Round numbers are not a sample

The settling table was grown from nine starting angles: 40, 60, 80, 100, 120, the golden angle, 150, 165 and 180 degrees. Every share this collection reports about how often a stem reaches a lattice is a share over those nine.

Eight more were added, at the midpoints between them. The nine settle at 40.6 per cent and the eight midpoints at 25.8 per cent, over the same rises and the same falloff exponents.

That was not the finding the extra angles were grown for. They were grown to halve an error bar and to give basins a width; the bias came out of counting the two groups separately, which was done only so that a refinement could be told from an extension.

That is not noise #

Two hundred and eighty-eight runs against two hundred and fifty-six, which puts a binomial standard error of about three per cent on each share. The gap is fifteen points.

It is also not a fluke of one cell. The nine settle more often at every rise and at every one of the four falloff exponents, which is thirty-two comparisons all running the same way.

And it is not the range #

Three more angles were added below the nine’s lowest — 10, 20 and 30 degrees — because the list started at forty for no stated reason. Those settle at 25.0 per cent.

So the three angles outside the original range behave like the eight inside it. If the fall were about extending the sampling into unfamiliar territory, the three below forty would look different from the eight midpoints, and they do not.

What the nine have that the midpoints do not #

Several of them sit next door to a destination. The table reaches a lattice at 101.6 degrees and one of the nine is at 100; it reaches one at 79.2 and one of the nine is at 80; it reaches one at 151.0 and one of the nine is at 150.

And the golden angle is itself a destination, to within a third of a degree: runs settle at 137.8 and the ninth starting angle is 137.508.

A run that starts next to where it would settle #

That is the account and it is not complicated. Settling is a stem’s arrangement stopping changing, and a stem started at an angle a lattice already sits at has almost nothing to change.

The measured settling times bear it out: the runs that settle fastest are the ones started nearest a destination, and several of the nine settle at organ zero — meaning the run was at its destination from the first organ after the seed.

Where a stem settles at all it does so within about 290 organs, so a settling time of zero is not a fast run; it is a run that had nowhere to go.

Which makes the nine a biased list #

Not deliberately, and not by much of a mechanism. Round numbers in degrees are what a person types, and the golden angle is in the list because the whole subject is about it.

Both are good reasons for those angles to be in a list and neither is a reason for them to be representative. What makes them unrepresentative is that the destinations this rule produces are themselves near round numbers — 79.2, 101.6, 137.8, 151.0 — for reasons that have nothing to do with anybody’s typing.

So the coincidence has to be checked #

It is worth pausing on, because round numbers are near destinations could be a coincidence of four out of nine. The measurement is how many of each group sit within two degrees of a destination.

Of the nine, four do. Of the eight midpoints, one does. That is the account in one comparison, and it is thin — nine and eight are small groups — which is why the settling times are quoted beside it.

How thin is worth a number, because “thin” is a word a reader can round either way. Four of nine against one of eight, on seventeen angles of which five are near a destination, would arise by chance about one time in five. That is not evidence; it is a direction.

So the two halves of this essay are not equally secure and the difference should not be smoothed over. The effect — 40.6 per cent against 25.8, over two hundred and eighty-eight runs and two hundred and fifty-six, with all thirty-two cells running the same way — is as solid as anything in this thread. The account for it, that the nine sit nearer destinations, rests on a four-against-one comparison that a coin produces one time in five.

That ordering matters for what follows from the essay. The finding that the nine are biased stands on its own and requires no mechanism: every share this collection has quoted from the nine is overstated, whatever the reason. The proposed reason is a hypothesis with a settling time argument beside it, and it would take a different sampling — angles chosen at random rather than at midpoints — to test it properly.

Which is also the cheapest way to have both. Twenty angles drawn at random from the range carry no round numbers and no midpoints, so their share is an unbiased estimate; and their distances to the destinations are a continuous variable rather than two groups of nine, which is a much better instrument than a two-by-two.

The destinations themselves are measured rather than assumed: eighteen of them at half a degree, of which five sit on a ladder sequence and thirteen do not.

What it does to every share #

Every settling share in this collection is about fifteen points too high, and the correction is uniform rather than selective: the shares fall at every rise and at every exponent by roughly the same amount.

That is the good case. A uniform bias cancels in any comparison — which is what the table is for — and survives only in a share quoted on its own.

And to the one number that is a share #

The wall is a share read against a level: the rise at which fewer than half the starting angles still reach a lattice. That is the one place in the thread where a share is doing work rather than being compared.

It survives, and more comfortably than before. The four exponents’ walls sit closer together at twenty angles than at nine, which is what the file’s own conclusion said they should — and the wall is a wall rather than a budget at both samplings, since tripling the run length changes no row of the table.

Why a uniform bias does not move a level crossing much #

Because the shares fall by about the same amount everywhere, and the rise at which a falling curve crosses a half moves by however far the curve shifts divided by its slope.

The curves here are steep near the wall — the share falls from 60 to 30 per cent over one step of the rise at the working exponent — so a fifteen-point shift moves the crossing by well under one rise of the eight.

What would not have survived #

A claim of the form a stem reaches a lattice about four times in ten. That is a share quoted on its own and it is now a share over a list of angles chosen partly because they are round.

Nothing in the collection makes that claim, which is luck rather than foresight. The file that grew the table read its own error bar and confined itself to comparisons for that reason, which turns out to have protected it from a bias it had not noticed.

The general shape of it #

A list of parameter values chosen for readability will sit near whatever the system’s own special values are, if the system’s special values are near readable numbers. That is not obvious in advance and it is not rare.

The check is cheap: place a second list between the first one’s members and compare. It costs one more run per cell and it is the only way the bias shows, since every cell of the original table is internally consistent.

What a fair list would look like #

Twenty angles drawn at random from the range would have no structure to be unrepresentative in, and it is not what was used. A random list cannot contain the old one, so every number would have been re-measured and the comparison would have become an argument about two designs.

The midpoint design keeps the old table inside the new one, which is what makes the nine settle more often than the eight between them a sentence about one table rather than two.

What the midpoints are not #

They are not a control in the strict sense. Eight angles halfway between nine others are themselves a structured list, and their structure is as far from the round numbers as possible, which could bias them the other way.

That is a real worry and the three angles below forty are what answers it. They are not midpoints of anything and they settle at the same rate as the midpoints, so the midpoints are not unusually far from destinations either.

The rate the two additions agree on #

25.8 per cent and 25.0 per cent, from two groups constructed differently, is the best estimate this table has of what an unbiased list would give.

It is still not an estimate of the circle. Eleven angles is eleven angles, and both groups avoid the round numbers by construction, so the honest reading is that the true rate is somewhere at or below the nine’s forty per cent and near the twenty-five the others give.

What this says about the angle itself #

There is a temptation to read the golden angle is a destination as a result and it is not one — it is what the whole settling thread found, and the angle is an output of the rule rather than an input to it.

What is new here is narrower: because the golden angle is a destination, a table that includes it among its starting angles is including one starting angle that cannot fail to settle. That is a sampling remark rather than a claim about the angle.

The same distinction has to be kept when reading what a counter says about the settled runs: a destination is where runs go, and a starting angle at a destination is a run that was already there.

And about the ones that never settle #

The other end of the table is worth a sentence. At the finest rises almost nothing settles at any starting angle — one of twenty at a rise of 0.004 — so the bias has nothing to work on there.

That is the wall, it is a property of the rise rather than of the sampling, and it is the part of the table the extra angles leave exactly as it was.

Why this is a selection effect and not an error #

Nothing was measured wrongly. Every one of the 288 original runs is a correct run, every share is a correct share of the angles it is a share of, and no number has to be withdrawn.

What is wrong is a reading: this share is about the rule when it is about the rule and a list. That is a selection effect, and selection effects are the class of problem that survives every check on the measurement itself, because the measurement is fine.

The collection has met one before. An ordering in a residual belonged to the method that measured it rather than to the heads it was measured on, and it too passed every check that asked whether the numbers were right.

What made it visible #

One design decision, and it was made for a different reason. The eleven new angles were split into eight midpoints and three below the old floor because a refinement and an extension answer different questions, and mixing them would have made it impossible to say which had moved a share.

Had all eleven been counted together, the answer would have been the share falls by eight points when eleven angles are added, which reads as an ordinary sampling correction and prompts nothing.

Splitting them turned one number into three, and the three do not agree with each other in the way an ordinary correction would.

What else is grown from a chosen list #

Worth asking, since the same shape could be elsewhere. The rises the table is grown at — 0.030, 0.020, 0.013, 0.008, 0.005, 0.0045, 0.004, 0.003 — are also round numbers, and the ladder’s rungs sit at particular rises.

That list is not obviously biased in the same way, because a rise is not a thing a run settles onto: the rungs are stretches rather than points, and a rise near a rung boundary is not a rise where anything special happens to a run. But nobody has checked it by putting eight rises between these eight, which would cost the same twelve minutes.

The three that agree, and what they are worth #

Two independent constructions gave 25.8 and 25.0 per cent. That agreement is the strongest thing in this reading and it is worth saying why rather than leaving it as a coincidence.

The eight midpoints were built to be as far from the nine as possible; the three below forty were built to extend a range and have no relation to the nine’s spacing at all. Two lists with different construction rules and no members in common landing within a point of each other is what a shared property looks like.

The property they share is that neither is near a destination. One of the eight is within two degrees of one and none of the three is, against four of the nine.

What it would take to be sure #

Forty angles, evenly spaced, with the nine somewhere inside them. That is another 640 runs and about twenty-five minutes and it would give the share over a list with no structure worth speaking of.

It is not obviously worth doing, because the share is not a number this collection uses. What it would settle is whether 25 per cent is the rate or whether the two additions are also unrepresentative in some way nobody has thought of, and the answer to that changes nothing that is currently claimed.

What a reader should carry #

That the nine starting angles the settling table was grown from settle half as often again as angles placed between them, and that four of the nine sit within two degrees of somewhere a stem could settle.

And that this cost nothing to find. Eight angles, three hundred and fifty-two runs, twelve minutes — and the check is the same one that has caught every instrument problem in this collection: put a second value beside the first and compare.

What the picture at the top shows #

Three bars. The top one is the nine starting angles the table was grown from and reaches 40.6 per cent. The second is the eight midpoints between them, at 25.8. The third is the three angles below the nine’s lowest, at 25.0.

The counts beside each bar are how many runs of how many settled. The nine and the midpoints cover the same rises at the same exponents, so nothing but the angles differs between the top two bars.

The one line #

The nine starting angles the settling table was grown from reach a lattice 40.6 per cent of the time; eight angles placed halfway between them reach one 25.8 per cent of the time, and three below the old floor 25.0 per cent.

Four of the nine sit within two degrees of a destination and one of the eight does — so the settling share this collection reports is biased upwards by the choice of angles, uniformly enough that every comparison built on it survives.