Measurement uncertainty fundamentals

Foundational vocabulary for bias, precision, and propagation, the layer beneath instrument-specific derivations such as the pneumatic probes overview, the 3-hole probe uncertainty note, and the narrative in experimental aerothermal measurement.

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Reference Metrology GUM-aligned practice
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Why visitors arrive: You need metrology basics before reading instrument-specific derivations.

Your question: What is the difference between bias and precision, and how do I propagate uncertainty?

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  • Where is a worked probe example?
  • How does this appear in publications?
  • What checklist applies to experiments?

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Uncertainty is part of the model

An experimental result without a stated uncertainty budget is not falsifiable. Before comparing CFD to data, transferring a calibration between tunnels, or publishing a loss coefficient, separate what is systematic (bias) from what scatters (precision), then propagate both through the algebra that converts raw readings into physical quantities.

Core vocabulary

Three schematics. Bias: a tight cluster of readings whose mean sits offset from the true value at the centre. Precision: readings scattered about the true value. Expanded uncertainty: a normal distribution with the interval from minus two to plus two standard uncertainties shaded, U equals k times u with k often 2 for about 95 percent.BIAS, PRECISION AND EXPANDED UNCERTAINTY, SCHEMATIC[01]Biasa persistent offset ofthe mean from the truth[02]Precisionscatter about the meanunder the same conditions[03]−2u+2uExpandedU = k u, with k often 2for about 95 percent
BIAS, PRECISION, EXPANDEDBiasa persistent offset ofthe mean from the truthPrecisionscatter about the meanunder the same conditions−2u+2uExpandedU = k u, with k often 2for about 95 percent
The cross is the true value, each dot a reading, the ring the mean of the readings. Schematic, not data.
Term Meaning Typical sources
Bias (systematic) Persistent offset in a measured value; repeatable if conditions unchanged Transducer calibration drift; misalignment; polynomial fit error; facility-specific flow non-uniformity
Precision (random) Scatter about the mean under ostensibly identical conditions Tunnel unsteadiness; electronic noise; sampling window; spatial averaging
Standard uncertainty Estimated standard deviation of a quantity, often denoted \(u(x)\) Type A from repeated observations; Type B from bounds, certificates, or models
Expanded uncertainty Interval about the estimate, typically \(U = k\,u\) with coverage factor \(k\) (often 2 for ~95%) Reporting in publications; comparison to simulation confidence intervals
Sensitivity coefficient Partial derivative \(\partial f / \partial x_i\) showing how output \(f\) responds to input \(x_i\) Essential when outputs are nonlinear functions of calibrated readings

Propagation methods

How the vocabulary fits together. Each input x i carries a standard uncertainty u of x i, from Type A repeated observations or Type B bounds. The model y equals f of x 1 to x n, with a sensitivity coefficient, the partial derivative of f with respect to x i, for each input. The uncertainties are propagated by first-order Taylor (GUM) or by Monte Carlo, and the result is stated as the value plus or minus the expanded uncertainty U, equal to k times u, with k often 2 for about 95 percent.FROM READINGS TO A STATED RESULTx1u(x1)x2u(x2)xnu(xn)each u(xi)Type A: repeatsType B: boundsTHE MODELy = f(x1, …, xn)sensitivity coefficients∂f / ∂xiPROPAGATEFirst-order Taylor (GUM)Monte CarloSTATE THE RESULTvalue ± U, with U = k uk often 2 for about 95%
FROM READINGS TO A STATED RESULTeach u(xi) from Type A repeats or Type B boundsx1u(x1)x2u(x2)xnu(xn)THE MODELy = f(x1, …, xn)sensitivity coefficients∂f / ∂xiPROPAGATETaylor (GUM)Monte CarloSTATE THE RESULTvalue ± U, with U = k uk often 2 for about 95%
The terms of the vocabulary table, in the order a calculation uses them. The table below says which propagation route suits which model.

When a measured quantity \(y\) is a function of several inputs \(x_1,\ldots,x_n\), choose a propagation route consistent with linearisation error and reporting needs.

Method When it applies Limitation
First-order Taylor (GUM) Smooth functions; uncertainties small relative to curvature Breaks down for strongly nonlinear maps unless inputs redefined
Root-sum-square of components Independent bias and precision budgets on a single output Cross-correlations between inputs must be included when present
Monte Carlo Nonlinear calibration surfaces; correlated inputs; asymmetric distributions Requires defensible input PDFs; heavier to document

For a scalar output \(a\) with bias \(B_a\) and precision \(S_a\) treated independently and both stated at about 95% coverage (if both are standard uncertainties, their root-sum-square is the combined standard uncertainty, and \(U_a\) is \(k\) times it), a common reporting form is:

\[ U_a = \sqrt{B_a^2 + S_a^2} \]

The probe note applies this structure to pressure coefficients and derived flow angles, with explicit partial derivatives at the operating point.

Practical workflow

  1. Define the measurand: the quantity reported (e.g. yaw angle, effectiveness, metal temperature)
  2. Write the model: algebraic map from raw readings to the measurand, including calibration polynomials
  3. List uncertainty sources: instrument, installation, facility, data-reduction choices
  4. Estimate component uncertainties: Type A repeats and Type B bounds
  5. Propagate: Taylor or Monte Carlo; document assumptions
  6. State results: value ± expanded uncertainty; archive enough detail for reproduction

This sequence is the backbone of the instrumentation & measurement theme and the Oxford probe publication track.

About this work

Lucas Rey, aerothermal systems engineer and academic tutor.

  • University of Oxford: DPhil Researcher, Thermofluids Institute.
  • University of Cambridge: Alumnus.
  • Rolls-Royce: Sponsored researcher (High-pressure turbine programme).

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