Theory — Introduction and Learning Goals

Short summary This chapter covers measurement theory: uncertainty concepts, best practices, measurement-system analysis, and worked Monte Carlo examples.

Learning objectives

• Understand types of measurement uncertainty and how to report them.

• Distinguish repeatability vs reproducibility, bias, and systematic errors.

• Apply basic uncertainty propagation (analytical and Monte Carlo).

• Recognize good-practice recommendations for lab notebooks and reporting.

Key concepts (brief)

• GUM-style uncertainty vs Type A/B estimates.

• Propagation of uncertainty for slopes and model parameters.

• Role of simulations (Monte Carlo) to validate analytical propagation.

Recommended notebooks to run

• uncertainty_example.ipynb

• uncertainty_of_a_slope.ipynb

• uncertainty_propagation_monte_carlo_gum.ipynb

• teaching_measurement_uncertainty.ipynb

• best_practice_summary.ipynb

Suggested exercises

• Compute and compare analytical and Monte Carlo propagation on a simple function.

• Prepare a short lab report following the best-practice notebook checklist.

Prerequisites Basic probability, calculus, and comfort with Python arrays.

Checklist for the 8-step uncertainty analysis process

See checklist.md for a detailed 9-step checklist to guide uncertainty analyses.

Pages in this chapter

• Sensitivity Coefficients in Uncertainty Budgets

• Short summary of the ``Measurement good practice guide ‘’ by NPL

• Mars Rover Temperature Measurement & Uncertainty Analysis

• Using AI tools to learn uncertainty

• Engineering Example: Uncertainty Analysis in Mechanical Measurements

• ## General measurement system diagram

• Uncertainty in simple terms from IAEA

• Laboratory Notebook

• Significant digits

• Simple example of mechanical measurement with uncertainty analysis

• Using simulations to explain uncertainty

• Standardization

• Teaching Measurement in the Introductory Physics Laboratory

• Teaching uncertainty in mechanical measurements

• Uncertainty 101

• Uncertainty Analysis

• uncertainty_example

• How to estimate the uncertainty of a slope for static calibration or regression

• Propagating uncertainty using Monte-Carlo simulations

Ordered reading (suggested)

Follow this sequence when teaching or self-studying. The order moves from foundational lab practice and best-practice guidance, to measurement-system analysis and elementary worked examples, then to uncertainty concepts and quantitative propagation methods (analytical & Monte Carlo), and finishes with advanced case studies and community presentations.

1. laboratory_notebook.ipynb — practical lab notebook practices and data recording.

2. best_practice_summary.ipynb — concise recommendations for reporting and reproducibility.

3. teaching_measurement_uncertainty.ipynb — pedagogical overview of uncertainty.

4. standartization.ipynb — standards and common terminology.

5. general_measurement_system_analysis.ipynb — system-level thinking and error sources.

6. simple_example.ipynb — a short worked example linking practice and theory.

7. example_from_best_practice.ipynb — illustrated application of best practices.

8. uncertainty_example.ipynb — basic uncertainty calculations and interpretation.

9. uncertainty_of_a_slope.ipynb — propagation for regression-derived quantities.

10. uncertainty_propagation_monte_carlo_gum.ipynb — Monte Carlo propagation following GUM ideas.

11. simulations_for_uncertainty.ipynb — simulation-driven exploration of uncertainty.

12. uncertainty_analysis_NASA.ipynb — applied example from NASA guidance.

13. iaea_uncertainty_presentation.ipynb — community presentation and advanced perspectives.

Rationale: this ordering lets students first acquire good lab habits and reporting skills, then build a conceptual toolbox for system analysis, then learn measurement uncertainty in increasing rigor (examples → slope propagation → Monte Carlo → case studies). Use the checklists added to notebooks to guide in-class or lab activities.