The spread between hands charges the mean, and an office's habit hides in the sheet

Averaging tracings reads the easing, until the hand repeats itself traced the same compiled coast many times by one drifting hand and averaged the tracings. The copy eased into the coast is on the sheet and in every tracing; the hand’s drift is fresh each time; so the mean keeps the easing and divides the drift, and a seam’s width that one tracing hides becomes readable — at the strong seam from two tracings, at the weak one from sixteen. It then found the limit. A hand that repeats part of its drift in every tracing keeps that part in the mean, and the spread between its tracings, which is how a digitiser would learn the hand, cannot see it: the repeated part cancels from every difference exactly as the sheet does. Charged at what the spread implies, a mean of sixteen collapsed into every piece the count was allowed.

It ended on the obvious repair and the doubt about it. A habit belongs to a hand, so a second hand does not share it, and a stack drawn from several hands divides the habits among the hands as it divides the wander among the tracings. The measurement that would show whether that works is the spread between hands’ means against the spread within each hand’s tracings. And it would fail in one way that no arrangement of hands can fix: if the hands share a habit too, because they learned in one office from the same sheets with the same pens.

Sixteen tracings by four hands carry half what sixteen by one do #

The model is the earlier essay’s with one more level. Each of H hands traces the sheet n times. A tracing’s error is the sum of three drifts with the same correlation length along the coast: an office habit common to every tracing by every hand, carrying a share o of one tracing’s variance; the hand’s own habit, common to its own tracings, carrying s; and a wander drawn afresh for each tracing, carrying the rest. One tracing therefore drifts exactly as the earlier trials’ did, whatever the split. The mean of all Hn tracings carries

$$\sigma^2\Bigl(o + \frac{s}{H} + \frac{1 - s - o}{Hn}\Bigr),$$

the office’s share whole, each hand’s habit divided by the number of hands, and the wander divided by the number of tracings.

So how sixteen tracings are arranged matters more than how many there are. With half of each hand’s drift its own habit and no office habit, sixteen tracings by one hand leave 74 per cent of one tracing’s drift in their mean, measured over 120 sheets — the earlier essay’s floor. Eight each by two hands leave 54 per cent; four each by four, 40; two each by eight, 31; one each by sixteen, 25. No number of tracings by one hand goes below 71 per cent, the square root of its habit’s half, so two hands of eight already do what one hand cannot do at all; and four hands of four are well under the 71 per cent at which an independent hand’s means read the strong seam’s easing, while sixteen hands of one reach the 25 at which they read the weak one.

The paler bars are the same arrangements when a quarter of each hand’s drift is its own and a quarter is shared by every hand in the office. The office’s quarter is in every tracing, so no arrangement takes the mean below its square root, 50 per cent: sixteen hands of one tracing each leave 54. The arrangement still helps — 73 per cent from one hand, 58 from four — but the weak seam’s 25 is out of reach of every arrangement.

What four hands look like on one sheet #

The arithmetic is a statement about averages over sheets, and a reader holding one sheet wants to know what it looks like. The figure draws a typical sheet, chosen by a stated rule rather than by eye: of 120 sheets, the first on which all three means sit within a quarter of their own medians and in the medians’ order.

The easing is the dip, a few ten-thousandths of the map deep, that the copy makes as it bends away from the host coast over eighteen points either side of the seam — the width an easing is a seam with a width, and a drifting hand hides the width found one tracing unable to read. One hand’s sixteen tracings follow it in outline and wander away from it by 3.0 ten-thousandths on average over the window, because half of that hand’s drift is in all sixteen and survives the averaging: the mean’s long excursions are the hand’s habit, drawn sixteen times. Four hands of four follow it within 1.8. Over the 120 sheets the four hands are closer than the one hand on 119.

The dashed line is four hands of four whose habit is half their own and half the office’s. It is closer than one hand and further than four independent hands — 2.4 on this sheet and in the median, and further than the independent hands on 110 sheets of 120 — and its long excursions do not average away because they belong to the office, which every hand in the stack repeats.

Two hands read a hand’s share of habit #

The measurement the earlier essay proposed reads two spreads from the tracings and nothing else. Within each hand, the scatter of its tracings about their own mean holds only the wander, $(1 - s - o)\,\sigma^2$, because the office’s habit and the hand’s own are the same in every one of them. Between hands, the scatter of the hands’ means about the grand mean holds each hand’s habit and what is left of its wander after n tracings, $s\,\sigma^2 + (1 - s - o)\,\sigma^2/n$; the office’s habit is again the same in every mean and cancels. Subtracting the within-hand spread over n from the between-hand spread leaves $s\,\sigma^2$, and the hand’s share is that over everything the two spreads see.

The within-hand spread is the one a second tracing tells a drifting hand from a copied stretch used, from a pair of tracings, to learn the drift’s length; the between-hand spread needs a second hand. Both are pooled over the 360 points of the coast and both coordinates, and the drift’s correlation length of eighteen points leaves about forty independent stretches in each tracing, so even two hands supply a usable estimate. With half of each hand’s drift its own habit, two hands of two tracings read the share as 0.49 in the median, the middle eight sheets in ten between 0.33 and 0.60. Four hands of four narrow that to 0.44–0.55 and eight of eight to 0.47–0.53. The share is not hard to read; the earlier essay’s single hand could not read it because it had no second hand to compare.

The right-hand panel is the first sign of what the office does. A quarter of each hand’s drift is its own, a quarter is the office’s, and the spreads read the hand’s share as 0.33 at every arrangement — a quarter out of the three quarters they can see. The office’s quarter has not been misattributed to the hand. It has vanished.

The between-hand spread charges the mean without reading the share #

The count does not need the share. It needs the charge — the size of the drift the pooled mean carries, which the earlier essay found a mean of sixteen by one hand undercharged by a factor of two and a half. And the charge has a simpler estimate than anything built from the share. Divide the between-hand spread by the number of hands: its expectation is $s\,\sigma^2/H + (1 - s - o)\,\sigma^2/(Hn)$, which is exactly the pooled mean’s variance with the office’s share left out. It is the standard error of a mean taken over groups, and it never asks how much of a hand is habit.

With no office habit it is right. At four tracings a hand it charges the pooled mean with 98 per cent of its true drift at two hands and 100 from three hands on, within a fifth of the truth on 96 sheets of a hundred at two hands and on every sheet from three. At two tracings a hand and at eight the figures differ by a point or two. The within-hand spread, divided by all Hn tracings, is what a digitiser who treated the pooled tracings as one hand’s stack would charge, and it charges 45 per cent of the truth at four tracings a hand, 57 at two and 33 at eight — falling as each hand traces more, which is the earlier essay’s failure carried over unchanged.

With an office habit the between-hand charge fails, and it fails in the same inverted way. At two hands of four it charges 65 per cent of the truth, at four hands 52 and at eight 40. Every hand added divides the hands’ own habits further and shrinks the between-hand spread, which is the part of the error it can see, while the office’s quarter, which it cannot see, stays in the mean whole. A digitiser who added hands to break the habit and charged the result from their spread would be charged more wrongly with each hand.

Four hands of four, counted #

The arithmetic says the between-hand spread is the right charge; the count is where that has to show. Four hands each traced the sheet four times, the sixteen tracings were pooled, and the mean was counted exactly as the earlier essay counted its stacks, with the criterion a residual cannot count the pieces of a map introduced, which charges each piece for the numbers it spends — whitened at the drift’s own length, offered up to six rigid seams and four eased — under four charges, on four sheets each.

Charged from the spread between hands, with half of each hand’s drift its own habit and no office habit, every sheet is counted as two seams, which is what was made. The charges came out at 0.84, 1.03, 0.91 and 1.09 of the pooled mean’s true drift, and the strong seam’s half-width reads 12, 20, 16 and 20 against eighteen drawn. That is better than the earlier essay’s single hand managed from sixteen tracings at the same share of habit even when charged at the truth — 8, 8, 16 and 26 — because four hands’ habits average where one hand’s could not. The weak seam, which an independent hand needed sixteen tracings to read, comes out at 5, 10, 4 and 20: read once in four, as a mean holding 40 per cent of one tracing’s drift should allow.

Charged from the spread within each hand — what a digitiser gets who pools sixteen tracings and learns the drift from their scatter about each hand’s mean — every sheet buys every piece it is offered: four eased seams and six rigid, on all four sheets. The charges were 0.43 to 0.46 of the truth, and the earlier essay’s collapse recurs exactly, because the within-hand spread is still blind to each hand’s habit.

With a quarter of every hand’s drift shared by the office, charged at the pooled mean’s true drift, the count is right on every sheet and the strong seam reads 10, 18, 16 and 18. The office’s habit does not stop the easing being read; what it stops is the charge being learned. Charged from the spread between hands, which sees 0.44 to 0.57 of the truth on these sheets, two sheets of the four are counted as four eased seams, the most offered, and the other two escape with two seams at 18 — the sheets whose charge came closest. Every rigid count under that charge is six. An office habit turns the right charge into the earlier essay’s wrong one, and a digitiser has no way to tell from the tracings which of the two charges is in hand.

An office’s habit is a property of the sheet as far as the tracings can tell #

The failure has a sharper form than a charge that falls short, and the trials state it exactly. Take the office’s sheet with its habit, and take a second sheet that differs from the first by precisely what the office’s habit adds — the coast as the office would draw it — traced by an office with no shared habit, whose hands drift by the remaining three quarters of the variance, a third of it each hand’s own. Every tracing of the first sheet is then a tracing of the second. The two offices’ spreads were read on 200 sheets each with the same draws, and they agree on every sheet to the last digit: the share read as 0.33 in the median, the charge, the within-hand spread, all identical.

That is not a weakness of the estimator. Nothing computed from one sheet’s tracings, by any number of hands, can distinguish a habit every hand shares from a feature of the coast, because the two produce the same tracings. The sheet moved before it was measured met the extreme case of this, a shrinkage of the paper itself that every tracing inherits; an office’s habit is the same kind of error arrived at through training rather than through humidity.

What can break it is something that is not a tracing of this sheet. A sheet whose coast is known independently — surveyed, or printed from a source the office did not draw — traced by the same office in the same way, shows the office’s habit as the difference between the office’s pooled tracing and the truth. That measures the office’s habit on that coast, which is not this one; in the model here the habit is a drift along the coast, as particular to a sheet as the wander is. What it can transfer is the habit’s size. If an office’s shared share is learned from control sheets and added to the between-hand spread, the charge is right again, and the count with it — though the office’s habit is still in the mean and still draws the coast where the office would draw it.

Groups within groups, measured for two centuries #

Splitting an error into a part each group shares and a part each member draws afresh, and estimating each part from the spread between groups and the spread within them, is the oldest variance-component problem there is. It is usually traced to George Airy’s treatise on errors of observations of 1861, in which an astronomer’s observations on one night share an error the night imposes and differ by errors of their own, and the spread between nights’ means is the measure of the first. Ronald Fisher’s analysis of variance put the arithmetic used here into its modern form in the 1920s, and the weights are a guess the solve believes met Friedrich Helmert’s estimator of 1907 for the weights of groups of observations in a survey.

The habit of tracing many times is older still, and so is the instruction to vary the observer. What the arithmetic adds to it is an order of priority. The between-hand spread is the right charge for the pooled mean, and two or three hands make it reliable; a hand’s share of habit is readable from the same two spreads, though nothing needs it; and the one error a stack of hands cannot see is the one they all share. A week of twilights learns what a sight is worth found the same order in a navigator’s sights, where the stars of one evening share the evening’s refraction and only other evenings can measure it; a season learns the stretch from where the streak was found it in a forecast ensemble, where a bias every member shares is invisible to the spread among them; and tonight’s residuals measure tonight’s air badly found how little one evening says about its own share.

What the hands were held to #

The between-hand charge must read the truth when nothing is shared by every hand. Over 120 sheets of four hands of four, with half of each hand’s drift its own, the median charge must lie within a tenth of the pooled mean’s true drift. It is 1.00 of it.

The office’s share must be invisible to it. With a quarter of every hand’s drift shared by the office, the same median must fall more than a third short. It is 0.52 of the truth.

The within-hand spread must hold only the wander. On eight hands of eight with a quarter own and a quarter shared, it must read half of one tracing’s variance to within fifteen per cent.

Independent tracings must be independent. The random drifts are seeded per office, hand and tracing, and seeds that differ by fixed multiples left the streams correlated by about one per cent, so that sixteen tracings averaged to 0.068 of one tracing’s variance instead of 0.0625. Every seed is now mixed before use; sixteen tracings average to 0.062, and the measured drift left in every arrangement’s mean over 120 sheets lies within three per cent of the formula. The earlier essay’s stacks used the unmixed seeds and read 0.254 of one tracing’s drift where 0.250 was due, inside the tenth its own check allowed.

The count is the earlier essay’s count. The stack is pooled, whitened at the drift’s true length and charged as stated, and the eased fit starts from the rigid count’s optimum and descends over seams, widths and sources from five starting widths, offered up to six rigid seams and four eased, exactly as before.

Where the hands stop #

Hands that differ only by chance. Every hand drifts by the same amount and repeats the same share of itself. Real draughtsmen differ in how steady they are, and a pooled mean weights a careless hand as heavily as a careful one; weighting hands by their own within-hand spread is the natural refinement, and it is not measured here.

Three levels and one length. The office, the hand and the tracing each drift along the coast with the same correlation length. A habit at the scale of the whole coast — an office that leans the same way on every long stroke — would be a slower departure than the wander and would be whitened wrongly at the wander’s length, as the earlier essay noted for one hand.

Four sheets a setting. The counts are four sheets for each charge, as the earlier essay’s were. The shares and charges are read over 120 or 200 sheets, and are far firmer than the counts.

One sheet’s habit, not a hand’s. A habit here is a drift along this coast. A habit that belongs to a shape — the same overshoot on every convex headland — would be shared between sheets, and could be learned from control sheets in a way a drift along one coast cannot.

Still open: whether control sheets can charge an office’s habit #

The between-hand spread charges everything but the office’s share, and the trials say what that share costs: a charge that falls from 65 per cent of the truth at two hands to 40 at eight. The repair the arithmetic allows is to learn the share elsewhere and add it. An office that traces a few control sheets — coasts known independently, drawn in the office’s ordinary way — can compare its pooled tracing of each with the truth and read how much of the error survives every hand.

Whether a handful of control sheets estimates the office’s share well enough to restore the count, whether the share learned on one coast’s shapes transfers to a coast of different shapes, and whether an office’s habit is better modelled as a drift along a coast or as a response to the coast’s own curvature — so that it would transfer exactly where the shapes agree — are questions one sheet’s tracings, by any number of hands, cannot ask.