Squared versus absolute deviation in envelope construction

Two widths from the same inputs

Bollinger Bands are a price envelope whose width is set from a standard-deviation measure of how far observations sit from a central average. The commodity channel index is a channel oscillator whose scale is built from mean absolute deviation rather than standard deviation.

When the same inputs are used and the constants are equalized, the main construction difference is that dispersion choice. Standard deviation squares deviations from the mean and therefore enlarges the influence of large moves. Mean absolute deviation averages unsigned deviations and does not magnify outliers by squaring.

Close near the average, apart at extremes

The two dispersion calculations stay close when prices hug their average and diverge sharply once prices move away from that average. Squaring deviations in the standard-deviation calculation accentuates large moves, while mean absolute deviation has no equivalent magnifying step.

The construction gap matters most at extremes, where a band-based reading of relatively high and low must stay aligned with a rapidly changing price structure.

A nonlinear construction gap

Changing constants cannot force the two series to match. The difference is a nonlinear construction gap: it comes from squaring deviations when standard deviation is computed, and it cannot be removed by changing constants because one path squares deviations and the other does not.

For a normally distributed series, mean absolute deviation is about 0.7979 of standard deviation. Because security prices are not normally distributed, the gap can be larger.

Percent-b is not a substitute channel oscillator

Percent-b is an oscillator that restates price as a position inside a Bollinger Band envelope. Because that envelope is a standard-deviation construction, the percent-b oscillator taken from Bollinger Bands and the commodity channel index should not be treated as interchangeable tools.