# Technical Studies Reference

### Standard Deviation Bands

This study calculates and displays Moving Averages and Standard Deviation Bands of the data specified by the Top Band Input Data and Bottom Band Input Data Inputs.

Let $$X^{(High)}$$ and $$X^{(Low)}$$ be random variables denoting the Top Band Input Data and Bottom Band Input Data, respectively, and let $$X^{(High)}_t$$ and $$X^{(Low)}_t$$ be their respective values at Index $$t$$. Let the Inputs Length and Multplication Factor be denoted as $$n$$ and $$v$$, respectively. Then we denote the Top Band and the Bottom Band at Index $$t$$ for the given Inputs as $$TB^{(\sigma)}_t\left(X^{(High)},n,v\right)$$ and $$BB^{(\sigma)}_t\left(X^{(Low)},n,v\right)$$, respectively, and we compute them in terms of Simple Moving Averages and Standard Deviations for $$t \geq n - 1$$ as follows.

$$TB^{(\sigma)}_t\left(X^{(High)},n,v\right) = MA_t\left(X^{(High)},n\right) + v\cdot\sigma_t\left(X^{(High)},n\right)$$

$$BB^{(\sigma)}_t\left(X^{(Low)},n,v\right) = MA_t\left(X^{(Low)},n\right) - v\cdot\sigma_t\left(X^{(Low)},n\right)$$

The Simple Moving Averages $$MA_t\left(X^{(High)},n\right)$$ and $$MA_t\left(X^{(Low)},n\right)$$ are also computed and displayed for $$t \geq n - 1$$.

Note: Depending on the setting of the Input Moving Average Type, the Simple Moving Averages in the above calculations could be replaced with Exponential Moving Averages, Linear Regression Moving Averages, Weighted Moving Averages, Wilders Moving Averages, Simple Moving Averages - Skip Zeros, or Smoothed Moving Averages. The types of all Moving Averages in the above calculations are determined by this one Input.