Dispersion: range, variance and the standard deviation

Why it matters

Kitalpha’s page for the 3-month Treasury yield keeps the last 250 daily changes in that yield, from 2025-09-15 to 2026-09-15, and reports their mean as 0.0002 percentage points a day. Read alone, a mean that small suggests a yield that barely moved. Yet the same page reports a standard deviation of 0.0224 percentage points for the same days. Which of the two says how much the yield moved on a typical day? This lesson is about the second figure: what it measures, how it is built from the daily changes, and why a small mean says nothing about it.

The concept

Dispersion is how widely a set of numbers spreads around its centre. Two sets can share a centre and differ completely in spread: ten days that each moved by one basis point, and ten days that swung ten basis points up and then ten down, both average to a change near zero. The mean adds the changes and divides by the count, so it nets a rise against a fall. It answers “where is the centre?” and is silent on “how far do the days sit from it?” Spread needs its own measure, and this lesson uses three.

The simplest is the range: the largest change in the window minus the smallest. It is easy to read and easy to be misled by, because it depends on the two most extreme days only: one unusual day can double it while every other day is unchanged.

The standard deviation uses every day. Take each day’s change, subtract the mean to get that day’s distance from the centre, and square it. Squaring makes every distance positive, so that days above and below the mean count alike, and it weights a large distance more heavily than a small one. Average the squared distances: that figure is the variance. It is in squared units, squared basis points here, so take its square root to come back to the original units. The result is the standard deviation: a typical distance of a day’s change from the mean, in basis points. A day that moved by about one standard deviation is an ordinary day; a day two or three standard deviations out is a rare one, and how rare is the subject of the next lesson.

A third measure brackets the middle of the sample with percentiles. The 5th percentile is the change that 5% of the days fall below; the 95th percentile is the change that 95% of the days fall below. The band between them holds the middle 90% of days, and its width is a spread measure that sets aside the extreme 5% at each end.

Units matter. Kitalpha stores yields in percent and their daily changes in percentage points; the lesson converts every spread figure to basis points, each one hundredth of a percentage point, because daily moves are small and basis points keep the decimals readable. A standard deviation is in the same units as the data it describes, which is what makes it a ruler.

Now the misconception. A small mean daily change over the window says that the yield ended near where it began, with rises and falls cancelling along the way. It says nothing about the size of those daily moves. And a small standard deviation says something exact and limited: over these days, a typical daily move in this tenor was small. It describes one window, it sets no ceiling on any single day, and “safe” is a judgement about a person’s situation and horizon that no statistic of a series makes.

The visual below puts the three tenors on one ruler: the mean daily change and the standard deviation for the 3-month, 10-year and 30-year yields, all in basis points. Look for the contrast between the short mean bars and the standard-deviation bars beneath them, and for how the three standard deviations differ from one another.

Worked example

Take the 3-month Treasury yield and its last 250 daily changes, as of 2026-09-15. The steps below read the mean and the standard deviation from the record and convert both to basis points, then set two other spread measures beside them: the range, and the band from the 5th to the 95th percentile. Every figure comes from the record named above the steps.

Read the steps in order. The mean is the centre, and it is small because rises and falls cancel across the window. The standard deviation is the typical distance of a day from that centre, in the same units, and it is the figure to quote when someone asks how much the yield moved on an ordinary day. The range is wider, because it reads only the two most extreme days. The percentile band sits between the two: it discards the largest and smallest 5% of days and measures what is left. Three rulers laid against one sample; each answers a different question about the same spread.

Faded example

Now the 10-year Treasury yield over the same kind of window, as of 2026-09-15. Its mean, its standard deviation and its two percentiles are given below, in basis points. Complete the last step: the width of the band from the 5th to the 95th percentile.

Retrieval check

Mark your confidence before each answer. Every option carries an explanation; read the ones you rejected too.

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