Measurement Errors

Error, uncertainty and repeatability

Indication error is measured value minus reference value. A correction has the opposite sign. Uncertainty describes the dispersion of values reasonably attributable to the measurand; it is not simply a known error and does not disappear after correction. Systematic effects can shift readings consistently; random effects cause scatter. Repetition can improve the estimate of a mean under suitable assumptions, but does not remove a persistent calibration bias. Precision describes agreement among repeated measurements and is distinct from closeness to the reference.

Basic calculations

For n independent comparable readings, the mean is Σxi/n and sample standard deviation is sqrt[Σ(xi − mean)²/(n − 1)]. A Type A standard uncertainty of the mean is s/√n under the applicable statistical assumptions. Type B contributions use other information such as calibration reports or specifications. For y = f(x1,…), first-order propagation for uncorrelated inputs is u(y)² = Σ[(∂f/∂xi)u(xi)]². Include covariance terms when inputs are correlated. Worst-case tolerance addition and statistical uncertainty combination are different methods.

Worked example

Measured V = 10 V and I = 2 A give R = V/I = 5 Ω. Suppose independent standard uncertainties are u(V) = 0.1 V and u(I) = 0.02 A. Relative standard uncertainty in R is sqrt[(0.1/10)² + (0.02/2)²] = 0.01414. u(R) = 5 × 0.01414 = 0.0707 Ω. If instead ±1% values were treated as worst-case bounds, the first-order resistance bound would be about ±2%, or ±0.1 Ω. Do not label that bound as the same standard uncertainty.

Common mistakes

State whether a specification is percent of reading or full scale. Keep the error sign convention explicit. Quote meaningful significant figures and the uncertainty interpretation.

Quick check

1. Can a precise instrument have bias? Yes.
2. Does averaging remove fixed bias? No.
3. When is root-sum-square propagation incomplete? When correlation or important nonlinear effects are ignored.