Standard Deviation: The Root of Bollinger Bands

Every volatility indicator you use traces back to one formula. Understand standard deviation and you understand them all.

Bollinger Bands, Keltner Channels, volatility stops, the ATR-normalized position size — all of them rest on the same statistical foundation: standard deviation. Strip away the marketing and you'll find σ doing the work.

Standard deviation refresher

σ measures dispersion around the mean:

σ = √( Σ(xi − μ)² ÷ n )

For a sample (which is what traders always have), we use the unbiased version:

s = √( Σ(xi − x̄)² ÷ (n − 1) )

The (n − 1) is the Bessel correction — it corrects the downward bias that comes from using the sample mean in place of the true mean.

How Bollinger Bands are built

A Bollinger Band is three lines:

Middle = SMA(price, n)
Upper  = Middle + k · σ
Lower  = Middle − k · σ

Where σ is computed over the same n-bar window and k is typically 2.

So the band width is 4σ. When volatility rises, σ rises, and the bands widen. When markets quiet down, σ falls and the bands squeeze. That "squeeze" is just σ collapsing to a multi-period low — a volatility regime change, not a directional signal.

What the bands actually tell you

1. Relative expensiveness: price near the upper band is statistically high for this window; near the lower, statistically low
2. Volatility regime: band width is a literal volatility gauge
3. Squeeze expansion: a low-σ period is often followed by expansion — but the band can't tell you which direction price breaks out

The critical caveat

The "95% of price stays within 2σ" rule is true only if returns are normally distributed. As we covered in the fat-tails article, market returns are not — they have negative skew and excess kurtosis. Real markets blow through 2σ Bollinger Bands far more often than 5% of the time. Don't treat the band as a hard wall; treat it as a soft, leaky envelope.

Practical uses

1. Mean-reversion entries: buy touches of the lower band in a range, exit at the middle band
2. Trend filter: in a strong trend, price "rides" the band — don't fade it
3. Volatility sizing: scale position size inversely with σ so each trade risks a constant dollar amount
4. Stop placement: set stops at entry ± k·σ, where k reflects your fat-tail tolerance

Summary

Bollinger Bands are not magic — they're standard deviation wrapped around a moving average. Once you see that, you can build your own volatility channels, normalize stops across instruments, and stop trusting the bands as a probability wall. σ is the unit; everything else is decoration.