# Technical Studies Reference

### Bill Williams Moving Average

This study calculates and displays a Bill Williams Moving Average of the data specified by the Input Data Input.

Let $$X$$ be a random variable denoting the Input Data, and let $$X_t$$ be the value of the Input Data at Index $$t$$. Let the Input Length be denoted as $$n$$. Then we denote the Bill Williams Moving Average at Index $$t$$ for the given Inputs as $$BWMA_t(X,n)$$, and we compute it for $$t \geq 0$$ as follows.

For $$t = 0$$: $$BWMA_0(X,n) = X_0$$

For $$t > 0$$: $$\displaystyle{BWMA_t(X,n) = \left(1 - \frac{1}{n}\right)BWMA_{t - 1}(X,n) + \frac{1}{n}X_t}$$

Note: Depending on the setting of the Input Moving Average Type, the Bill Williams Moving Average in the above calculation could be replaced with a Smoothed Moving Average.

#### Inputs

• Input Data
• MovAvg Length
• Moving Average Type: This is a custom Input that can be set to either Bill Williams EMA (documented above) or Smoothed Moving Average.