Description
The Bollinger Bands were introduced by J. Bollinger. They provide a visual channel of upper and lower bounds that prices tend to stay between. The channel calculation is based on variation about a statistical mean over a certain look back period. The channels are defined by the calculation of the standard deviation (sigma) of the input value. The upper band is some multiplication factor of Sigma added to a simple moving average of the input value for the same period as the Sigma calculation. The lower band is the value minus Sigma times a multiplication factor. The SMA is plotted as a dashed line and the upper and lower Bollinger Bands are plotted as thin, solid lines. A text note is added to the plot in the upper left corner showing the look back period and the upper and lower Bollinger Band Values.
Bollinger Bands bring statistics into the problem of determining stock trading ranges and extremes. Standard deviation can be used to determine when a price move is statistically out of the ordinary. Using a multiplication factor of 2 means that approximately 95% of the time price should remain between the Bollinger Bands. The distance between the upper and lower bands is also an indication of the volatility of the issue. Usually when the bands tighten a relatively substantial price move is pending.
The purpose of Bollinger Bands is to provide a relative definition of high and low. By definition prices are high at the upper band and low at the lower band. This definition can aid in rigorous pattern recognition and is useful in comparing price action to the action of indicators to arrive at systematic trading decisions.
Bollinger Bands consist of a set of three curves drawn in relation to securities prices. The middle band is a measure of the intermediate-term trend, usually a simple moving average, that serves as the base for the upper and lower bands. The interval between the upper and lower bands and the middle band is determined by volatility, typically the standard deviation of the same data that were used for the average. The default parameters, 20 periods and two standard deviations, may be adjusted to suit your purposes:  

Middle Bollinger Band = 20-period simple moving average
Upper Bollinger Band = Middle Bollinger Band + 2 * 20-period standard deviation
Lower Bollinger Band = Middle Bollinger Band - 2 * 20-period standard deviation
Two important tools are derived from the Bollinger Bands: Band width, a relative measure of the width of the bands, and %b, a measure of where the last price is in relation to the bands.

BandWidth = (Upper Bollinger Band - Lower Bollinger Band) / Middle Bollinger Band
%b = (Last - Lower Bollinger Band) / (Upper Bollinger Band - Lower Bollinger Band)

BandWidth is most often used to quantify The Squeeze, a volatility-based trading opportunity. %b is used to clarify trading patterns and as an input for trading systems.