Why a six-sigma day in daily data should come once in 1.4 million years
In a normal distribution about 68 per cent of values fall within one standard deviation of the average, 95 per cent within two and 99.7 per cent within three. The shortcut is handy, but it quietly assumes thin tails. When real data throw up six-sigma days, the bell curve is usually the thing that is wrong.
The rule, also called the empirical rule or the three-sigma rule, is a memory aid for the normal curve. More precisely, the shares are about 68.27, 95.45 and 99.73 per cent, figures that come from the normal distribution's cumulative distribution function. Many sciences treat the 99.7 per cent band as practical certainty, so nearly every value is expected to lie inside three standard deviations.
Different fields draw the line for a real effect in different places. Social scientists often treat a two-sigma result, around 95 per cent confidence, as worth a closer look. Particle physicists are far stricter, demanding five sigma, or 99.99994 per cent confidence, before announcing a discovery.
The big caveat is shape. Lots of real-world distributions look bell-like but have fatter tails than the normal curve, which packs an unusual share of probability near its centre. For such data the three bands capture less than advertised. Weaker guarantees still hold: Chebyshev's inequality promises at least 88.8 per cent of any distribution within three sigma, and for single-peaked distributions the Vysochanskij-Petunin inequality raises that floor to 95 per cent.
Practitioners also run the rule backwards as a diagnostic. Convert each observation into a number of standard deviations, by standardising if the true parameters are known or studentising if they are estimated, and compare the counts with what a normal curve predicts. Points beyond three sigma are probable outliers unless the sample is huge, and several four-sigma moves in 1,000 observations should cast doubt on normality. The odds fall away exponentially: a six-sigma event has roughly a two in a billion chance, so for daily data it should appear about once every 1.4 million years. Seeing one far sooner means the normal model is failing.
Source: 68–95–99.7 rule
