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.
Not sure where to start? 4 places to go
Start with your question
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?
You may also be asking
- Where is a worked probe example?
- How does this appear in publications?
- What checklist applies to experiments?
Where to go next
- 3-hole probe note: Instrument-specific derivation
- Experimental measurement article: Campaign-level practice
- Instrumentation theme: Full probe workflow index
- CFD validation article: Uncertainty overlap with simulation
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.
- Instrument example: Pneumatic probes overview · 3-hole probe uncertainty analysis
- Thermal example: HPT transient thermography
- Validation context: CFD-experiment validation
Core vocabulary
| 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
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:
The probe note applies this structure to pressure coefficients and derived flow angles, with explicit partial derivatives at the operating point.
Practical workflow
- Define the measurand: the quantity reported (e.g. yaw angle, effectiveness, metal temperature)
- Write the model: algebraic map from raw readings to the measurand, including calibration polynomials
- List uncertainty sources: instrument, installation, facility, data-reduction choices
- Estimate component uncertainties: Type A repeats and Type B bounds
- Propagate: Taylor or Monte Carlo; document assumptions
- 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.
Related resources
- Pneumatic probes overview: choose probe type and workflow
- 3-hole probe uncertainty analysis: full derivation
- Instrumentation & measurement: thematic index
- Experimental aerothermal measurement: methods article
- Dimensionless groups reference: similarity and scaling
- CFD-experiment validation: simulation pairing article
- Engineering hub: full platform index
- Technical notes hub
Part of
This page sits within the broader knowledge structure on lucasrey.com:
- Engineering - Central knowledge platform - tools, curriculum, notes, and research assets.
Connected work
Related content from the same research and engineering work:
- Technical notes index
- Pneumatic probes overview
- 3-hole probe uncertainty note
- Instrumentation theme
- Uncertainty fundamentals - Referenced from Additive heat-exchanger low-order models
- Uncertainty fundamentals - Referenced from Aerothermal engineering methods
Part of Engineering
This page is part of the engineering knowledge platform on lucasrey.com.
- Engineering hub - Full knowledge platform index
- Engineering notes - Reference material and methods documentation - instrumentation and turbomachinery clusters.
- Pneumatic probes overview - 3-hole vs 5-hole selection, calibration workflow, and facility transfer - orientation before instrument-specific derivations.
- 3-hole probe uncertainty - Uncertainty propagation for multi-hole pneumatic probes.
- HPT transient thermography - Transient IR data-reduction and uncertainty workflow for ECAT turbine campaigns.
- Engineering notes - Reference material and methods documentation - instrumentation and turbomachinery clusters.
- 3-hole probe uncertainty - Uncertainty propagation for multi-hole pneumatic probes.
- Instrumentation & measurement - Pneumatic probing, calibration transfer, and uncertainty propagation.
- Experimental aerothermal measurement - Methods article on traceability, similarity, and facility transfer - links instrumentation, curriculum, and tools.
- CFD-experiment validation - Methods article on closing the loop between RANS simulations and transient aerothermal measurements.
- HPT transient thermography - Transient IR data-reduction and uncertainty workflow for ECAT turbine campaigns.
- Technical notes index
- Pneumatic probes overview
- 3-hole probe uncertainty note
Engineering knowledge platform
More tools, curriculum, notes, and research from the same body of work:
- Engineering - Central knowledge platform - tools, curriculum, notes, research, and applied engineering work.
Where to go from here
Ways to take this further, in the order they usually happen.
- Get air properties at your own conditions - Conductivity, viscosity, specific heat, gamma and Prandtl for dry air from a temperature alone, and density, thermal diffusivity and kinematic viscosity once you add a pressure, each beside the correlation it came from and its valid range
- The rest of the engineering work - Tools, notes and the software behind them, indexed by topic
- How an engagement is scoped - Defined remit, timeline and deliverables, and which path suits which kind of work
- Discuss a piece of technical work - Scoped advisory, aerothermal review, instrumentation design and R and D roles
I take a small number of advisory engagements alongside the doctorate.