Accuracy, uncertainty & reproducible DC workflow

A display with many digits can have fine resolution and still have poor accuracy. Repeated readings can be very precise yet biased. Uncertainty is not a synonym for error; it expresses the remaining dispersion attributed to the measurand after the measurement model, corrections and information have been considered.

A result is meaningless until the quantity intended to be measured is defined. “Measure the voltage” is incomplete if the circuit changes when the meter is connected.

For the unloaded Thévenin voltage from Chapter 02, one possible measurement model is

The output quantity is the desired unloaded voltage. The input quantities are the loaded indication plus source and meter input resistances. This model makes a crucial point explicit: loading correction is not an afterthought; it is part of the definition of how the measurand is obtained.

If the intended measurand is instead the actual terminal voltage with the meter connected, no unloading correction belongs in the model. The model must match the question.

Suppose five repeated DC readings under repeatability conditions are:

Their arithmetic mean is

The sample standard deviation is approximately

If the mean of these n independent observations is used as the estimate and the repeatability model is appropriate, the standard uncertainty of the mean is

This is a Type A evaluation because it is obtained by statistical analysis of repeated observations. It does not mean “all random effects belong to Type A” or “all systematic effects belong to Type B.” Type A and Type B classify methods of evaluating uncertainty components, not error mechanisms.

Consider a deliberately hypothetical meter specification used only to teach the calculation:

Assume the reading is 5.000 V and the displayed least significant digit is 1 mV. The specification bound is

A bound is not automatically a standard uncertainty. If, and only if, a rectangular distribution within ±a is justified by the available information, then

Another distribution would give another standard uncertainty. The probability model must be justified from the specification, calibration information or engineering knowledge.

For a measurement model y=f(x₁,…,x_N), the first-order law of propagation uses sensitivity coefficients

For uncorrelated input quantities,

If relevant input quantities are correlated, covariance terms must be included rather than silently set to zero:

For the simple teaching example where the Type A mean uncertainty 0.707 mV and the hypothetical Type B specification uncertainty 2.598 mV are treated as independent contributions to the same voltage estimate,

The larger component dominates. Collecting many repeated readings cannot compensate for a much larger specification/calibration contribution.

An expanded uncertainty is commonly written

For the teaching example, choosing k=2 gives

But k=2 is not a universal mathematical synonym for exactly 95% coverage. Coverage depends on the probability distribution represented by the result, effective degrees of freedom and the method used. If a coverage probability is claimed, the basis must be stated.

A defensible teaching-style report could therefore be: V = 5.0000 V ± 0.0054 V, U, k=2, followed by the conditions and budget used. The number of reported digits should be consistent with the uncertainty rather than copied mechanically from the display.

Calibration establishes, under specified conditions, a relation between reference quantity values with uncertainties and indications with associated uncertainties, then uses that information to obtain a measurement result from an indication. Calibration does not necessarily change the instrument.

Adjustment is a set of operations performed on a measuring system so that it provides prescribed indications corresponding to given values. Adjustment can change the instrument and may require recalibration afterward.

Metrological traceability is a property of a measurement result whereby it can be related to a reference through a documented unbroken chain of calibrations, each contributing to measurement uncertainty.

Therefore “the meter is calibrated” is not enough for a reproducible record. We also need the calibration status/date or interval context, the applicable certificate or specification information, and whether the conditions of use fall within that information.

AoN treats measurement evidence as first-class technical data. A minimum reproducible record should capture:

1. Measurand: what quantity is intended, between which nodes, under what circuit state and environmental conditions.
2. Circuit state: source settings, loads, relevant topology, power state and any loading/correction model.
3. Instrument identity: manufacturer/model, asset or serial identifier when appropriate, and calibration status/context.
4. Configuration: function, range, terminals, integration/filter settings if relevant, probe/lead arrangement and polarity/reference convention.
5. Raw observations: individual readings, not only their final average.
6. Corrections: loading, offsets, lead compensation or other corrections, with sign and source of correction.
7. Uncertainty budget: each input quantity, estimate, standard uncertainty, distribution/evaluation method, sensitivity coefficient and contribution.
8. Result statement: estimate, uncertainty type, coverage factor/probability when used, units and significant digits.
9. Provenance: date/time, operator or automated workflow, raw-data file, script/version and links to calibration or instrument documentation when needed.

This transforms “I saw 5.000 V” into evidence another engineer can inspect, recompute and reproduce.

1. Five readings are 4.998, 5.001, 4.999, 5.000 and 5.002 V. Compute mean, sample standard deviation and standard uncertainty of the mean.
2. A hypothetical specification is ±(0.05% of reading + 2 digits), reading 5.000 V, least significant digit 1 mV. Find the bound and, assuming a justified rectangular distribution, the standard uncertainty.
3. Combine the Type A and Type B standard uncertainties from exercises 1 and 2 as independent contributions.
4. Using k=2, calculate expanded uncertainty. Why may you not write “exactly 95%” without further justification?
5. Explain in one sentence each the difference among accuracy, precision and uncertainty.
6. A technician adjusts a meter until its indication changes, then says “calibration is complete.” What conceptual step is missing?
7. For VTH=Vmeas(1+RTH/RIN), list the three input quantities whose uncertainty can contribute to the corrected unloaded voltage.
8. Design the minimum record you would require before accepting a 5 V DC measurement into AoN's Real Measurement Library.