Why the median for a lopsided sample: robustness to outliers

Why it matters

Kitalpha’s 30-year Treasury par yield page carries the yield’s history, and from it the record used here takes the last 250 one-day changes, from 2025-10-30 to 2026-10-30. The middle value of those changes, once they are sorted, is 0.000 percentage points. The largest of them is 0.130. A reader who wants one figure for “a typical day” has two candidates, the mean and the median, and on real records they can disagree. This lesson is about why they disagree, and which of the two a handful of extreme days can move.

The concept

A sample is a set of numbers taken from a record: here, the one-day changes in a yield over a window of trading days. Two figures describe where such a sample centres. The mean adds every value and divides by the count. The median sorts the values from smallest to largest and takes the middle one, or, with an even count, the average of the two middle values. On a sample that is symmetric about its centre the two coincide. On real market data they often do not, because of how each treats the values far from the rest.

An outlier is a value far from the bulk of the sample. Consider five daily changes, in basis points: −2, −1, 0, 1 and 2. The mean is 0 and the median is 0. Now replace the 2 with 20. The mean becomes (−2 − 1 + 0 + 1 + 20) ÷ 5 = 3.6 basis points; the median is still 0, because the sorted list still has 0 in the middle. The mean takes every value at its full size, so one value can carry it anywhere. The median only asks how many values sit above and below it, so the size of the largest value is irrelevant once it is the largest. A figure that a few extreme values cannot move far is called robust, and the median is robust in exactly this sense while the mean is not. A bigger sample dilutes the effect without removing it: with 250 days rather than five, one outlier shifts the mean by its distance from the mean divided by 249. The shift is smaller in basis points, but it still grows without limit as the outlier grows, and the median still moves at most one place.

That difference is what makes the gap between the mean and the median useful when reading a record. The distribution of a sample is the pattern of how often each size of value occurs, and its tails are the thin ends where the rare, large values sit. When the tail on one side reaches further out than the tail on the other, the sample is said to be skewed toward that side. The extreme values in the longer tail pull the mean toward them; the median stays with the bulk. So the mean sits on the side of the longer tail, and the sign of mean minus median says which side that is. The size of the gap depends on the units and on how widely the sample is spread, so analysts scale it. Dividing the gap by the standard deviation, the typical distance of a value from the mean, gives a figure with no unit, comparable between series whose days typically move by different amounts. Kitalpha’s series records report this scaled gap as a skew proxy, and the same scaling expresses any single day as a number of standard deviations from the mean.

The histogram below draws the 30-year record’s sample as bars, one per range of daily change, with the mean and the median marked. Look for the thin bars at the far left and far right: the side that reaches further is the longer tail, and the mean’s line sits on that side of the median’s.

Worked example

Take the record from the opening: the 30-year par yield‘s last 250 daily changes. The record’s figures are in percentage points; the steps convert them to basis points. Steps one to three read the two centre measures and their difference. Steps four and five divide that difference by the standard deviation, giving the unit-free skew proxy. Steps six and seven place the largest single day in standard deviations above the mean; the last step measures its pull on the mean.

The gap in step three carries a sign, and the sign is the reading: the mean sits on the side of the longer tail, or on neither if the gap is near zero. Step five restates that gap in units of a typical day’s move, the form that travels between series. Step seven says how far out the largest day sits: a bell-shaped sample keeps about 95% of its values within two standard deviations, so a figure well beyond two marks a day the bulk does not resemble. The last step is the robustness test: dropping that day shifts the mean by that many basis points, and the median by at most one place in the sorted list. At the other end of the sample the smallest single-day change was -0.130 percentage points; the retrieval check asks how far below the mean it sits.

Faded example

Now a second record: the daily changes of the 2s10s spread, the 10-year par yield minus the 2-year, over its own last 250 trading days. The mean, the median and the standard deviation are given below, in basis points, as the worked example’s steps read them. Complete the worked example’s fifth step yourself: the mean minus the median, divided by the standard deviation, which puts this record’s gap on the same unit-free scale as the 30-year’s.

Retrieval check

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