Thresholds

Half Of Them, Which Admits Most People

Four of eight optional items admits one hundred and sixty-three profiles in two hundred and fifty-six. The rule that sounds like a coin toss is not one.

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A list of eight optional items requiring four of them sounds like a fifty-fifty proposition. It admits one hundred and sixty-three profiles in two hundred and fifty-six, which is about sixty-four per cent.

A clear majority. The rule that reads as "half" is substantially more generous than half.

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Why the majority

Because "four or more" is not "exactly four". The rule admits every profile meeting four items, five items, six, seven or all eight, and those add up to more than the profiles meeting three or fewer.

The asymmetry comes from where the single remaining profile sits. There is exactly one way to meet none of the eight and exactly one way to meet all eight, and the threshold of four includes the all-eight case and excludes the none case — along with everything else above and below.

Counting it out: the profiles meeting four or more number one hundred and sixty-three, and those meeting three or fewer number ninety-three. The split is not even because the midpoint of nine possible counts is four, and a threshold of "four or more" takes five of the nine bands rather than four and a half.

Threshold on eight items	Admits	Reads as
4 of 8	163/256	"half"
5 of 8	93/256	"most"

A threshold of five, which most people would describe as "more than half", admits ninety-three in two hundred and fifty-six — about a third. The two rules sit adjacent on the staircase and land on opposite sides of a half.

The practical consequence

Anybody choosing a threshold by how it sounds will be badly calibrated.

"Half of these" is a wide filter admitting two profiles in three. "Most of these" is a narrow one admitting one in three. The step between them is a single item and a factor of nearly two, and nothing in the language signals it.

The threshold that sounds moderate is wide, and the one that sounds slightly stricter is narrow. There is no vocabulary for the difference, so it has to be computed.

Where the midpoint actually is

If the intention is to admit about half of the possibility space, the threshold to set on eight optional items is somewhere between four and five, and since thresholds are whole numbers it is not available.

Four admits sixty-four per cent. Five admits thirty-six per cent. Nothing in between exists, and a list of eight cannot express "half" exactly however it is written.

Lists of other lengths behave the same way. The threshold is a whole number, the staircase has flat treads, and the intake jumps from one tread to the next with nothing in between — which is the structural reason that fine-tuning a threshold is not possible and choosing one is a discrete decision between a small number of quite different filters.

The same effect at other lengths

The gap between what a threshold sounds like and what it admits is not peculiar to eight items. It appears at every length, with the same direction.

On six optional items, three-of-six admits twenty-one in thirty-two — about two thirds, again a clear majority for the rule that reads as half. Four-of-six admits eleven in thirty-two, about a third.

So the pattern holds: the rule at the apparent midpoint is comfortably wide, and the rule one step above it is comfortably narrow, with nothing available in between.

Why there is no middle to find

Because the distribution of how many items a profile meets is symmetric and peaked, and a threshold takes a running total from one end.

For an even number of optional items there is a central band — profiles meeting exactly half — and that whole band falls on one side of any threshold. On eight items that band holds seventy profiles out of two hundred and fifty-six, more than a quarter of everything, and it goes entirely to whichever side the rule puts it on.

That is the structural reason the staircase jumps across the middle rather than passing through it, and it is why "half of them" is never available as a filter admitting half.

What the sixty-four per cent is not

It is sixty-four per cent of configurations, not of candidates.

Under the convention this record uses, each item is counted as a separate coin, so the two hundred and fifty-six profiles are the ways a list of eight can be met and unmet. One hundred and sixty-three of those pass a threshold of four.

How many actual people that corresponds to depends entirely on how often each item is held and on how the items relate to one another, neither of which is on the page. The figure describes the document. It does not forecast an intake, and the section on what this record cannot do returns to the distinction at length.

What to do with that

Pick the tread, not the adjective.

Look at the two or three thresholds that plausibly fit the work, read off what each admits, and choose. On eight items the realistic candidates are three, four and five, admitting two hundred and nineteen, one hundred and sixty-three and ninety-three profiles respectively — a range of more than two to one across three adjacent rules.

That is a real choice with a real consequence, and it takes under a minute to make properly. Made by vocabulary instead, it is a coin toss between filters that differ by a factor of two.