Every volatility band you have ever seen on a crypto chart was probably built the same way: take the daily standard deviation, multiply by the square root of the horizon, and draw. It is one line of code, it is what the textbook says, and for the horizons people actually care about it is wrong in both directions — sometimes too wide, sometimes too narrow, with the sign flipping as the horizon grows.
We can show this, because we had to build the alternative for a product that gets scored in public. What follows is the measurement, the sample-size correction that most published versions of it skip, and what we changed after seeing the result — including a claim of our own that the data killed.
What sigma times root-t assumes
The scaling rule falls out of one assumption: that log returns are independent draws from the same distribution. Under that assumption, variance adds linearly with time, so standard deviation grows with the square root of it. Nothing about the rule is specific to finance; it is the random walk, and it is a good first approximation of a great many things.
Crypto returns are not independent draws from the same distribution. Volatility clusters — quiet weeks follow quiet weeks and violent days arrive in packs. The distribution is fat-tailed at short horizons and skewed at most of them. Every one of those violations bends the scaling rule, and the interesting question is not whether it bends but which way, and by how much, at the horizons anyone actually forecasts over.
The measurement
Binance daily closes for BTCUSDT and ETHUSDT, from listing in August 2017 to July 2026 — 3,261 candles each. For each horizon we take every overlapping window of that length, compute the log return, and read the empirical quantiles straight off the sample. Against that we put the Gaussian band implied by the daily standard deviation scaled by root-t.
The daily standard deviation is 3.56% for BTC and 4.60% for ETH over the full history.
**BTC, width of the 10th-to-90th-percentile band:**
1 day empirical 7.3% root-t 9.1% ratio 0.80 7 days empirical 21.3% root-t 24.1% ratio 0.88 30 days empirical 50.1% root-t 50.0% ratio 1.00
**ETH, same:**
1 day empirical 9.6% root-t 11.8% ratio 0.82 7 days empirical 27.3% root-t 31.2% ratio 0.88 30 days empirical 67.0% root-t 64.6% ratio 1.04
At one day the Gaussian band is a fifth too wide. At thirty days it has converged, and for ETH it has crossed over and is now slightly too narrow. The error does not shrink monotonically toward the truth; it changes sign.
Why the sign flips
The one-day result looks paradoxical next to the fattest tails in the sample. Excess kurtosis of daily BTC returns is 15.8. That is an enormous number — and it is entirely a statement about the extremes, not about the middle.
Fat tails at fixed variance have to be paid for somewhere, and they are paid for in the shoulders. A distribution with a handful of enormous days and many more small ones has the same standard deviation as a Gaussian with a fatter middle. The 10th and 90th percentiles sit inside the shoulders, not out in the tails, so the empirical 80% band comes out narrower even though the 99.9th percentile is far beyond anything a Gaussian would predict.
That has a practical consequence worth stating plainly: a root-t band drawn at 80% is too conservative day to day and too aggressive in the extremes. It is not "safe because it is wide". It is wide in the region where nothing happens and narrow in the region that hurts.
As the horizon lengthens, sums of returns pull toward the central limit theorem and kurtosis collapses — 15.8 at one day, 3.5 at seven, 1.6 at thirty. The empirical middle stops being unusually thin. Meanwhile volatility clustering, which the scaling rule cannot represent at all, has had time to accumulate, and the empirical band catches up and passes the Gaussian one.
Skew moves too, and in a direction that surprises people: daily BTC returns are negatively skewed at -0.96, but thirty-day returns come out slightly positive at +0.15. The shape of the distribution is not fixed and then scaled. It is a different shape at every horizon, and a single sigma cannot carry that information no matter what you multiply it by.
The part almost everyone skips
Here is where most published versions of this analysis quietly overstate themselves, and where we had to correct our own charts.
Those 3,231 thirty-day windows are not 3,231 observations. Consecutive windows share 29 of their 30 days. The sample contains roughly 107 genuinely independent months, and the standard error of a quantile estimate scales with the independent count, not the overlapping one. Reporting 3,231 next to a percentile invites the reader — and the author — to believe the estimate is thirty times more precise than it is.
The correction is arithmetic, not clever: overlapping windows divided by horizon length.
1 day 3,260 windows 3,260 independent 7 days 3,254 windows 464 independent 30 days 3,231 windows 107 independent
Overlapping windows are still the right thing to use — throwing away 96% of your data to get non-overlapping samples is a worse trade — but the honest sample size is the small number, and it belongs next to the estimate. We now print it on every chart we publish: *band from 107 independent 30d windows*. Not because it flatters us. Because the alternative is a number that reads as evidence and is not.
What this killed
Our own engine carried a comment, written earlier and believed since, saying that BTC's 30-day empirical tails come out **wider** than the normal band and that root-t was therefore understating risk. The full measurement says the ratio is 1.00 for BTC and 1.04 for ETH. The effect we had written down as a finding was, at that horizon, essentially nothing.
The claim survived as long as it did because it was checked once on a shorter history and never re-checked. That is the ordinary way wrong things persist in a codebase — not through dishonesty, but because nobody re-runs the measurement after the comment is written. The comment is fixed now.
Why we do not publish a yearly horizon
The same arithmetic settles a question we get asked often. A one-year band from nine years of BTC history rests on nine independent observations. From ETH, the same. Paging the exchange back to listing does not fix this; nothing fixes it except waiting a decade.
Nine observations cannot support a quantile estimate that anyone should act on, and dressing the result in a chart does not change that. There is a second reason too: a yearly forecast returns nothing to a public scoreboard for twelve months, which makes it unfalsifiable in practice for exactly as long as it is interesting. So there is no yearly horizon, and there will not be one.
How a band like this gets scored
A distribution is not scored the way a point call is. The question is not whether the median was close; it is whether outcomes land inside the stated band as often as the band claims.
Over many forecasts, a 50% core zone should contain the outcome about half the time. Both directions are failures. Too few and the band was overconfident. Too many and it was cowardice — a band wide enough to always be right carries no information, and there is no scoring rule under which that should be rewarded. Coverage catches the second failure mode, which is the one that flatters the forecaster and is therefore the one that goes unreported.
One more caveat belongs on any multi-asset version of this: eight coins forecast on the same morning are not eight independent tests. When Bitcoin leaves its band, most of the others leave theirs on the same candle. A coverage statistic pooled across correlated assets inflates the apparent sample the same way overlapping windows do — the same mistake, one level up.
No method reliably beats a liquid market, and anyone promising that is selling something. What a method can do is state its uncertainty in a form that can be checked, publish the sample size that uncertainty rests on, and then submit to being counted. All three of those are unglamorous, and all three are missing from most of what gets published with a band drawn on it.
Educational content — not financial or betting advice.