CogoKit

Error propagation

Combining a systematic correction with a random error

A tape 0.012 ft short and thirteen settings of it. The systematic part is removed by arithmetic and the random part is carried as an uncertainty — and the two must never be combined in quadrature, which is the mistake the problem exists to prevent.

Apply· about 20 minutes by hand· 4 steps

Given

  • A line was taped and recorded as 1247.60 ft. It was laid off in 12 full tape lengths of a nominal 100 ft tape plus a partial length of 47.60 ft — 13 independent settings of the tape in all.
  • The tape was subsequently standardised and found to be 99.988 ft long under the tension, temperature and support used in the field. It is therefore 0.012 ft short of its nominal 100.00 ft.
  • The random error of each setting — marking, plumbing and reading — is ±0.006 ft.
  • All other systematic effects (slope, temperature, tension, sag) have already been corrected out of the recorded 1247.60 ft.
  • Distances in US survey feet.

Required

  • The corrected length of the line.
  • The random uncertainty of the corrected length.
  • A statement of the result, and a demonstration of why the two quantities do not combine in quadrature.

Work it through yourself before reading on — the solution below shows every step, so there is no way to skim it without giving the answer away.

Worked solution

The systematic correction

A tape that is short reads long. Each time it is laid down it covers only 99.988 ft of ground while 100.00 ft is written in the book, so the recorded length exceeds the true length by 0.012 ft for every tape length used.

The correction is therefore proportional to the recorded distance, and it has a known sign. Recorded distance divided by nominal tape length gives 12.476 tape lengths, and 12.476 times 0.012 ft is 0.150 ft, to be subtracted.

Get the direction right by reasoning about it rather than by memorising a rule. Short tape, too many tape lengths recorded, recorded distance too long, correction negative. For a tape that is long the whole chain reverses.

tape error per tape length = 99.988 − 100.000 = −0.012 ft number of tape lengths = 1247.60 / 100.00 = 12.476 C = 12.476 × (−0.012) = −0.1497 ft → −0.150 ft corrected length = 1247.60 − 0.150 = 1247.450 ft

The random error

The random error is a different animal entirely. It arises fresh at each of the 13 settings, is as likely to be positive as negative, and its sign is unknowable. It therefore propagates in quadrature, giving 0.006 × √13 = 0.0216 ft.

Count the settings and not the tape lengths: the partial length of 47.60 ft was marked and read exactly like a full one, so it contributes its own ±0.006 ft. Thirteen, not twelve.

The random error is unaffected by the standardisation. Correcting the tape length removes a bias; it does nothing to the scatter of the marking and reading, which is what this term describes.

σ_random = E √n = 0.006 × √13 = 0.006 × 3.6056 = 0.0216 ft the systematic effect used 12.476 tape lengths; the random effect uses 13 settings the two counts differ because one is a proportion and the other is a count of independent events

Why they do not combine in quadrature

The temptation is to treat 0.150 ft and 0.0216 ft as two errors and combine them: √(0.150² + 0.0216²) = 0.1516 ft. That number is meaningless, and it is meaningless for a reason worth stating plainly.

Quadrature combination is a statement about independent random variables whose signs are unknown. The 0.150 ft is not a random variable at all. Its sign and magnitude are both known — the tape was measured — so it is not an uncertainty, it is a correction. Corrections are applied; uncertainties are carried.

Adding them linearly, 0.150 + 0.022 = 0.172 ft, is equally wrong and for the same reason. Both errors treat a known bias as though it were unknown, and both leave the reported distance 0.150 ft too long while attaching an uncertainty large enough to make the mistake look like caution.

The correct treatment has two separate steps. Remove the systematic part from the value. Attach the random part to it as an uncertainty. The result is 1247.450 ± 0.022 ft, and the ±0.022 ft says nothing whatever about the tape's standardisation.

Three treatments, one of which is right
TreatmentReported distance (ft)Stated uncertainty (ft)Verdict
Quadrature: √(0.150² + 0.0216²)1247.60±0.152wrong — bias left in, disguised as uncertainty
Linear sum: 0.150 + 0.02161247.60±0.172wrong — same bias, larger disguise
Correct and carry1247.450±0.022correct
No standardisation at all1247.60±0.022wrong — 0.150 ft too long, and confidently so

What the two terms are worth, separately

Expressed as relative precisions the two are wildly different. The uncorrected systematic bias alone is 1:8,330 — a figure that would fail an ordinary boundary specification, produced entirely by a tape 0.012 ft short. After correction the random uncertainty gives 1:57,700.

That contrast is the practical argument for tape standardisation. No amount of care in marking and plumbing recovers what an unstandardised tape gives away, because care reduces the random term and the systematic term is untouched by it. Seven times better random performance would still leave the line 0.150 ft long.

The corresponding argument in the other direction: once the tape is standardised, further standardisation buys nothing. The systematic term is gone and the random term is all that remains, so the next improvement has to come from procedure — more settings measured more carefully, or a different instrument.

systematic bias as a relative error: 1247.60 / 0.1497 = 1 : 8,330 random uncertainty after correction: 1247.450 / 0.02163 = 1 : 57,700 E95 = 1.96 × 0.0216 = 0.042 ft 95 % interval: 1247.450 ± 0.042 ft, i.e. 1247.408 to 1247.492 ft

Answer

  • Systematic correction = 12.476 tape lengths × (−0.012 ft) = −0.150 ft. The corrected length is 1247.450 ft.
  • Random uncertainty = 0.006 × √13 = ±0.0216 ft, or ±0.022 ft.
  • Report the line as 1247.450 ± 0.022 ft at one sigma, or ±0.042 ft at 95 per cent.
  • The two do not combine. √(0.150² + 0.0216²) = 0.152 ft is not a valid uncertainty: a systematic error of known sign and magnitude is removed from the value, never carried as scatter about it. Combining them linearly, at 0.172 ft, is wrong the same way.
  • Left uncorrected, the tape bias alone is a relative error of 1 : 8,330 — worse than most boundary specifications — while the corrected line's random uncertainty is 1 : 57,700.

Check

Compute the true length directly instead of by correction: 12.476 tape lengths at the tape's actual 99.988 ft is 12.476 × 99.988 = 1247.4503 ft, matching the corrected value of 1247.450 ft. This route never forms a correction at all, so it independently confirms both the magnitude and the sign.

Check the sign by an extreme case. Suppose the tape were 90 ft long but recorded as 100. A line recorded as 1000 ft would be ten tape lengths, so really 900 ft. The correction formula gives 10 × (90 − 100) = −100 ft, and 1000 − 100 = 900 ft. The formula reproduces the extreme case exactly and with the right sign, which is what makes it trustworthy on the 0.012 ft case where the answer cannot be seen by inspection.

Check the random term against the general sum formula: √(13 × 0.006²) = √0.000468 = 0.0216 ft. The series shortcut and the explicit sum agree.

Check that the two terms really are of different kinds by asking what repeating the measurement would do. Taping the line a second time and meaning the two results would reduce the random term to 0.0216/√2 = 0.0153 ft, and would leave the 0.150 ft systematic error exactly where it was. That asymmetry is the operational definition of the distinction, and it is why the two cannot be combined.

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