Distribution shape and tails: why a big move is an ordinary day

Why it matters

The Kitalpha record for the Henry Hub natural-gas spot price holds 250 daily changes, from 2025-09-04 to 2026-09-09, and reports their mean, their standard deviation and one more figure: the number of days that sit more than two standard deviations from the mean, 5 of them. The largest single-day rise in the window was a change of 22.30 dollars per million BTU, and the largest fall a change of -7.85. A reader who has learned that almost everything sits within two standard deviations might call such days freaks, errors, or once-in-a-lifetime events. Whether they are is the question this lesson answers.

The concept

A distribution is the set of values a series has taken, sorted by size rather than by date. For a price series the natural values to sort are its daily changes: today’s price minus yesterday’s. A histogram draws that sorted set as bars, one per band of values, with each bar’s height counting the days that fell in the band. The centre of the picture is the mean, the plain average of the changes; the standard deviation is the typical distance of a day from that mean, computed from every day in the window. Here it becomes a ruler: a day is described by how many standard deviations it sits from the mean, which puts a natural-gas price and a Treasury yield on one scale.

The bell curve, or normal distribution, is the shape textbooks reach for first. It is symmetric, tallest at the mean, and thins out on either side in a fixed way. Under it, 68.3% of observations sit within one standard deviation of the mean, 95.45% within two, and 99.73% within three. Turn those figures round and they describe the tails, the two thin ends of the curve: about 4.55% of observations lie beyond two standard deviations, about 0.27% beyond three, and about 0.006% beyond four. In trading days, a three-standard-deviation day arrives about once in 370, a four-standard-deviation day about once in 16,000 (roughly sixty years), and a five-standard-deviation day about once in 1.7 million. That is the source of the intuition that extreme moves are impossibly rare.

Market series do not follow the bell curve, and the failure is concentrated in the tails. Prices move on news, and news arrives in lumps: a storage report, a pipeline outage, a cold snap, a policy decision. Calm stretches alternate with turbulent ones, so the days come from a mixture of quiet and wild periods rather than one steady process. The result is a distribution with a taller, narrower centre than the bell curve and more days far out in the tails. That shape is called fat-tailed, or heavy-tailed: the tails carry more of the days than the bell curve allows. Under it, a day four or five standard deviations from the mean is unusual but expected to appear within a few years of daily data, rather than once in a lifetime. Fat tails do not mean that big moves are frequent; they mean that the bell curve badly understates how often the big ones arrive.

The record gives two ways to see this. The first is its count of days beyond two standard deviations, which becomes a share when divided by the number of days; set that share beside the bell curve’s 4.55%. One caution: the extreme days are inside the standard deviation that measures them, so they stretch the ruler and push the two-standard-deviation fence outward; a fat-tailed series can show a share close to the bell curve’s while still holding days the curve treats as impossible. So the second, sharper test is the farthest day itself, measured in standard deviations, read against the once-in-370 and once-in-16,000 figures above.

The histogram below is the Henry Hub record’s distribution of daily changes. Look for two things: how crowded the bars are around the mean marker, and how far the outermost bars sit from it on each side.

Worked example

Take the Henry Hub record: 250 daily changes in the spot price, in US dollars per million British thermal units, with a mean of -0.001 and the standard deviation shown in the first step. The steps read the record’s tail count, turn it into a share of the window, set that share beside the bell curve’s 4.55% and the count the curve implies for a window of this length, and finish with the farthest single day from the mean, measured in standard deviations.

Read the third and fourth lines together first: the record’s share against the curve’s, and the second and fifth lines as the same comparison in days. Then the last line, which is the sharper test. Under the bell curve a day beyond three standard deviations is expected about once in 370 trading days and a day beyond four about once in sixty years; the record’s farthest day sits at the distance shown, inside a window of 250 days. Every figure comes from the named record, extreme days included, and none of them was set aside.

Faded example

Now a second record: the 10-year Treasury par yield, whose daily changes are measured in percentage points rather than dollars. Its standard deviation, its tail count and its number of days are given below. Complete the last step, the tail days as a share of the window, then set it beside 4.55%.

Retrieval check

Mark your confidence before each answer. Every option carries an explanation; read the feedback for the options you rejected as well as the one you chose.

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