Central tendency: where a set of numbers sits

Why it matters

Kitalpha’s page for the 10-year Treasury yield, the 10-year tenor, records a change every trading day. Over its last 250 trading days, from 2025-09-15 to 2026-09-15, those changes averaged 0.0038 percentage points a day. Is that what an ordinary day looked like? An average is one number standing in for hundreds, and it is tempting to read it as “the typical day”. Whether that reading holds depends on which centre the number describes, and on whether the days it summarises were evenly balanced or lopsided. This lesson gives three ways to locate the centre of a set of numbers and shows, on this record, how to tell when they agree.

The concept

This lesson reads one set of numbers: the daily changes in the 10-year Treasury par yield. A daily change is the yield on one trading day minus the yield on the day before, measured in percentage points; one basis point is a hundredth of a percentage point, so a change of 0.05 percentage points is 5 basis points. Central tendency is the question of where such a set sits: which single number best stands in for all of them. Three answers are in common use.

The mean is the sum of all the changes divided by their count. It is the balance point: if each change were a weight placed on a ruler at its value, the mean is where the ruler balances. Every value pulls on the mean, and a value far from the others pulls hardest, because the balance point has to move a long way to offset it. Since the mean times the count gives the sum back, the mean daily change times the number of days is the net move in the yield across the window.

The median is the middle value. Sort the changes from the smallest to the largest and take the one in the middle position; with an even count, take the average of the two middle values. The median answers a different question from the mean: half the days had a change at or below it and half at or above. Because it is defined by position rather than by size, the largest day in the window could double or treble and the median would barely move.

The mode is the most common value. For measured values that rarely repeat exactly, such as yield changes recorded to four decimal places, the mode is read from a histogram instead. A histogram sorts the changes into bins of equal width and draws a bar for each bin whose height is the number of days that fell in it; the tallest bar is the modal bin, the range of changes that occurred most often.

On a symmetric set, one that spreads out evenly on both sides of its centre, the three measures all but coincide, and “the average” is a fair description of an ordinary day. On a lopsided set they part. A handful of days with large changes on one side pull the mean towards that side, while the median stays with the crowd of ordinary days and the mode sits under the tallest bar. The gap between the mean and the median is therefore the first thing an analyst checks. A gap near zero says the balance point and the middle day agree. A clear gap says the set has a longer tail on the side the mean leans towards, and that the mean lands where few actual days sit. The error this lesson targets is reading the mean as the typical day without making that check.

The histogram below is drawn from the live 10-year record. Look first for the tallest bar, which marks the mode; then for the two vertical lines marking the mean and the median; and finally for how far apart those two lines are.

Worked example

Take the record from the opening: the daily changes in the 10-year yield over its last 250 trading days, ending 2026-09-15. The steps below count the changes, add them up, divide to get the mean, read the median from the sorted list, and measure the gap between the two in basis points. Every figure is computed from the record’s own list of changes on the date shown.

Read the steps in order. The sum is the net move in the yield across the window; dividing it by the number of days gives the mean, usually a small number because up days and down days largely cancel. The median comes from position, not from adding: it is the value with as many days below it as above. The last line converts the difference between the two to basis points. A figure near zero says the two centres agree and the window is close to symmetric; a figure clearly away from zero says a few large days on one side pulled the mean off the middle day. Check it against the histogram above: the mean line sits on the side of the median line where the bars run out further, and the tallest bar need sit on neither.

Faded example

Now the second record: the daily changes in the 2-year yield over its last 250 trading days, ending 2026-09-15. The count, the mean and the median are given below in percentage points. Complete the last step yourself, the mean minus the median in basis points, then reveal the answer and compare.

Retrieval check

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

Your summary