A thickness is a maximum, not a mean

Every standard for measuring the thickness of a fabric specifies two things that look like housekeeping: the area of the presser foot, and the pressure under it. Twenty-five square millimetres and a kilopascal are typical, and a reader coming to the standard fresh would take both for the sort of detail that exists so that two laboratories can agree.

They are not that. Both of them are in the answer, and the first one is in the answer for a reason that has nothing to do with the cloth at all.

A presser foot does not average the fabric under it. It rests on whatever is highest, so the number it reads is a maximum over the crossings it covers — and a maximum over a larger area is a larger number, without anything about the fabric having changed.

The claim #

A fabric’s measured thickness exceeds the thickness of its average crossing by about a quarter, the excess grows with the area of the foot, and the number of independent chances a foot has of finding a thick place is the number of threads under it rather than the number of crossings.

At an ordinary cotton construction with a yarn varying by fifteen per cent, a twenty-five square millimetre foot reads 26 per cent above the geometry. A square millimetre reads 8 per cent above. A hundred square centimetres — which is nobody’s presser foot but is roughly what a hand feels — reads 50 per cent above.

The argument, which is one identity #

The thickness at a crossing is decided by its end and by its pick. Approximating it as the sum of their two diameters, the largest crossing under a foot is

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

The maximum of a sum over a grid separates. That is worth pausing on, because it demolishes the obvious count. A foot covering twelve ends and eleven picks covers a hundred and thirty-two crossings, and the natural thing to say is that it has a hundred and thirty-two chances of landing on a thick place. It has twenty-three. Every crossing along one end shares that end’s diameter, so the crossings are not independent draws — the threads are.

The difference matters because an extreme grows with the logarithm of the number of tries, roughly, and a hundred and thirty-two against twenty-three is a factor of six in the count and only a few per cent in the answer. Treating the crossings as independent reads the cloth about four per cent thicker than the identity does, which is the sort of error that survives indefinitely because it is small, plausible and in the right direction.

What was counted, and how #

The population is a lognormal on the diameter at the yarn’s stated coefficient of variation, and the expected largest of n draws is computed by integrating the density of the order statistic rather than by simulating it — then checked against a simulation that knows none of that arithmetic, because a closed form applied to the wrong quantity is silent and a simulation that disagrees with it is not.

The identity above is checked the same way and more directly: a grid of ends and picks is drawn, every crossing’s sum is formed, the largest is taken, and it is required to equal the largest end plus the largest pick to within nothing at all. That check would catch the failure that matters — a model in the figure that is not the model in the arithmetic.

And the naive count is computed as well as the right one, so that the size of the mistake is a number rather than a caution. Treating the hundred and thirty-two crossings as independent draws from a distribution with the correct mean and the correct spread overstates the reading by 4.4 per cent.

How fast an extreme grows #

The growth is slow enough to be worth putting a number on, because “it depends on the sample size” is often taken to mean “it is unbounded and therefore hopeless”, and it is neither.

The expected largest of n draws from a normal population sits above the mean by roughly the standard deviation times the square root of twice the logarithm of n. A logarithm under a square root is about as slow as growth gets: quadrupling the area adds one to the count of threads’ logarithm and a few per cent to the answer.

Two things fall out of the table. The first is that the standard foot is not in a special place: nothing about twenty-five square millimetres is a plateau, and the curve is as steep there as anywhere. The second is that the whole range from a pinhead to a hand’s breadth spans a factor of six in reading only because the underlying population is narrow — at a coarser yarn the same table would run from ten per cent to seventy.

So the sample size cannot be argued away and does not need to be. It needs to be stated, which is precisely what a standard does when it names an area, and which is precisely what a number quoted without one fails to do.

The pressure, which does not do what it looks like it does #

The area is in the answer because a thickness is a maximum. What about the pressure?

It looks like a corrective. Press hard enough and the thick places should flatten more than the thin ones — a thick crossing has more to give — so the reading ought to come back towards the mean, and a standard specifying a kilopascal would be specifying how much of the population’s tail to squash out.

Run it and the opposite happens.

The mechanism is lateral, and it is one this collection has met before. A thread flattens by spreading sideways, and what it can spread into is its neighbours’ room. A cloth of thicker yarn at the same sett has less of that room — it is nearer its own jam across the width — so it resists flattening more, not less. The thick cloth is stiffer in compression exactly because it is thicker.

So the pressure in the standard is doing something else entirely. It is there to define which thickness is being reported, on a quantity that has no natural value: a fabric under no load at all has a thickness that depends on how it was last handled, and the pressure removes that history. It is a definition, not a correction, and a reader who takes it for a correction will expect two fabrics measured at high pressure to agree better than two measured at low pressure. They will agree worse.

The consequence for everything computed from a thickness #

A thickness is not usually wanted for itself. It is an input, and this collection has used it as one repeatedly.

• A thermal resistance goes as the thickness, near enough, because warmth is a thickness of still air. A measured thickness a quarter high makes a predicted warmth a quarter high.
• A bending rigidity goes as a higher power still, so a fabric’s stiffness inferred from its measured thickness inherits the excess amplified.
• A channel through the cloth has the thickness as its length, and the flow through it depends on that length — but here the excess works the other way, because the passage a jet of air takes is not through the thickest crossing.

The pattern is worth naming: an extreme used as though it were a mean propagates as a bias, and the bias inherits the exponent of whatever it is fed into. Nothing in the chain is measuring anything wrongly; the number is simply an answer to a different question from the one the next equation asks.

Which of the two numbers a question wants #

There are two thicknesses here and they are both correct. The confusion is not about which is right but about which question is being asked, and the two questions separate cleanly.

The gauge’s maximum is the right answer whenever something else has to rest on the cloth, or pass over it, or be stacked with it. How high a bolt of a thousand metres stands, whether a fabric will feed through a nip set to a gap, how much a coating has to bridge, what a hand feels when it pinches — all of those are decided by the high places, because the high places are what touch. A calender roll set to the mean crossing would crush the tail of the population and leave the rest untouched.

The solved geometry is the right answer whenever the thickness appears inside an equation about the whole cloth. A thermal resistance is an average over the area, not a maximum; so is a mass per unit area, so is a channel length, so is the volume fraction a composite is quoted at. Each of those integrates over the fabric, and integrating wants a mean.

The failure mode is measuring the first and substituting it into the second, which is what happens by default because the first is the one an instrument produces. And the error is not small: a quarter, entering a thermal calculation as a quarter and a bending one as considerably more.

The size of the gap is set by the yarn’s evenness alone, which is the useful part. A filament yarn at eight per cent gives about half the excess of a staple yarn at twenty, at the same construction and the same foot, so two fabrics that differ only in the regularity of their threads will differ in measured thickness while agreeing exactly in every quantity computed from their construction. Nothing in a specification sheet records that, and the number that predicts it — the coefficient of variation — is measured on every delivery of yarn and never travels with the cloth.

Where the model stops #

The crossing’s thickness is taken as the sum of two diameters. The real geometry is Peirce’s and the thickness of a crossing is not simply d₁ + d₂; what is done here is to scale the solved thickness by the ratio of the extreme sum to the mean sum, which keeps the geometry and gives the variation to the part of it that is yarn. A fully solved extreme crossing would need the whole locus recomputed at every draw, and it would move the number rather than the argument.

The two limits of the pressing result are far apart and only one is computed. A single thick end among ordinary ones can push its neighbours aside and flatten more than the curve above says; a cloth made wholly of thicker yarn cannot, because every neighbour is thick too. The solver reaches the second, which is the conservative limit for the claim being made — the true local answer is somewhere between, and this collection has no instrument for a cloth with one thick end in it.

The extreme is quoted at a stated law, and the tail is a choice. The two laws carried here differ by a couple of per cent at the largest of a hundred and by considerably more at the largest of ten thousand, so the hundred-square-centimetre reading in the curve above is the softest number in the essay.

Correlation along a thread is ignored. A thick place recurs along its own end at whatever period the spinning gave it, so two crossings a few millimetres apart are not independent even after the identity above has been applied. That would reduce the effective number of tries again and lower the reading slightly.

The conversion, which is arithmetic rather than calibration #

The essay is careful to say the gap between the two thicknesses is not a calibration, and it is right — but it is a conversion, and the conversion can be written down and applied by anybody holding a specification sheet.

The expected largest of n draws sits above the mean by the standard deviation times a coefficient that depends only on n. Writing a_n for that coefficient and applying the identity — the largest end plus the largest pick — the reading exceeds the mean crossing by

CV × (aₙ₁ + aₙ₂) ÷ 2,

where n₁ and n₂ are the ends and picks the foot covers.

At a standard foot on a muslin — twelve ends by eleven picks — the coefficients average 1.61, so a fifteen per cent yarn gives 24 per cent. The essay’s own solve gives 26, the difference being the lognormal’s tail against a normal’s, and the normal form under-reads by more as the foot grows.

So the practical recipe is one division. To recover the geometric thickness from a gauge reading, divide by 1 + CV × a_n, with n read off the sett and the foot. Nothing about the instrument enters, and nothing has to be measured that is not already printed: a thread count, a foot area, and the yarn’s coefficient of variation.

That it is a conversion rather than a calibration matters for how it is used. It does not make the gauge more accurate — the gauge was already exactly right about the question it asks — and it must not be applied to the uses in the left column of the essay’s own table. It applies only when a reading is about to be substituted into an equation that wanted a mean.

Where the identity itself starts to fail #

The limits section notes that thick places recur along a thread and that this would reduce the effective number of tries. The identity already assumes the strongest possible version of that: one diameter per end, perfectly correlated over the whole foot.

Whether that is right is a comparison of two lengths. A thick place recurs at the drafting roller’s own circumference, typically thirty to a hundred millimetres, and a standard foot is about five millimetres across. So the foot sits comfortably inside one period, every crossing along one end really is the same diameter, and the identity is exact rather than approximate.

The row where it stops being exact is the last one. A hundred square centimetres is a hundred millimetres across — one to three spinning periods — so each end contributes two or three independent diameters rather than one, and the effective number of tries is two or three times the thread count.

Two or three times the tries adds about 0.2 to a_n, which is three per cent on the reading. So the fifty per cent in the essay’s table should be nearer fifty-three, and the essay’s own caution that the largest foot is its softest number is right for a second reason it does not name.

Below a foot of about thirty millimetres the identity is exact and above it the count has to be corrected, which is a clean boundary and puts every real presser foot on the exact side of it. It is only the hand — which is not an instrument and has no standard — that sits beyond.

The generalisation #

When a measurement rests on the largest thing in its window, the size of the window is part of the result, and no amount of care with the instrument removes it. The thickness gauge is a clean case because the mechanism is so plain, but the shape is everywhere: a peak reading, a worst case, a maximum load, a longest delay. Each of them is an order statistic, each grows with how much was sampled, and each is routinely quoted as though it were a property of the thing measured.

The diagnostic is a single question: would this number change if the same object were measured over twice as much of itself? If the answer is no, it is a mean and it can be corrected, compared and propagated in the ordinary way. If the answer is yes, the sample size has to travel with the number wherever it goes — and a standard that fixes the sample size, as the thickness standards do, is not being pedantic. It is the only thing making the number mean anything.

The second lesson is about correctives that are not. The pressure in the thickness standard looks like a way of removing the population’s tail and is nothing of the kind. Testing that assumption cost one sweep of a solver that already existed; believing it would have cost an argument built the wrong way round, in which two laboratories pressing harder were supposed to agree better.

Who found it, and when #

That a thickness reading depends on the presser foot is not news to anyone who has run the test — it is why every standard fixes both the area and the pressure, and why comparing thicknesses across standards is discouraged. What appears not to be written down is why the area matters by that amount: the separation of the maximum into the largest end plus the largest pick, and the consequence that a foot has as many chances as it covers threads rather than crossings.

The pressing result is this collection’s own and was a surprise to the argument that produced it. It came out of a solver built two ladders ago for the crossing squashed, asked a question it had not been built for, and returned the opposite of the expected sign — which is the most useful thing a piece of machinery can do.

Where the ladder goes next #

The other two extremes in this ladder run the same argument to different ends. A warp jams at its thickest pair, which makes the closest sett a property of the loom’s width; and at the other end of the distribution, a bundle breaks at something well below its mean thread, while a tear asks an even smaller sample and scatters accordingly.

Sideways, the same population that makes the thickness a maximum makes the holes a distribution, and what a cloth passes belongs to the widest of them.