Bollinger Bands: Formula, Signals, and What Tests Reveal

Bollinger Bands place an upper and lower band around a moving average. The bands widen when the prices in their calculation become more spread out and narrow when those prices become more similar.

Traders use them to read relative high and low prices, watch volatility, and define trading rules. Across 226,064 configuration evaluations, a longer lookback could produce more trades, while extending it further could reduce the count again. The current price also helps determine the boundary it is trying to cross.

What do Bollinger Bands show?

The middle line is usually a simple moving average. The outer lines sit a chosen number of standard deviations above and below it. Standard deviation measures how far the prices in the window spread around their average.

The familiar setting is 20 periods and two standard deviations, often written 20/2. Twenty periods means twenty candles: it covers different amounts of time on a two-hour chart and a daily chart. These are defaults, not universally best settings. Fidelity’s guide explains the conventional construction.

A high price relative to its recent window may be close to the upper band. A relatively low one may be close to the lower band. Neither description tells us whether the next move will continue or reverse.

%b expresses that relative position as (close − lower band) ÷ (upper band − lower band). When the width is nonzero, 0 is the lower band and 1 is the upper band; values outside 0–1 lie outside the bands.

How to calculate Bollinger Bands

For a lookback of N closing prices and a multiplier of k:

Middle band = SMA of the last N closes
Upper band  = middle band + k × standard deviation
Lower band  = middle band − k × standard deviation

Here we use population standard deviation: average the squared distances from the window’s mean, then take the square root. The current close belongs to that window.

For example, four closes of 100 followed by 110 have an average of 102 and a standard deviation of 4. With a multiplier of two, the upper band is 110 and the lower band is 94. The newest price has reached the upper band exactly.

The calculation uses the levels of the closing prices, rather than the standard deviation of percentage returns. Keeping that distinction clear helps explain why a steady trend can also create wide bands.

Common Bollinger Band signals

There are several familiar ways to use the same lines:

• A breakout rule follows strength beyond the upper band, looking for continuation.
• A fade rule buys weakness below the lower band, looking for a return toward the average.
• A recovery rule waits for the close to return above the current lower band before buying.
• A squeeze describes unusually narrow bands. Traders then watch for a subsequent move and its direction.

The bands also move, so returning above a lower band need not mean the close rose from the previous candle. A touch made by a candle’s high or low also differs from a close outside a band.

John Bollinger explicitly warns that a band tag alone is not a buy or sell signal. Prices can keep moving along a band during a trend. The use of two standard deviations also does not promise that 95% of future prices will remain inside.

Why a longer lookback can produce more activity

With the bands fixed at two standard deviations, one breakout comparison produced 452 executions at six closes, 608 at ten, and 432 at twenty. Extending the lookback first made the rule more active, then quieter. The same rise-then-fall pattern repeated across the studied samples for both breakout and fade rules.

Here, breakout means buying above the upper band and exiting below the middle line. Each buy or sell counts as one execution. The curve is not a count of every candle outside a band.

These comparisons hold the multiplier and trading rule fixed while changing the lookback. They measure rule activity, rather than the quality of the resulting trades.

The interaction between length and width is established. Bollinger’s own rules suggest changing the multiplier when the average length changes if the goal is similar price containment. Holding it fixed here lets us measure how a lookback change affects these exact trading rules.

The band’s lookback changes both its center and its width. A longer window can make an outside close easier to achieve near the short-window boundary, while also changing how long an existing position lasts. The formula alone does not predict the resulting execution count.

Changing the entry rule changes the comparison again. Buying above the upper band and buying below the lower band are separate strategies, even when both use the familiar 20/2 display. Their default-setting results and broader summaries are retained with the evidence; none of these activity comparisons establishes a profitable rule.

Why very short bands can stop signaling

The latest close helps determine the boundary it is trying to cross. As that price moves away from the other closes, it pulls the average toward itself and increases the standard deviation.

Each point above is a separate constructed calculation. Keep the latest close at 110 and the older closes at 100, then change the number of closes in the window.

With five closes, 110 lies exactly on the two-standard-deviation upper band. With six, the upper band is about 109.12, so 110 is outside it. The latest price has not changed; its influence on the calculation has.

There is an exact limit behind this example:

Largest possible distance from the mean, in standard deviations
= square root of (N − 1)

For three closes that limit is about 1.41. For five it is 2. A rule requiring a close strictly beyond a two-standard-deviation band therefore cannot trigger with five or fewer closes.

This is established finite-sample geometry, related to Samuelson’s inequality. It applies when the same window, including the current close, supplies both the average and population standard deviation. It does not impose this limit on a candle’s high against close-based bands, or on a new price compared with yesterday’s bands.

With six closes, a two-standard-deviation crossing becomes possible. Possibility still tells us nothing about its frequency or profitability.

Why bands can narrow while the latest price barely changes

Every new close adds one price to the window and removes another. Sometimes the price leaving matters much more than the latest change.

Select The older high remains in the window, then A small price change, much narrower bands. The close falls by about 0.07%, while BandWidth narrows from 2.07% to 0.50% as an older high leaves the twenty-close window.

BandWidth is the distance between the outer bands divided by the middle band, expressed here as a percentage. The chart shows the changing indicator values without execution markers.

The window has become much more compact. That is useful information, but a sudden narrowing does not by itself establish a new change in the market’s direction. Bollinger’s own rules already describe this window effect; the example makes its arithmetic visible.

How we checked the behavior

The 226,064 main configuration evaluations varied lookback and band width for breakout and fade rules. The full settings, sources, and paired comparisons are recorded below.

To explore the same effect, open TrainBard with the supplied script. Hold the multiplier and rule fixed while changing the lookback. Then keep the bands fixed and compare the entry rules: the indicator supplies the boundaries, while the rule determines the decisions.

For another measure of volatility, see our ATR guide. Related indicator guides cover the simple moving average, RSI, and MACD.