A cloth is a population, not a thread

Everything this collection has computed about a cloth has been computed from one thread. A cover factor comes from the diameter, a jammed sett from the diameter, a hole from the spacing less the diameter, a bending rigidity from the diameter to the fourth power. The thread has been a number throughout.

No yarn is a number. A cotton yarn spun from staple fibres has a diameter that varies along its own length by something between eight and twenty per cent, and that variation is not a defect to be designed out — it is the single most measured quantity in the whole industry, quoted as a coefficient of variation on every delivery note.

So there is an obvious question, and it turns out to have three different answers rather than one.

The claim #

Putting the distribution back changes a quantity in one of exactly three ways, and which way is decided by the shape of the function rather than by the size of the spread.

• Nothing at all. A quantity linear in the diameter is unaffected — not approximately, exactly. A cover factor is threads per centimetre times diameter, and the average of a sum is the sum of the averages whatever the distribution.
• A little, in a direction the curvature sets. A quantity with a second derivative is biased by half the variance times that derivative. A mass per unit area goes as the square and comes out 2.25 per cent high at a fifteen per cent spread; a bending rigidity goes as the fourth power and comes out 14.3 per cent high.
• By a lot, and by more the more cloth is asked. A quantity that is a maximum or a minimum is not an average of anything. Where a warp jams is decided by its thickest pair of neighbours; what a thickness gauge reads is decided by the highest crossing under its foot; where a fabric tears is decided by the weakest thread at the tip of the cut. Each of those grows — or shrinks — without limit as more cloth is included, and none of them has a value that a mean can be corrected into.

The three are not degrees of the same thing. They are three different kinds of quantity, and the first job is to say which is which.

The argument, which is one derivative #

Take any quantity f computed from a diameter. The population has mean μ and standard deviation σ. Expanding f about the mean and averaging kills the first-order term, because the deviations average to nothing, and leaves

$$\mathbb{E}[f(X)] \approx f(\mu) + \tfrac{1}{2}\sigma^2 f''(\mu).$$

That is the whole of the second answer. The sign of the correction is the sign of the second derivative, so a convex quantity is under-reported by the average thread and a concave one is over-reported; the size is the variance times the curvature, so it grows as the square of the spread rather than in proportion to it. And a linear function has no second derivative at all, which is why the first answer is exact rather than approximate.

For the common case where f is a power, the correction has a closed form and no expansion is needed. If the diameters are lognormal — which is the law a positive quantity has to have, since a diameter cannot be negative — then

$$\frac{\mathbb{E}[X^k]}{\mathbb{E}[X]^k} = (1 + \mathrm{CV}^2)^{k(k-1)/2},$$

so every moment ratio is a power of one number. At a fifteen per cent CV that number is 1.0225, and the ratios for the square, cube and fourth power are 1.0225, 1.0690 and 1.1428.

The law has to be named, because this collection’s fourth rule requires it and because the two laws do not answer the same question equally well. For the second answer — the curvature correction — the choice barely matters: a normal and a lognormal with the same mean and CV differ by half a per cent in their fourth moment at CV 0.15. For the third answer it matters completely, because an extreme is a question about the tail and the two laws have entirely different tails.

What was counted, and how #

Seven quantities this collection already computes were taken as they stand, wrapped as functions of the one diameter, and pushed through a population whose mean is an ordinary cotton warp’s and whose CV is fifteen per cent. Nothing was re-derived. The functions are the ones the earlier essays argued about, and the bias is a property of those functions rather than of any new modelling.

Two of the rows deserve their own sentence.

The hole is the interesting one, because it is where the amplification lives. A hole between two ends is the spacing less the diameter, and the spacing is set by the reed and does not vary at all. So a yarn at fifteen per cent leaves holes at fifteen per cent times d/w — a factor of one in an open cloth and a factor of four in a close one. The variation in the thing that matters is several times the variation in the thing that was measured, and it gets worse the closer the cloth is set, because the quantity in the denominator is exactly what closing the cloth shrinks.

The rigidity is the largest exponent on the site. A fourth power at fifteen per cent is a fourteen per cent understatement, which is larger than the difference between two of the fibre moduli this collection has argued about — and it arrives without anybody measuring anything wrongly.

The third kind, which is not a correction at all #

The first two answers are about averages. The third is about a different question being asked.

A warp does not jam where its average pair of neighbours would touch. It jams wherever its thickest pair is, anywhere across the width, because a cloth that cannot be beaten up at one place in the reed cannot be woven at that sett at all. That is an extreme of a moving sum, and it grows with the number of ends — which means, uncomfortably, that the closest sett a yarn will take depends on how wide the loom is.

The same shape governs a thickness. A presser foot rests on whatever is highest beneath it, so a thickness measurement is a maximum over the crossings under the foot — and the arithmetic of that maximum is prettier than it has any right to be. The thickness at a crossing depends on its end and on its pick, so

$$\max_{i,j}\,(d_{1i} + d_{2j}) = \max_i d_{1i} + \max_j d_{2j},$$

which changes the count entirely. A twenty-five square millimetre foot covers a hundred and thirty-two crossings, and a first guess treats those as a hundred and thirty-two chances of finding a thick place. They are twenty-three chances — the threads — because every crossing along one end shares that end’s diameter.

And the third instance runs the other way. A bundle of threads pulled together does not break when its threads reach their average strength: the weakest goes first and hands its load to the rest, so the bundle’s peak is reached before every thread is at its own limit.

A spread is measured over a length, and the length is part of the number #

There is a question hidden inside every coefficient of variation, and it has to be asked before any of the arithmetic above means anything: variation over what?

A yarn’s thickness is not a single random draw per thread. It varies along its own length continuously, so the spread between one-centimetre pieces of it is larger than the spread between one-metre pieces, which is larger again than the spread between bobbins. Quoting fifteen per cent without saying over what length is quoting half a number.

And the three kinds of quantity above want three different lengths.

• The jam wants the diameter over a few millimetres, because what has to fit through a dent of the reed is the thread as it is at that instant, beside its neighbour as it is at that instant.
• The bundle wants the strength over the gauge length, which is a different quantity again: a thread tested over half a metre breaks at its own worst place over half a metre, so its measured strength falls as the specimen lengthens even before any bundle is made.
• The cover factor wants the diameter averaged over the whole cloth, which is the one length at which the spread does not matter — because the quantity is linear, and averaging a linear function over a longer piece changes nothing.

So the linear case is doubly safe: it is insensitive to the spread and insensitive to the length the spread was measured over. Everything else inherits both sensitivities at once, and a number carried from one to another without saying which length produced it is the commonest way this arithmetic goes wrong in practice.

Which exponents are safe, and the one range where the bias reverses #

The closed form for a power gives the correction as (1 + CV²) raised to k(k − 1)/2, and that exponent is worth reading as a function of k rather than evaluating case by case, because its shape sorts every quantity in this collection at a glance.

k(k − 1)/2 is a parabola through zero at k = 0 and k = 1. So:

The exponents that are exact are nought and one, and no others. A quantity independent of the diameter is unaffected, and a quantity linear in it is unaffected — the cover factor’s zero is not a coincidence of that function but a statement about its exponent.

Everything outside that interval is biased upward, because the parabola is positive there. A mass at k = 2, a rigidity at k = 4, a permeability between two and four: all convex, all understated by the average thread.

And a reciprocal is biased exactly as a square is. At k = −1 the exponent is 1, the same value it takes at k = 2 — so a quantity going as one over the diameter carries precisely the 2.25 per cent a mass does at a fifteen per cent spread. That is worth having because reciprocals are everywhere in this collection and none of them looks like a convex quantity: a wicking height goes as one over a pore radius, a resistance goes as one over a flow, a count per unit length goes as one over a spacing. Every one of them is understated by the mean thread by the same amount a mass is.

And between nought and one the bias reverses. The parabola dips below zero on that interval, so a quantity going as a fractional power of the diameter is overstated by the average thread — the one direction nothing else in this essay goes.

There is such a quantity in this collection and it is worth naming. A wicking front travels as the square root of the pore radius times the time, so it is a half power, its exponent is −1/8, and a population’s front travels 0.28 per cent slower than the mean thread’s would at a fifteen per cent spread. The size is negligible and the sign is not: it is the only place in the whole list where the correction goes the other way, and a reader who has learnt the rule “the population beats the mean thread” would apply it wrongly.

So the practical recipe is three lines and needs no integration. Take the log-slope of the quantity with respect to the diameter. If it is nought or one, do nothing. If it is outside that interval, multiply by (1 + CV²) to the k(k − 1)/2. If it is inside, divide. The last case is rare, small, and the only one where remembering the rule is worse than looking it up.

Where the model stops #

The variation is treated as independent from thread to thread, and it is not. A yarn’s thickness varies along its own length as well as between one bobbin and another, and a thick place in a warp end recurs at whatever period the spinning frame gave it. That correlation is invisible to everything above and it is the whole subject of what happens when the error has a period, where a spread of exactly the same size behaves entirely differently.

Only the diameter is given a distribution. A real yarn also varies in twist, in strength, and in how much it has been stretched, and those are not independent of each other or of the diameter. Giving one quantity a spread and holding the others at their means is a first step rather than a model of a yarn.

The extremes are quoted at a stated law, and a lognormal’s tail is a choice. At a fifteen per cent CV the largest of a hundred threads differs by a couple of per cent between the two laws carried here, which is small; at the largest of ten thousand it is not small, and nothing in this collection measures which tail a real yarn has.

The quadratic estimate is an estimate. It is quoted beside the true value in every table above precisely so that the reader can see where it stops working: at the spreads a yarn has it is good to a part in a thousand, and at the spreads a hole has in a close cloth — which is where the interesting arithmetic is — it is out by several per cent and the direction of the error is not constant.

And nothing here is about a fault. A thread that is not there at all, or a pick made in the wrong shed, is not a large deviation from a mean; it is a different object, and it is priced by a different argument entirely.

The generalisation #

The habit worth taking away is smaller than the arithmetic and more useful.

Before correcting a number for a spread, ask which of the three kinds it is. A linear quantity needs no correction and any correction offered for it is a mistake. A curved quantity needs one, and the correction is knowable from the function alone — take the log-slope, look up the moment ratio, and the answer is within a part in a thousand without any integration at all. An extreme cannot be corrected: it needs a different question, which is over how much, and the answer changes when the sample changes.

The failure this prevents is common and quiet. Someone measures a mean, computes with it, finds the answer disagrees with the measurement, and adjusts a constant until the two agree. If the disagreement was a curvature bias, the fitted constant now carries the spread of that particular yarn inside it and will be wrong for the next one. If it was an extreme, the fitted constant carries the size of the sample as well and will be wrong for the next loom.

The second lesson is about which direction the errors go. All the biases above are in the same direction — the population beats the mean thread — because almost everything a cloth does is convex in its threads. The exceptions are the ones that are minima: a strength, a tear, a weakest link. So a cloth computed from mean threads is systematically predicted to pass less air, be less stiff, be thinner and be stronger than it is. Three of those four are quietly benign and the fourth is not.

Who found it, and when #

The moment identities are ordinary probability and are applied here rather than derived; the same is true of the distribution of an order statistic and of the bundle result, which belongs to Daniels and dates from 1945. What this collection contributes is the list: which of its own functions are linear, which are curved, and which are extremes — and that list could not have been written without the functions, which is why it arrives now rather than at the beginning.

The yarn-variation measurement itself is far older than any of the arithmetic. Evenness testing was industrialised in the 1940s and the coefficient of variation has been printed on yarn specifications ever since, which makes the situation slightly comic: the one number everybody measures about a yarn is the one number that none of the geometry uses.

Where the ladder goes next #

Straight into the three extremes, each of which turns out to be a different essay. The jam decides what sett a warp will actually take and makes it a property of the loom’s width; the maximum under a foot decides what a thickness gauge reads and explains why every standard specifies the foot; and the minimum decides what a bundle breaks at and, in a smaller sample still, what a tear costs.

Sideways, the curvature argument has a consequence for finishing that nobody appears to have written down: an operation multiplies a spread by its own log-slope, so a process with an exponent below one makes a more even cloth whether or not it was chosen for that.