# Technical Studies Reference

### Kurtosis

This study calculates and displays the Kurtosis of the data specified by the Input Data Input. Kurtosis is a statistical measure of the peakedness or flatness of a distribution relative to the Normal Distribution, which has a Kurtosis of $$3$$. More precisely, this study calculates what is known as Excess Kurtosis, which is the Kurtosis in excess of $$3$$.

Let $$X$$ be a random variable denoting the Input Data, and let $$X_t$$ be its value at Index $$t$$. Let the $$n$$ be the Length Input. We denote the Kurtosis at Index $$t$$ as K_t(X,n), and we compute it over a moving window of Length $$n$$ for $$t \geq n - 1$$. The method of computation depends on the setting of the Kurtosis Type Input. There are four such settings, and we explain each one below. In each case, the function $$SMA_t(X,n)$$ refers to the Simple Moving Average.

Case 1: Population Excess Kurtosis

$$\displaystyle{K_t(X,n) = \frac{E_t\left[(X - SMA(X,n))^4\right]}{\left(E_t\left[(X - SMA(x,n))^2\right]\right)^2} - 3}$$

In the above formula, the function $$E_t[]$$ refers to the Expected Value at Index $$t$$. The numerator and denominator are the Fourth and Second Central Moments of $$X$$, respectively, and they are given explicitly by the following formulas.

$$\displaystyle{E_t\left[(X - SMA(X,n))^4\right] = \frac{1}{n}\sum_{i = t - n + 1}^t(X_i - SMA_t(X,n))^4}$$
$$\displaystyle{E_t\left[(X - SMA(X,n))^2\right] = \frac{1}{n}\sum_{i = t - n + 1}^t(X_i - SMA_t(X,n))^2}$$

Case 2: Sample Excess Kurtosis

$$\displaystyle{K_t(X,n) = \frac{(n - 1)(n + 1)}{(n - 2)(n - 3)}\cdot\frac{E_t\left[(X - SMA(X,n))^4\right]}{\left(E_t\left[(X - SMA(x,n))^2\right]\right)^2} - \frac{3(n - 1)^2}{(n - 2)(n - 3)}}$$

The Sample Excess Kurtosis is an unbiased estimator of the Population Kurtosis.

Regardless of whether one is calculating Excess Kurtosis for populations or samples, the value is interpreted as follows.

• If $$K_t(X,n) > 0$$, then the distribution is more peaked than a Normal Distribution with the same mean and variance.
• If $$K_t(X,n) = 0$$, then the distribution is a Normal Distribution.
• If $$K_t(X,n) < 0$$, then the distribution is less peaked than a Normal Distribution with the same mean and variance.

#### Inputs

• Input Data
• Length
• Kurtosis Type: This is a custom Input that determines the method of calculation of the Kurtosis.