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Metrology and Uncertainty — Introduction and Learning Goals

This chapter covers measurement theory or metrology: uncertainty concepts, best practices, analytical measurement system analysis, examples of measurement systems.

Learning objectives

  • Understand types of measurement uncertainty and how to report them.
  • Distinguish repeatability vs reproducibility, bias, and systematic errors.
  • Apply basic uncertainty propagation.
  • Recognize good-practice recommendations for lab notebooks and reporting.

Key concepts

  • GUM-style uncertainty vs Type A/B estimates.
  • Propagation of uncertainty in complex measurements

Prerequisites

Basic probability, calculus, and comfort with Python, Numpy, Matplotlib, Scipy, Jupyter.

Laboratory notebook

Keep a good laboratory notebook for the course: date, name, instruments and their calibration stickers, temperature, and every reading, laid out so outliers stand out. Guides: MIT, Science Buddies, UBC.

Ordered reading (suggested)

Foundations first, then the core lecture and workflow, then worked examples of increasing difficulty, finishing with two full real-world budgets and capstones.

  1. Significant digits — how many digits to keep, and how to report \(x \pm u\) (1 sig. fig. for \(u\)).
  2. Standardization and Traceability — standards, traceability chains, common terminology.
  3. Short summary of the ``Measurement good practice guide '' by NPL — error vs uncertainty, Type A/B in one precis.
  4. The Generalized Measurement System — system-level thinking: stages, signals, error sources at each block.
  5. Basic error analysis — systematic vs random errors, RSS combination, \(P\)/\(B\) notation, small-sample \(t\) note.
  6. Uncertainty 101 — the core lecture: Type A/B, divisors, combined and expanded uncertainty on the caliper example (\(\ell = 21.49 \pm 0.05\) mm, \(k=2\)).
  7. The Engineer's 9-Step Uncertainty Analysis Checklist — the workflow (with the NASA 5-step variant as sidebar).
  8. Engineering Example: Uncertainty Analysis in Mechanical Measurements — full 8-step shaft measurement.
  9. Simple example of mechanical measurement with uncertainty analysis — string and tape measure, plus how to (carefully) use AI tools on the guides.
  10. Sources of Uncertainty in Measurement for Every Uncertainty Budget — the 8-source catalog with instrument examples.
  11. Comparing Two Methods: Cylinder Volume — geometric vs gravimetric, dominant contributors, fractional-form appendix.
  12. How to estimate the uncertainty of a slope for static calibration or regression — interactive: drag \(u(I)\) and watch the slope interval change.
  13. Propagating uncertainty using Monte-Carlo simulations — interactive: non-normal outputs, cube/thermal/cosine/density examples, best practices.
  14. Sensitivity Coefficients in Uncertainty Budgets — partial derivatives as weights, building-height example.
  15. Introduction to uncertainty analysis — full surface-roughness budget with Welch–Satterthwaite.
  16. Full Uncertainty Budget of a Hot-Wire Anemometer for Isothermal Turbulent Air Flow in a Wind Tunnel — complete 24-source real-world budget.
  17. Mars Rover Temperature Measurement & Uncertainty Analysis — capstone: instrument selection, static/dynamic calibration design, full 8-step exam answer.
  18. Teaching Measurement in the Introductory Physics Laboratory — pedagogy capstone: plain-language guide plus instructor material.

Rationale: students first acquire reporting habits and vocabulary, then the propagation toolbox, then the canonical workflow — each worked example reuses the same 8 steps on harder systems until the capstones.

What comes next?

Now that you understand what uncertainty is and where it comes from, the Statistics chapter teaches you how to quantify it from real measurement data. You'll learn to use histograms, distributions, and statistical tests to characterize the variability in your measurements — which feeds directly into Type A uncertainty budgets.