9.6 Measurement Error, Propagation, Positional Accuracy, and Units

Three Classes of Error

Every measurement carries error, and the FS exam expects you to classify it because each class demands a different response:

The critical insight is that averaging only helps random error. Repeating a measurement with a mis-calibrated tape ten times yields a very precise — but still wrong — result, because the systematic component does not cancel. Systematic errors must be eliminated by procedure (balancing backsight/foresight distances in leveling) or by applying corrections (temperature, tension, and sag corrections to taped distances). Blunders are not 'errors' in the statistical sense at all; they are mistakes that adjustment math will spread harmfully unless first located through redundant observations and residual inspection.

Accuracy vs. Precision, and Error Propagation

Accuracy is how close a result is to the true value; precision is how closely repeated measurements agree with each other. A tight cluster of shots offset from the bullseye is precise but inaccurate (a systematic error is present). A scattered cluster centered on the bullseye is accurate on average but imprecise. Good surveying seeks both: small random scatter (precision) and no systematic bias (accuracy).

When a result is computed from several measured quantities, their errors propagate. For a sum or difference of independent quantities, errors combine in quadrature (root-sum-of-squares):

E_series = √(E₁² + E₂² + ... + Eₙ²)

Worked propagation — a traverse leg sum. Three taped segments have standard errors of ±0.02, ±0.03, and ±0.01 ft. The error in the total length is √(0.02² + 0.03² + 0.01²) = √(0.0004 + 0.0009 + 0.0001) = √0.0014 = ±0.0374 ft, not the simple sum 0.06 ft. For n equal-error segments of error E each, E_total = E·√n, and the error of the mean of n observations is the inverse: E_mean = E/√n. For a product or quotient, relative (fractional) errors add in quadrature instead of absolute errors. Recognizing which rule applies — sums use absolute errors, products use relative errors — is the heart of FS propagation questions.

Positional Accuracy and Units

Positional accuracy describes how well a point's coordinates are known, and it is meaningful only with its confidence basis. Standards such as the National Standard for Spatial Data Accuracy (NSSDA) report horizontal accuracy at the 95% confidence level (radius of a circle, often written as CE95). A 'horizontal accuracy of 0.10 ft at 95%' means 95% of well-defined points fall within 0.10 ft of their true position — a far stronger claim than a bare '0.10 ft.' Always pair a positional tolerance with its probability level before comparing it to a specification.

Units are part of measurement science because a correct formula with wrong units produces a wrong magnitude. Memorize these FS conversions:

Worked conversion. A parcel of 2.75 acres equals 2.75 · 43,560 = 119,790 ft². The U.S. survey foot versus international foot distinction is small (about 2 ppm) but accumulates over State Plane coordinate distances of hundreds of thousands of feet — at 500,000 ft the difference is about 3 ft, which is why FS problems specify which foot to use.

Propagation Through Products and Mixed Computations

When a result is a product or quotient rather than a sum, errors propagate through relative (fractional) terms. For R = x·y, the relative error of R is √((Ex/x)² + (Ey/y)²), and the absolute error is that fraction times R. This is why area, computed from two measured sides, has a larger fractional error than either side alone.

Worked area propagation. A rectangular lot is measured as 200.0 ft (± 0.05 ft) by 150.0 ft (± 0.04 ft). The area is 30,000 ft². Relative errors are 0.05/200 = 0.00025 and 0.04/150 = 0.000267. Combined relative error = √(0.00025² + 0.000267²) = √(6.25e-8 + 7.11e-8) = √1.336e-7 = 0.000366. Absolute area error = 0.000366·30,000 = ±11.0 ft². So the area is 30,000 ± 11 ft² — the uncertainty grew because two independent measurement errors both feed the product.

The general law of propagation unifies all cases: for a function R = f(x₁, x₂, ...), E_R = √[ Σ (∂f/∂xᵢ)²·Eᵢ² ]. The partial derivative ∂f/∂xᵢ is the sensitivity of R to each input, exactly the differential idea from the calculus section. For a sum the partials are all 1, recovering the root-sum-of-squares rule; for a product the partials reproduce the relative-error rule. Mastering this one formula — and remembering that independent errors combine in quadrature while a systematic error adds directly — lets you handle any FS propagation question, including angle-and-distance combinations in coordinate computations.