Bias — where it appears
Named by 11 essays across 6 fields — each of them below, with the objects they name alongside it.
A scale chain leans rather than wanders
A camera driven 200 m along a street, each pair's scale handed to the next through the points they share. Across sixteen streets the scale does not spread the way a random walk would — 7.3 per cent after 199 links, where independent ties would give 108 — and its average leans by an amount the choice of average decides, from +28.6 per cent to −12.2.
What survives being copied
A workshop copying a drawing from a drawing is a random walk — the spread across lineages grows as the square root of the generation, with a fitted exponent of 0.5001. A workshop copying the method is not, and its exponent is 0.012, which is no growth at all, and after forty generations two lineages started from different originals end up 5 × 10⁻¹⁵ apart. Copying the marks loses the picture; copying the recipe loses the original and keeps the recipe.
Which reference to measure from
Given four candidate scale bars in one photograph, the best is not the longest and not the nearest — it is the longest in the picture. A five-point-two metre bar near the horizon is the longest thing in the scene and the worst reference in it; a four metre bar close to the camera is the best. Walk one bar outward and the term it controls falls as one over its length in pixels, with a fitted exponent of −1.08.
The floor a better camera cannot reach
Sweep the marking error from four pixels down to a hundredth and the measurement's error falls thirty-fold and then stops — at 6.0 per cent, which is exactly the six per cent the reference's assumed shape was wrong by. With the closure exact the same sweep keeps falling to 0.06 per cent. The crossing is at half a pixel, and it can be computed before the photograph is taken, which makes it a decision about equipment rather than a discovery about it.
The workshop that throws drawings away
Adding a rule that discards a drawing which looks wrong turns the mark copyist's spread from a growing random walk into a stationary process — the fitted exponent falls from 0.542 to 0.022, indistinguishable from the method copyist's 0.033 — and the two mechanisms then separate only by where they settle, 0.7002 against 0.8000, seven and a half spreads apart.
Five facts that close the same gap
The gloss that one length has to come from outside the photograph names a single option, and there are at least five — a length on the ground, the camera's own height, a repeated object of unstated size, a standing object of known height, and the focal length with the horizon. All five give an answer on the same picture with the same noise; their spreads run from 0.36 to 3.60 per cent, a factor of ten, and which one is available depends on the photograph rather than on the arithmetic.
A rectification's free shift is free only near the pair
Sliding a rectified pair's principal points apart adds a constant to every disparity, and a constant changes nothing about where points are — on paper. A whole-pixel reading is not paper. The same slide moves every depth the reading can report, and for a wall ten metres away the choice between the best shift and the worst is 1.71 metres. Near the pair it is millimetres. A matcher with half a pixel of its own error erases the choice, and pays more than the worst shift did.
A straight stick tests the index only across the sight
Told that each stick is straight, three eyes fitting a water surface's height and its index still slide down the valley between them — when the stick leans away from them. A wrong index carries such a stick into another straight stick, off its line by 0.016 mm and 116 mm short. Turned to lean across the line of sight, the same wrong index bows it by 2.2 mm, and the straight-line fit's slide falls from 0.058 to nothing. A tape on the stick, read only to ±50 mm, stops the slide outright.
A bowed stick is read as a wrong index
Holding a stick to a straight line cured the slide in a refraction fit — but only for a stick leaning across the sight, where a wrong index bows the read stick by two millimetres. A stick that is itself bowed by a millimetre in the same plane is, to that constraint, a straight stick seen through the wrong index: read without error, it moves the fitted index by 0.074 one way or 0.062 the other, and at a quarter pixel the constraint's bias overtakes the slide it was removing at about three-quarters of a millimetre of bow. Its misfit is a fiftieth of a pixel, so nothing in the residuals says so. A tape, which measures length, does not notice the bow at all.
Crooked sticks cancel each other's bows slowly, and eight of them match a tape
A stick bowed by a millimetre fools a refraction fit that holds it to a straight line: the fit reads the bow as a wrong index. Several sticks each bowed in its own random direction pull the index both ways, and the pulls cancel — but not as one over the square root of the count, because the sticks do not pull equally: a near stick pulls two and a half times harder than a far one, and a stick leaning just off the line of sight tens of times harder than one leaning straight along it. Eight crooked sticks leave the index ±0.016 out where one leaves it ±0.037, and with a quarter pixel of reading error they read it to ±0.037 — exactly what one straight stick with its length taped does. The earlier guess was that matching a tape would take a large number.
The bias out of reach
A single-view height recovered from the cross-ratio has an error with two terms. The spread of an average of m readings falls as one over the root of m, by a fitted exponent of −0.52 over three decades. The bias does not fall at all — and it is eight microns, against a single-shot spread of six millimetres, so the two cross at 535,000 measurements. There is a floor here and no experiment anybody will run can see it.
Named alongside it
The objects these essays reach for when they reach for this one.
error propagationModel errorConditioningleast squaresTriangulationCollinearityDriftrandom walkReference lengthRefractionRefractive indexAttribution