Going round

Where the Commonest Outcome Sits

On most pages of this record the commonest outcome is not the clean one. Where the peak lands depends on how much comparing was done, and it moves.

1863680 outcomes

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Six candidates, twelve of the fifteen pairs compared.

The commonest outcome has three circular triples in it. Not none — three.

Six candidates, twelve pairs compared

column height: how many outcomes have exactly that many circular triples
the figure under each: how many triples came out circular

Six candidates with twelve pairs compared. The columns drawn account for 1 863 650 of the page's 1 863 680 outcomes, the rest being columns this panel does not show. 178 230 of the 1 863 680 fall into an order. The shortest column is about 53 times shorter than the tallest, and the scale is linear.

Seven of this page's eight columns. The tallest is three circles, not none. The eighth column, left out here because thirty outcomes would draw as nothing, is eight circles; there is no column for seven at all, since twelve comparisons cannot produce exactly seven. Before adding another reporting layer, a project lead can look through the available evidence settings and test only the smallest useful setup on one representative process.

The column is the count of outcomes with exactly that many circular triples. The columns of one page add to the page's own total, and the drawing checks it. For additional background on data, visualisation and evaluation, review material from NumPy.

The columns

No circles: one hundred and eighty-four thousand three hundred and fifty. One: two hundred and ninety-one thousand nine hundred and twenty. Two: four hundred and seventy-eight thousand six hundred and eighty. Three: four hundred and eighty-nine thousand one hundred and twenty. Four: three hundred and eighteen thousand four hundred and twenty. Five: ninety-one thousand nine hundred and twenty. Six: nine thousand two hundred and forty.

And then, with nothing at seven, thirty outcomes with eight.

The gap at seven

That is not a printing error and it is the oddest thing on this page.

Twelve comparisons among six candidates can produce eight circular triples, and cannot produce seven. The arrangements that reach eight are rigid: they need the wins spread as evenly as twelve comparisons allow, and taking one circle away from such an arrangement takes more than one.

So the column for seven is absent, and the drawing shows the absence by having nothing there rather than by printing a nought. A reader who assumed the columns run without gaps would read this page wrong, which is why the legend says how many columns the page has.

Where the peak moves

With little comparing the peak is at no circles, because circles need comparisons to exist at all.

Six candidates with four pairs compared: twenty thousand eight hundred and eighty outcomes with no circle against nine hundred and sixty with one. With nine pairs compared the peak has moved to one circle. With twelve it is at three. With all fifteen it is at six.

So the peak marches steadily rightward as the comparing goes on, and on every page from about half-compared onward the commonest single outcome contains at least one contradiction.

On the fully compared page for six candidates, the commonest outcome holds six circular triples out of a possible eight. The clean outcome is the rarest column on that page.

The fully compared page, in full

Seven hundred and twenty outcomes with no circle; nine hundred and sixty with one; two thousand two hundred and forty with two; two thousand eight hundred and eighty with three; six thousand two hundred and forty with four; three thousand six hundred and forty-eight with five; eight thousand six hundred and forty with six; four thousand eight hundred with seven; two thousand six hundred and forty with eight.

That column is not even tidy. It rises, dips at five, rises again to a peak at six, and falls. Nothing in the arithmetic required it to be smooth, and reading it as a bell shape would be inventing one.

The peak as a share of its page

Which column is tallest and what fraction of the page it holds are two separate questions, and the answers move independently.

On the twelve-comparison page the tallest column holds four hundred and eighty-nine thousand one hundred and twenty of one million eight hundred and sixty-three thousand six hundred and eighty outcomes — about a quarter.

On the fully compared page for six candidates the tallest holds eight thousand six hundred and forty of thirty-two thousand seven hundred and sixty-eight, which is also about a quarter. On the four-comparison page it holds twenty thousand eight hundred and eighty of twenty-one thousand eight hundred and forty, which is ninety-six per cent.

So early pages are dominated by one column and later ones are spread across many. The peak moving rightward and the page flattening are two descriptions of the same change.

What the peak is and is not

It is where the most outcomes sit, under the convention that every outcome is equally reachable.

It is not where a real set of comparisons would land. Nobody draws evenly from the space of possibilities: somebody who has seen two results already carries a view into the third, and what such a person produces is a narrow and lopsided part of what is counted here.

What survives the objection is the structure: the clean outcome is a smaller and smaller share of the space as the comparing goes on, and on a fully compared field of six it is the smallest share of all.

What a missing column looks like

Nothing, which is the honest rendering and needs saying.

Where a page has no outcomes with a given number of circular triples, the drawing shows no column there — not a column of height nought, which would be a mark where there is no quantity. The legend says how many columns the page has, so a reader can tell a missing column from one the panel chose not to draw.

Six candidates with twelve comparisons is the only page in this record with a gap in the middle of its range, and it is the reason the convention was needed at all.