Bollinger Bands: The Sigma Is a Population Standard Deviation (Divide by n, Not n-1)

Implementations disagree on one detail that changes every number on the chart: **whether the standard deviation divides by n or by n-1.** The source settles it in a footnote: ★**"Since Bollinger Bands use the population method of calculating standard deviation, the proper divisor for the sigma calculation is n, not n - 1."**^[1]

★**This is not a rounding difference.** The sample form (n-1) divides by a smaller number, so it produces a **larger** sigma and therefore **wider** bands for the same prices. ★**Wider bands mean a touch of the lower band is registered later, or not at all** — so two charts of the same instrument can disagree about whether a signal occurred. ★**This page does not compute the size of that gap for any particular window**; it records which divisor the definition specifies.^[1]

A second footnote covers a related trap. **If the simple moving average is changed to an exponential or weighted moving average, "it must be changed for both the calculation of the middle band and the calculation of standard deviation."** ★**Switching only the centre line to an EMA and leaving the sigma on a simple average is a mismatch the source explicitly warns against.**^[1]

The bands are three lines. The middle is an **N-period moving average**; the upper is **MA + Kσ** and the lower **MA - Kσ**, where σ is an N-period standard deviation. The method was **propounded by John Bollinger in the 1980s**, and he **registered "Bollinger Bands" as a U.S. trademark in 2011**.^[1]

**"Typical values for N and K are 20 days and 2, respectively."** The default centre line is a **simple** moving average, with exponential averages **"a common second choice"**, and **"usually the same period is used for both the middle band and the calculation of standard deviation."** ★**20 and 2 are described as typical, not as derived optima** — the article calls them two parameters **"chosen independently by the user."**^[1]

Two derived measures come up constantly and are worth stating exactly. **%b = (last - lowerBB) / (upperBB - lowerBB)**, which **"equals 1 at the upper band and 0 at the lower band"** — so values above 1 or below 0 mean price is outside the bands. **Bandwidth** normalises the width of the bands over time. ★**%b is a position measure; bandwidth is a width measure.** They answer different questions.^[1]

The stated aim is modest. **"The purpose of Bollinger Bands is to provide a relative definition of high and low prices of a market. By definition, prices are high at the upper band and low at the lower band."** ★**That is a definition, not a prediction** — it says where price sits relative to recent volatility, not where it goes next.^[1]

★**Traders read it in opposite directions, and the source says so.** **"Some traders buy when price touches the lower Bollinger Band and exit when price touches the moving average"** — mean reversion. **"Other traders buy when price breaks above the upper Bollinger Band or sell when price falls below the lower"** — breakout. ★**These are contradictory rules drawn from the same lines**, and the article presents both without choosing.^[1]

On width: **"When the bands lie close together, a period of low volatility is indicated"**, and **"as the bands expand, an increase in price action/market volatility is indicated."** Options traders reportedly **sell options when the bands are historically far apart and buy when they are close together**, expecting volatility to revert. ★**Narrow bands describe present quiet; they do not date the move that follows.**^[1]

The article has an Effectiveness section, and it does not read as promotion. **"Various studies of the effectiveness of the Bollinger Band strategy have been performed with mixed results."** ★**Lento et al. (2007)**, analysing trades **"spanning a decade from 1995 onwards"** across markets including the Dow Jones and forex, **"found no evidence of consistent performance over the standard 'buy and hold' approach."**^[1]

★**The same study found the opposite rule worked better.** The authors **"did, however, find that a simple reversal of the strategy ('contrarian Bollinger Band') produced positive returns in a variety of markets."** A separate study of the Chinese market reported **"significant positive returns on buy trades generated by the contrarian version"** of several rules including the Bollinger Band rule, **"after accounting for transaction costs of 0.50%."**^[1]

★**This page does not adjudicate those results and carries no backtest figures of its own.** It records that the reference source itself publishes findings of **no consistent advantage over buy-and-hold**, alongside findings that the **reversed** rule performed better. ★**Whether any Bollinger Band rule produces a profit is not addressed here.**^[1]

★**This page rests on one institution — a single Wikipedia article — with no independent cross-check.** ★**The underlying studies (Lento et al. 2007 and the Chinese market study) were not read directly**; they are reported as the article summarises them, and their methods and sample definitions were not examined. ★**Bollinger's own book was not consulted.** ★**Investopedia, a usual second reference here, returned HTTP 402 on 2026-09-23 and could not be read.**^[1]

★**Not established:** how much the n versus n-1 choice moves the bands for a given window length, which charting platforms use which divisor, whether 20/2 outperforms other settings, and how the bands are seeded for the first N-1 periods. **As of 2026-09-23.**^[1]