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### Bollinger Bands

This study calculates and displays Bollinger Bands for the data specified by the **Input Data** Input, as well as a Moving Average of the **Input Data**.

Let \(X\) be a random variable denoting the **Input Data**, and let \(X_i\) be the value of the **Input Data** at Index \(i\). Let the Inputs **Length** and **Standard Deviations** be denoted as \(n\) and \(v\), respectively. Then we denote the **Bollinger Bands** at Index \(t\) for the given Inputs as \(TB^{(B)}_t(X,n,v)\) (Top Band) and \(BB^{(B)}_t(X,n,v)\) (Bottom Band), and we compute them for \(t \geq n - 1\) in terms of a Simple Moving Average and a Standard Deviation as follows.

Top Band: \(TB^{(B)}_t(X,n,v) = SMA_t(X,n) + v \cdot \sigma_t(X,n)\)

Bottom Band: \(BB^{(B)}_t(X,n,v) = SMA_t(X,n) - v \cdot \sigma_t(X,n)\)

The band in the middle is the graph of \(SMA_t(X,n)\).

**Note**: Depending on the setting of the Input **Moving Average Type**, the Simple Moving Average in each of the above formulas could be replaced with an Exponential Moving Average, a Linear Regression Moving Average, a Weighted Moving Average, a Wilders Moving Average, a Simple Moving Average - Skip Zeros, or a Smoothed Moving Average.

#### Inputs

#### Spreadsheet

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 Thursday, 13th June, 2019.