Every measurement is wrong. The useful question is not whether a value contains error but what kind of error, how large it plausibly is, and what happens to it when the value is combined with others. Error theory is the small body of arithmetic that answers those three questions, and it is a great deal smaller than its reputation.
The reason it matters in practice is that the three kinds of error behave completely differently when measurements are combined. One must be found and removed, one accumulates in direct proportion to the number of measurements, and one partially cancels. Treating them alike produces error estimates wrong by large factors, usually in the optimistic direction.
What follows covers the classification, the statistics of repeated observations, the propagation rules, weighted means and level-loop closure, with worked numbers throughout.
Three kinds of error
- Blunder (mistake)
- An outright error: a misread tape, a transposed digit, an angle turned from the wrong backsight, a point number entered twice. Blunders are not part of the statistical model and cannot be adjusted away. They must be detected by redundancy and removed.
- Systematic error
- An error with a consistent sign and magnitude under given conditions: a tape of the wrong length, an uncorrected temperature or sag effect, an uncalibrated prism constant, a missing scale factor. Systematic error accumulates linearly with the number of measurements and is removed by calibration or by modelling, never by averaging.
- Random error
- The small, unpredictable variation that remains after blunders and systematic effects are removed. Positive and negative departures are equally likely and small ones are more likely than large ones. Random error cannot be eliminated, only estimated and reduced by redundancy.
The practical importance of the distinction is what averaging does to each. Averaging reduces random error, does nothing to systematic error, and is actively harmful in the presence of a blunder because it spreads the blunder across the whole set. Hence the sequence: find blunders first, correct systematic effects second, and only then treat what remains statistically.
Accuracy and precision
Precision describes how closely repeated measurements agree with one another. Accuracy describes how close they are to the true value. They are independent: a set can be tightly grouped and badly wrong, which is exactly the signature of an uncorrected systematic error, and it is the most dangerous combination because the internal agreement inspires confidence.
Every statistic below measures precision. None measures accuracy, which can only be assessed against something independent and better. A standard deviation describes the scatter of a set and says nothing about whether the set is centred on the truth.
Statistics of a set of observations
A distance was measured six times: 284.98, 285.02, 285.06, 284.99, 285.05 and 285.01 ft.
mean x̄ = Σx / n
residual vᵢ = xᵢ − x̄
σ = √( Σv² / (n − 1) )
σ_x̄ = σ / √n| Observation (ft) | Residual (ft) |
|---|---|
| 284.98 | −0.0383 |
| 285.02 | +0.0017 |
| 285.06 | +0.0417 |
| 284.99 | −0.0283 |
| 285.05 | +0.0317 |
| 285.01 | −0.0083 |
| Quantity | Value |
|---|---|
| Mean | 285.018 ft |
| Range | 0.080 ft |
| Standard deviation (n − 1) | ±0.032 ft |
| Standard error of the mean | ±0.013 ft |
| 95 per cent error (1.96σ) | ±0.063 ft |
Two of those figures are routinely confused. The standard deviation describes the scatter of individual observations, so a further single measurement would be expected to fall within about ±0.032 ft of the mean two times in three. The standard error of the mean describes the uncertainty of the mean itself, and it is smaller by a factor of the square root of the number of observations.
Propagation of random error
Measurements are rarely used alone. The question that matters is what happens to the uncertainty when several are combined, and for independent random errors the answer is that they add in quadrature - as the square root of the sum of squares - rather than arithmetically.
sum of independent quantities: E = √(E₁² + E₂² + … + Eₙ²)
n repetitions of one measurement: E = E₁ · √n
product a · b: E = √((a·E_b)² + (b·E_a)²)A 500 ft line taped in five 100 ft lengths, each good to ±0.01 ft, is good to ±0.022 ft overall, not ±0.05 ft. Random errors partially cancel; only systematic ones accumulate linearly. If the same tape is 0.01 ft short, however, the five lengths accumulate to a full 0.05 ft of systematic error and the quadrature result does not apply at all.
For a product, take a rectangle measured as 200.00 ± 0.02 ft by 350.00 ± 0.03 ft. The area is 70,000 sq ft and its standard error is √((200 × 0.03)² + (350 × 0.02)²) = ±9.22 sq ft. That is roughly one part in seven thousand six hundred, tighter than either individual measurement in relative terms - a result that surprises people the first time they compute it and follows directly from the quadrature.
Weighted means
When observations of the same quantity differ in quality, averaging them equally throws away information. Weight each by the inverse of its variance.
w = 1 / σ² x̄_w = Σ(w·x) / ΣwThree measurements of a line: 452.31 ft with σ = 0.02, 452.36 ft with σ = 0.04, and 452.28 ft with σ = 0.01.
| Observation (ft) | σ (ft) | Weight w = 1/σ² |
|---|---|---|
| 452.31 | 0.02 | 2,500 |
| 452.36 | 0.04 | 625 |
| 452.28 | 0.01 | 10,000 |
| Σw | 13,125 |
The weighted mean is 452.2895 ft, against a simple mean of 452.3167 ft - a difference of 0.027 ft, which on a boundary line is not nothing. The weighted value sits close to the best observation because that observation carries more than three-quarters of the total weight. Halving a standard deviation quadruples the weight, and that quadratic relationship is the part people get wrong when they weight by intuition.
Level-loop closure
A closed level loop returns to its starting benchmark, so the observed elevation differences must sum to zero. What they actually sum to is the misclosure, and levelling error accumulates with the number of instrument setups, which tracks distance - so the correction is distributed in proportion to leg length rather than to the size of the elevation difference.
misclosure = Σ (elevation differences)
correctionᵢ = −misclosure · (lengthᵢ / Σlength)
allowable = C · √(distance)| Leg | Length (mi) | Observed Δelev (ft) | Correction (ft) | Adjusted Δelev (ft) |
|---|---|---|---|---|
| BM1-TP1 | 0.42 | +12.417 | +0.0050 | +12.4220 |
| TP1-TP2 | 0.35 | −4.882 | +0.0042 | −4.8778 |
| TP2-TP3 | 0.51 | +7.109 | +0.0061 | +7.1151 |
| TP3-BM1 | 0.48 | −14.665 | +0.0057 | −14.6593 |
| Totals | 1.76 | −0.021 | +0.0210 | 0.000 |
The misclosure is −0.021 ft over 1.76 miles. The allowable misclosure at C = 0.05 is 0.05 × √1.76 = 0.066 ft, so the loop passes and the corrections may legitimately be distributed. Had the misclosure been 0.30 ft, distributing it would have been the wrong response entirely: that is blunder territory, and the correct action is to find the setup where it happened.
The practical summary
- Classify before you compute. Blunders are removed, systematic errors are corrected, only random error is treated statistically.
- Precision is agreement; accuracy is truth. Every statistic here measures the first.
- Use n − 1 for a standard deviation, and keep the standard deviation of a single observation distinct from the standard error of the mean.
- Independent random errors combine in quadrature; systematic errors combine linearly.
- Weight by the inverse of the variance, so halving a standard deviation quadruples the weight.
- Redundancy has square-root returns. Past a handful of repetitions, an independent check is worth more than another observation.
- Take tolerance coefficients from the specification governing the work, not from a general table.