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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.


  • 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.


The spreadsheet below contains the formulas for this study in Spreadsheet format. Save this Spreadsheet to the Data Files Folder.

Open it through File >> Open Spreadsheet.


*Last modified Wednesday, 18th April, 2018.