What Bollinger Bands are

The three lines, and where each comes from 

The middle line is a simple moving average of closing prices, conventionally over twenty periods. Nothing about it is new: it is the mean of a stated count of recent closes, recalculated when each bar closes, with the drop out effect and the lag that construction guarantees. Everything said in the moving average lesson about the middle line remains true here, because it is the same object under a different name.

The outer two lines are the part that is new, and they are computed from exactly the same closes as the middle one. The standard deviation of those closes is worked out, multiplied by a stated number, conventionally two, and the result is added to the middle line to draw the upper band and subtracted from it to draw the lower one. The envelope is therefore not an independent measurement laid over the average. It is derived from the same window, the same input price and the same bars, which is the single fact most of the rest of this lesson follows from.

Two settings fully specify a set of bands: the length of the window and the multiple applied to the standard deviation. Twenty and two are the numbers the construction was published with, and they are inherited rather than derived, exactly as the moving average lengths in circulation are. Changing either one changes every line on the chart and therefore changes every date on which a close appears to have reached a band. A set of bands quoted without both settings, and without the timeframe they are drawn on, does not describe anything specific.

What the standard deviation measures 

A standard deviation is a measure of spread, and the procedure that produces it is short. The mean of the values is taken. Each value's distance from that mean is measured, and each of those distances is squared, which removes the sign so that distances above and below the mean count the same. The squared distances are averaged, and the square root of that average is the standard deviation. Squaring and then taking the root returns the answer to the units of the original data, so a standard deviation of closing prices is quoted in the same units the prices are quoted in.

One property of that procedure matters more than the rest. Because the distances are squared before they are averaged, a single close far from the mean contributes disproportionately to the result. A window containing one large excursion and many small ones returns a larger standard deviation than a window whose closes are evenly scattered by the same total amount. The measure is not a description of the typical bar. It is a description of the spread of the whole window, weighted towards its extremes.

One step in that table is a genuine and unresolved difference between charting packages. The squared distances can be divided by the count of closes, which treats the window as the entire population being described, or by one less than the count, which treats it as a sample drawn from something larger. The second divisor returns a slightly larger standard deviation and therefore slightly wider bands. Neither convention is wrong, and software does not always state which it uses, so identical settings on identical prices in two applications can draw bands in marginally different places.

What the width of the envelope records 

The distance between the two outer lines is the standard deviation multiplied by twice the chosen multiple, and nothing else enters it. The middle line does not affect the width, the direction of the market does not affect it, and the absolute level of price does not affect it. Width is a direct rendering of one quantity: how far the closes in the current window sat from their own mean.

Width is what people mean when they say the bands measure volatility, and the claim is accurate provided volatility is understood in its narrow statistical sense: the dispersion of a set of closes that have already printed, over a window that has already elapsed, on one timeframe. It is not a measure of how far price travelled inside each bar, since only closes are used and the highs and lows never enter the calculation. It is not a measure of anything expected. A wide envelope is a record of a scattered window and a narrow envelope is a record of a bunched one.

What a close at a band describes 

A close sitting on the upper band states one thing exactly: that close is the chosen multiple of standard deviations above the mean of the closes in the current window. A close on the lower band states the same in the other direction. That is a description of where one number sits relative to a small set of recent numbers, and it is complete. There is no second statement folded into it about the level being high or low, about a move being finished, or about what the next bar contains.

The reason the two standard deviation multiple was chosen is worth knowing, and so is the reason it does not carry the meaning it appears to. In a normal distribution, roughly ninety five observations in a hundred fall within two standard deviations of the mean, which makes an observation outside that distance an unusual one for that distribution. A rolling window of closes is not a normal distribution, and three separate features of the construction break the analogy. The mean and the standard deviation are recomputed every period, so the reference point moves. The window is short, so both are estimated from very few observations. And the close being judged is itself one of the observations used to compute the mean and the deviation it is being compared against.

That last feature has a consequence visible in arithmetic. Because an unusually distant close enters the window it is being measured against, it raises the standard deviation of that window, which pushes the band outward at the same moment the close is reaching for it. The band is not a fence that price arrives at. It is a line that moves in response to the same close that appears to be testing it.

Contraction, expansion, and walking a band 

When successive closes cluster tightly around their own mean, the standard deviation falls and the envelope narrows. Practitioners call a period of unusually narrow bands a squeeze, and the word is a description of the chart to the left of it: it records that closes have been bunched, over the window, up to now. Some traditions hold that a narrow envelope is followed by a wide one, and rest the claim on the observation that width is bounded below by nothing and unbounded above, so a series cannot sit near its floor forever. Critics of that reasoning point out that it establishes only that width eventually changes, which was never in question, and fixes neither when nor by how much.

Two limits are conceded by the practitioners who make that argument. The arithmetic dates nothing: a window can stay narrow for as long as closes keep clustering, and the construction contains no clock. And the width of the envelope is direction free by definition, because squaring the distances discarded the sign, so a contraction says nothing whatsoever about which way a subsequent expansion would run. An expansion is also visible only after the closes that produced it have printed and entered the window, which is the same lag the moving average lesson derived, inherited intact.

The mirror situation is the one most often misread. During a sustained move in one direction, closes cluster near the band on that side while the whole envelope travels with the mean, and successive closes touch or sit just outside the same band for a long run of bars. Practitioners call this walking the band. It is not a malfunction and not a contradiction of the construction: a directional move is precisely a set of closes that keep sitting on one side of their own recent mean, and that is what the arithmetic is built to report. Repeated contact with one band and a sustained move in that direction are the same fact described twice.

Where the description breaks down 

• A step change in price. After a gap or an abrupt repricing, the window carries closes from both sides of the change for its whole length. The standard deviation is inflated by a difference that occurred once, so the envelope reports elevated dispersion for as many periods as the window is long, and then collapses when the pre change closes fall out of it.
• The settings are not fixed by anything. The window length and the multiple are both chosen, and every touch, every squeeze and every walk on a chart is a consequence of that pair. Adjusting them until past touches look convincing is fitting the setting to the chart it will be judged on.
• The divisor convention. Dividing by the count of closes or by one less than the count produces different bands from identical prices, and packages do not always say which they use.
• Session breaks and data conventions. The bands are built from closes, so wherever two providers define a session differently or fill a non trading period differently, the mean and the dispersion both differ, and so do all three lines.
• The timeframe. A twenty period window covers hours on an hourly chart and a month on a daily one. The same instrument carries contracting bands on one timeframe and expanding bands on another at the same moment, and neither is the instrument's volatility.
• The highs and lows are absent. Only closes enter the calculation, so a period that travelled a long way and came back contributes nothing to the standard deviation, while a period that barely moved but closed away from the mean contributes a great deal.

Where practitioners disagree 

The largest disagreement is about what a close at a band is taken to indicate, and the two positions are flat contradictions of each other. One tradition reads a touch as a stretched condition and expects closes to return towards the middle line. Another reads the same touch as evidence of a move with enough force behind it to push closes to the edge of their own recent distribution, and expects continuation. The two schools look at the same line on the same chart and draw opposite conclusions from it, which is itself the most useful thing to know about the touch: it does not carry a direction. The arithmetic states where a close sits relative to a recent window, and both readings are interpretations laid on top of that statement rather than consequences of it.

The second disagreement is statistical. Critics of the construction argue that applying a normal distribution's two standard deviation convention to a short rolling window of prices borrows the authority of a statistical result whose conditions are not met, since the observations are not independent, their distribution is not normal, and the observation being tested is inside the sample that defines the test. Defenders answer that the envelope was never presented as a probability statement, only as a relative measure of position that adapts to the instrument and the period rather than using a fixed distance, and that its value is legibility rather than inference.

The third concerns the middle line. Some practitioners treat it as a reference in its own right, on the argument that it is a widely watched average and the whole envelope is anchored to it. Others treat it as scaffolding whose only job is to give the standard deviation something to be measured from, and pay attention to width alone. Nothing in the calculation settles that, because the construction defines the middle line as an input to the outer two and says nothing about what either is for.