Bollinger Bands: Mean, Deviation and Width

State the calculation convention

A common form uses a simple moving average as the middle line, with upper and lower bands a chosen number of standard deviations away. Fidelity describes the construction and adjustable parameters. A label such as 20, 2 usually means a 20-observation lookback and a two-deviation multiplier, but the price source and deviation convention still need checking.

Standard deviation describes dispersion around the mean. It does not count transactions or measure a gap from the previous close in the same way as ATR. Changing the band input from closes to typical prices, or changing the averaging method, changes the actual indicator being tested.

An original four-price calculation

Take hypothetical closing prices of 98, 100, 100 and 102. Their mean is 100. Squared deviations are 4, 0, 0 and 4, totaling 8. Using the population divisor of four gives variance 2 and standard deviation approximately 1.4142.

With a multiplier of two, the upper band is 102.8284 and the lower band is 97.1716. Band width is approximately 5.6568 price units. Dividing that width by the middle line and multiplying by 100 produces a relative width of approximately 5.6568%.

Using the sample divisor of three instead produces deviation approximately 1.6330 and wider bands at about 103.2660 and 96.7340. This is not a rounding disagreement. It is a different statistical convention. A spreadsheet and chart should be compared only after their input and denominator choices match.

Do not infer an automatic probability

A two-deviation envelope around recent prices is not automatically a 95% prediction interval for the next price. The familiar normal-distribution coverage statement requires assumptions that a rolling series of dependent market prices does not establish. The mean and deviation also change as observations enter and leave the window.

Likewise, touching the upper band does not mean price must fall. A sustained move can repeatedly place price near or beyond a changing band. A narrow envelope describes recent compression under the selected calculation; it does not identify which direction a later expansion will take.

Make any proposed rule observable

If a rule refers to a band touch, specify whether the high crossing is sufficient or the close must finish outside. Decide whether the band is frozen at the previous close or recomputed during the current bar. Otherwise, the threshold can move while price is being tested against it, making a historical example difficult to reproduce.

Compare Keltner Channels, whose range-based width answers a different question. Review standard-deviation conventions before comparing values across software. Then use the backtesting protocol to evaluate a precise rule with trading costs, rather than assigning a reversal probability to an attractive envelope.