistake: Confusing kurtosis with excess kurtosis. Many software packages report one or the other without labeling clearly.
Correction: Regular (standard) kurtosis of a normal distribution equals 3. Excess kurtosis subtracts 3 so that a normal distribution equals 0. Always check whether your tool reports kurtosis or excess kurtosis.