Science Education Systems · Article 96. Maya, Jia Jun, Hana and Ethan are fictional learners used to make scientific reasoning visible. This article owns one distinct scientific job: uncertainty propagation—following uncertainty in input quantities through calculations, measurement models and simulations to determine uncertainty in outputs. It does not replace measurement uncertainty, sensitivity analysis or confidence intervals. It explains how uncertainty moves.
The 50-second parent route
Science rarely starts with perfectly known numbers. A length has uncertainty. A calibration has uncertainty. A fitted parameter has uncertainty. When those quantities enter a calculation, their uncertainty enters too.
The route is:
measurand → measurement model → uncertain inputs → probability or interval description → sensitivity coefficients → covariance → propagation method → output distribution → standard uncertainty → coverage interval → decision consequence → updated measurement plan
The fastest diagnostic is to ask: If each input moved within its plausible uncertainty, how much could the final answer move—and which input matters most?
This article extends How Scientific Uncertainty Works, How Scientific Measurement Works, How Scientific Sensitivity Analysis Works and How Scientific Confidence Intervals Work.
1. Uncertainty propagation begins with a measurement model
Suppose an output quantity Y depends on inputs X1, X2 and X3 through Y = f(X1, X2, X3). If the inputs are uncertain, Y becomes uncertain even when the equation is exact.
2. Maya’s first error is propagating only rounding
She thinks uncertainty comes from decimal places. Her repair is to identify physical measurement variation, calibration uncertainty, model assumptions and parameter uncertainty separately.
3. Jia Jun’s first error is adding every uncertainty directly
He adds ±1, ±2 and ±3 to obtain ±6 regardless of the equation. His repair is to let the measurement model determine how each input affects the output.
4. Hana’s first error is assuming inputs are independent
Two measurements share the same calibration reference. Her repair is to include covariance because common errors can move together.
5. Ethan’s first error is reporting a precise output from uncertain inputs
He calculates 18.374921 and reports every digit. His repair is to match numerical precision to output uncertainty.
6. Uncertainty is not the same as error
Error is the difference between a measured value and a reference or true value when that difference is conceptually defined. Uncertainty describes the dispersion or doubt associated with the value assigned to the measurand.
7. A measurement can have zero known correction and non-zero uncertainty
An instrument may be calibrated so no correction is applied, yet calibration uncertainty remains.
8. Correcting known bias does not remove uncertainty
If a scale reads 0.5 g high on average, subtracting 0.5 g removes the estimated bias. Uncertainty remains in the correction and measurement process.
9. Inputs should be quantities, not vague “errors”
Length.
mass.
temperature correction.
calibration factor.
background signal.
Each uncertain input should have a scientific meaning.
10. The measurand must be defined first
“The temperature” is incomplete. Temperature of what, where, when and under what averaging rule? Uncertainty cannot be evaluated coherently until the quantity itself is defined.
11. Model uncertainty begins with measurand definition
If the object is non-uniform, defining one temperature may require a spatial average. Variation inside the object can become part of uncertainty.
12. Primary Science can learn propagation through repeated measurement
Measure length and width, then calculate area. If both measurements vary slightly, calculated area varies too.
13. Primary 3 can learn “uncertain in, uncertain out”
Two rulers give slightly different lengths. Any derived perimeter inherits that measurement variation.
14. Primary 4 can compare which input matters more
For a rectangle, changing a long side by 1 cm may affect area more than changing a short side by 1 cm, depending on dimensions.
15. Primary 5 can learn repeated calculation
Use several plausible input values and recalculate the output. The spread shows propagation without advanced formulas.
16. Primary 6 can learn dependence
If two lengths are measured using the same stretched ruler, both may shift together. Common measurement conditions create correlated uncertainty.
17. Secondary Science can formalise standard uncertainty
An input uncertainty can be expressed as a standard deviation-like quantity under an adopted probability model.
18. Type A evaluation uses statistical information from repeated observations
Repeated measurements provide sample variation that contributes to uncertainty evaluation.
19. Type B evaluation uses other information
Calibration certificates, instrument specifications, prior data, resolution limits or expert knowledge can contribute uncertainty without repeated measurement in the current experiment.
20. Type A and Type B describe evaluation method, not quality
A carefully evaluated Type B component can be more reliable than a poorly estimated Type A component.
21. Probability distributions encode uncertainty more fully than one ± number
Normal.
uniform.
triangular.
lognormal.
discrete.
The appropriate distribution reflects available information and physical constraints.
22. Uniform uncertainty can represent bounded ignorance
If a quantity is known only to lie between a and b with no reason to favour one value, a rectangular distribution may be used as a model.
23. Triangular distributions encode central plausibility
If central values are judged more plausible than boundary values, a triangular model may be more reasonable than uniform.
24. Lognormal distributions protect positivity and skew
Multiplicative quantities can have asymmetric uncertainty where large upward deviations are more plausible than equally large negative ones.
25. Distributions are models, not observations
Choosing a normal distribution does not make uncertainty truly normal. The distribution summarises knowledge under assumptions.
26. Linear propagation uses local sensitivity
For small uncertainties and a sufficiently smooth model, a first-order Taylor approximation can estimate output variance from input variances and sensitivities.
27. Sensitivity coefficients tell how strongly each input moves the output
The partial derivative ∂Y/∂Xi describes local change in Y per unit change in Xi.
28. Units matter in sensitivity coefficients
A sensitivity coefficient has output units divided by input units. It converts input uncertainty into output-scale contribution.
29. Independent variance contributions add in quadrature under linear assumptions
If inputs are independent, output variance approximately sums squared sensitivity-weighted standard uncertainties.
30. This is why standard uncertainties are squared
Variance, not standard deviation, is additive for independent linear combinations.
31. Simple addition is a worst-case bound in some contexts, not ordinary statistical propagation
Adding absolute uncertainty limits can be conservative when all errors could align, but it answers a different question from probabilistic combination.
32. Multiplication does not mean “add percentage errors” automatically
That rule is a small-error approximation under certain forms. The measurement model and dependence structure still matter.
33. Logarithms can simplify multiplicative propagation
Products become sums on the log scale, making relative uncertainty structure easier to analyse when quantities are positive.
34. Nonlinearity can break first-order propagation
If uncertainties are large or the model is strongly curved, a local linear approximation may distort the output distribution.
35. Second-order terms can matter near curvature
Ignoring curvature can bias the estimated mean and uncertainty of the output.
36. Boundaries create asymmetric propagation
A probability constrained between 0 and 1 cannot have symmetric uncertainty near zero or one without violating the boundary.
37. Ratios become unstable near zero denominators
Small denominator uncertainty can create huge, skewed output uncertainty. Linear approximation may fail badly.
38. Threshold functions create discontinuous uncertainty
A tiny input change can switch a binary output. Derivative-based propagation may be uninformative at the threshold.
39. Monte Carlo propagation handles nonlinear models flexibly
Sample plausible input values from their joint uncertainty distribution, compute the output repeatedly and inspect the resulting output distribution.
40. Monte Carlo makes propagation conceptually visible
Each simulation is one plausible world consistent with input uncertainty. The cloud of outputs shows what the model allows.
41. Monte Carlo requires correct input distributions
Garbage distributions produce polished garbage output. Simulation does not remove the need to justify uncertainty models.
42. Monte Carlo requires dependence modelling
Sampling correlated inputs independently destroys the true joint structure.
43. Copulas or multivariate distributions can represent dependence
Advanced workflows model marginal uncertainties and dependence separately where appropriate.
44. Covariance enters linear propagation explicitly
When two inputs move together, cross-terms can increase or reduce output uncertainty depending on sensitivities and covariance sign.
45. Positive correlation can increase uncertainty
If two inputs both push the output upward and tend to rise together, their uncertainty reinforces.
46. Negative correlation can partially cancel uncertainty
If one input rises when another falls and both affect the output in the same direction, combined output variation can shrink.
47. Shared calibration creates correlation
Two lengths measured with the same biased ruler inherit common scale uncertainty.
48. Common environmental conditions create correlation
Temperature affects several sensors simultaneously. Treating their uncertainties as independent underestimates shared movement.
49. Common data preprocessing creates correlation
Several derived variables depend on the same baseline measurement. Their uncertainties share that source.
50. An uncertainty budget lists contributions explicitly
Input quantity.
estimate.
standard uncertainty.
distribution.
sensitivity coefficient.
correlation.
output contribution.
The budget turns uncertainty into an auditable system.
51. Uncertainty budgets guide improvement
If one input contributes 80% of output variance, improving a tiny 1% contributor is unlikely to matter.
52. This is sensitivity analysis with measurement meaning
Sensitivity identifies influential inputs; uncertainty budgets combine sensitivity with how uncertain those inputs actually are.
53. A highly sensitive input can contribute little if known precisely
Sensitivity alone does not determine uncertainty contribution.
54. A weakly sensitive input can matter if extremely uncertain
Contribution depends on both sensitivity and input uncertainty.
55. Combined standard uncertainty summarises output spread
After propagation, the output can be expressed with a standard uncertainty analogous to a standard deviation under the adopted model.
56. Expanded uncertainty widens the interval
A coverage factor may multiply standard uncertainty to produce a wider interval associated with a stated coverage convention.
57. Coverage factor is not always exactly two
The appropriate factor depends on desired coverage, distribution shape and degrees-of-freedom considerations.
58. Coverage intervals should state coverage probability or convention
“±5” without defining what the interval represents is incomplete.
59. A 95% uncertainty interval is not always a frequentist confidence interval
Measurement uncertainty frameworks and inferential confidence intervals can have different interpretations even when both use 95% language.
60. Output distributions can be asymmetric
Report asymmetric intervals when the model produces skew rather than forcing ± symmetry.
61. Median can be more useful than mean for skewed outputs
Summary choice should fit the output distribution and decision.
62. Quantile intervals are natural under Monte Carlo
Take lower and upper percentiles of simulated output to form a coverage interval under the simulation model.
63. Monte Carlo convergence should be checked
Too few simulations create noisy uncertainty estimates. Increase runs until relevant summaries stabilise.
64. Simulation precision is not scientific precision
One million Monte Carlo samples can estimate the wrong uncertainty model very precisely.
65. Bootstrap propagation uses observed-data resampling
Resample observations, recompute fitted quantities and derived outputs, then inspect the distribution.
66. Parametric bootstrap simulates from a fitted probabilistic model
It incorporates an assumed data-generating distribution rather than resampling raw rows directly.
67. Bootstrap inherits identifiability limits
If model parameters are structurally ambiguous, resampling cannot create information that does not exist.
68. Parameter uncertainty should propagate into predictions
A forecast based on fitted parameters should not pretend those parameters are exact.
69. Process noise and parameter uncertainty are different
Future randomness can remain even with perfectly known parameters. Parameter uncertainty represents imperfect knowledge of the model values.
70. Measurement noise is another layer
Observed future values may differ from the latent process because the sensor itself is noisy.
71. Model discrepancy is another layer
The equation may omit real mechanisms. Parameter intervals alone do not capture structural model error automatically.
72. Total predictive uncertainty has multiple components
Parameter uncertainty.
input uncertainty.
future process variation.
measurement noise.
model discrepancy.
Scenario uncertainty.
73. Combining every uncertainty into one number can hide meaning
Decision-makers may need to know whether uncertainty is reducible by better measurement or irreducible future variability.
74. Aleatory and epistemic language can help cautiously
Aleatory is often used for inherent variability; epistemic for lack of knowledge. Real problems can blur the distinction.
75. Reducibility is an operational question
If more data or calibration can shrink uncertainty, measurement investment may help. If variability is intrinsic, better decision strategies may matter more.
76. Worked case: rectangle area
Length L = 10.0 ± 0.1 cm and width W = 5.0 ± 0.1 cm. Area A = LW. Width uncertainty contributes more strongly in relative terms because 0.1 is a larger fraction of 5 than of 10.
77. The rectangle case teaches sensitivity
∂A/∂L = W and ∂A/∂W = L. Each input’s absolute uncertainty is scaled by the other dimension.
78. The rectangle case teaches correlation
If both dimensions are measured with the same miscalibrated scale, the errors may move together. Independent propagation would be incomplete.
79. Worked case: density
Density ρ = m/V. Mass and volume uncertainty propagate through a ratio. If volume is small and uncertain, density uncertainty can become large.
80. The density case teaches denominator sensitivity
When V approaches zero, small absolute uncertainty creates large relative uncertainty in ρ.
81. Worked case: speed
Speed v = distance/time. Stopwatch reaction time and distance measurement both contribute to uncertainty.
82. Short timing intervals amplify reaction-time uncertainty
If timing uncertainty is 0.2 s, it is a much larger fraction of a 1 s experiment than a 20 s experiment.
83. Better experimental design can reduce propagated uncertainty
Measure a longer travel interval so timing uncertainty becomes a smaller fraction, provided the motion remains comparable.
84. Worked case: calibration curve
A sensor reading is converted to concentration through a fitted calibration line. Uncertainty comes from the new reading and uncertainty in slope and intercept.
85. Calibration parameters can be correlated
Slope and intercept estimates often covary. Ignoring covariance can misstate concentration uncertainty.
86. Extrapolating beyond calibration range increases uncertainty
Model boundary and parameter uncertainty combine. Output intervals should widen or the prediction should be rejected.
87. Worked case: time-series forecast
Tomorrow’s demand forecast depends on model coefficients and tomorrow’s temperature forecast. Both are uncertain.
88. Nested forecasts propagate uncertainty
Temperature uncertainty enters demand through the demand model’s sensitivity to temperature.
89. Worked case: engineering load
Maximum stress depends on load, geometry and material properties. Manufacturing tolerances and load variability propagate to stress uncertainty.
90. Safety margin should reflect uncertainty
A design operating barely below nominal failure stress can be unsafe if propagated uncertainty crosses the boundary.
91. Probabilistic design treats failure as a distribution
Compare load distribution with strength distribution rather than only nominal values.
92. Reliability estimates inherit model uncertainty
If tail behaviour is poorly known, calculated rare-failure probabilities can be far more uncertain than the displayed number suggests.
93. Worked case: learner score estimate
A tutor predicts examination performance from recent practice. Task difficulty, day-to-day variation and limited sample size all contribute uncertainty.
94. One score should not become a deterministic future
A forecast of “around 75” should preserve plausible range and conditions rather than becoming “the learner will get 75.”
95. Educational uncertainty can guide intervention
If uncertainty is wide because too few unseen tasks were sampled, collect more independent tasks. If uncertainty is wide because performance is genuinely unstable, work on stability.
96. Primary 3: use repeated values
Measure the same object three times. Use the spread to explain why the derived answer is not perfectly exact.
97. Primary 4: compare uncertainty contributions
Which measurement would you improve first to make the final calculation more dependable?
98. Primary 5: run a simple scenario table
Use low, middle and high plausible inputs and calculate low, middle and high outputs.
99. Primary 6: add correlated error intuition
Two measurements share the same instrument. Ask whether their errors might move together.
100. Secondary Science: use derivative-based propagation
Students can calculate approximate contributions using sensitivity coefficients for simple models.
101. Secondary Science: compare linear and Monte Carlo methods
Use a nonlinear ratio or exponential model and observe when simulation produces asymmetry the linear approximation misses.
102. Uncertainty propagation and sensitivity analysis are different
Sensitivity asks how output responds if an input changes. Propagation combines sensitivity with how uncertain that input actually is.
103. Uncertainty propagation and identifiability are different
Identifiability asks whether parameters can be learned. Propagation asks how uncertainty in those parameters affects outputs.
104. Uncertainty propagation and confidence intervals are different
A confidence interval may quantify sampling uncertainty in an estimated parameter. Propagation may then carry that parameter uncertainty through another model.
105. Uncertainty propagation and measurement uncertainty are nested
Measurement uncertainty describes uncertain quantities; propagation describes how those uncertainties combine through a model.
106. Uncertainty propagation and robustness are connected
If plausible input uncertainty hardly changes the conclusion, the decision is robust.
107. Uncertainty propagation and boundary conditions are connected
A distribution that crosses a model boundary cannot be propagated safely using one regime-specific equation without accounting for regime change.
108. Piecewise models need regime-aware propagation
Monte Carlo samples may fall into different physical regimes and require different equations.
109. Threshold decisions turn uncertainty into risk
If an output is compared with a safety limit, calculate the probability or plausible extent of crossing rather than only the nominal distance from threshold.
110. Decision uncertainty can differ from measurement uncertainty
A ±2 measurement interval may be irrelevant when the decision threshold is 100 units away, yet critical when the threshold is 1 unit away.
111. Value of information can prioritise measurement
Which uncertain input, if measured better, is most likely to change the decision? That is more useful than reducing uncertainty everywhere equally.
112. Uncertainty reduction has cost
Higher-precision instruments, more samples and better calibration require resources. Scientific design should reduce the uncertainty that matters.
113. Correlated uncertainty can make more measurements less useful than expected
Ten measurements sharing one calibration error do not average away that common component.
114. Repetition reduces random components, not systematic shared components
This is why uncertainty budgets separate sources.
115. Calibration chains propagate uncertainty too
A field instrument is calibrated against a reference instrument, itself linked to another standard. Uncertainty accumulates through the traceability chain.
116. Traceability does not mean zero uncertainty
It means the result is linked through documented calibrations to references, with uncertainty at each step.
117. Unit conversion can be exact or uncertain
Defined conversion constants can be exact. Empirical conversion factors may have uncertainty. Do not assign uncertainty automatically to every arithmetic constant.
118. Physical constants may have published uncertainty
When a constant is experimentally determined rather than defined exactly, its uncertainty can contribute to high-precision calculations.
119. Rounding should happen at the end
Premature rounding can create additional numerical error. Carry sufficient digits internally and round final reported values consistently with uncertainty.
120. Significant figures are not a full uncertainty method
They are a reporting convention. Proper uncertainty evaluation requires a model of measurement and information sources.
121. Interval arithmetic offers guaranteed bounds under certain assumptions
Inputs are represented as intervals and arithmetic produces output intervals. Dependence can cause over-wide results if the same variable appears multiple times.
122. Probability bounds can be used when distributions are poorly known
Chebyshev-like inequalities provide conservative limits using limited information such as mean and variance.
123. Polynomial chaos and surrogate models can accelerate expensive propagation
When one simulation takes hours, approximate models can represent input-output uncertainty more efficiently.
124. Surrogate error should be included
An emulator adds approximation uncertainty beyond the original physical model.
125. Global sensitivity analysis supports uncertainty attribution
Variance-based methods can estimate how much each input and interaction contributes to output variability across the full uncertainty range.
126. Sobol-like indices separate main and total effects
Main effects capture individual input contribution; total effects include interactions.
127. Interaction uncertainty can dominate
Two inputs may be harmless individually but highly uncertain in combination.
128. Local sensitivity can miss interactions
Derivatives near one point do not describe the entire uncertainty region when the model is nonlinear.
129. Scenario uncertainty may not be probabilistic
Future policy, technology or behaviour may have several plausible scenarios without defensible probabilities.
130. Keep scenario uncertainty separate when appropriate
Report outputs under each scenario rather than inventing precise probabilities unsupported by evidence.
131. Deep uncertainty changes decision strategy
When probabilities themselves are poorly known, robust decision-making may be more appropriate than optimising expected value under one distribution.
132. AI systems need uncertainty propagation
Input uncertainty, retrieval relevance, model uncertainty, tool output uncertainty and downstream calculations can compound.
133. AI confidence is not automatically calibrated uncertainty
A language model’s verbal confidence or token probability should not be treated directly as a probability that a scientific claim is true.
134. Retrieval uncertainty can propagate
If the wrong document is retrieved, later reasoning may be internally flawless yet externally wrong.
135. Tool uncertainty can propagate
A weather API, sensor, calculator or database can return uncertain or stale values. Tool provenance matters.
136. Compound AI workflows need uncertainty checkpoints
At each stage ask: what could be wrong here, how would that affect later output, and which uncertainty deserves verification?
137. AI can help learners simulate propagation
Useful prompts include: “Generate uncertain inputs and Monte Carlo outputs,” “Show why correlation changes combined uncertainty,” “Compare linear propagation with simulation for a nonlinear equation,” and “Build an uncertainty budget.”
138. AI can fabricate an uncertainty budget convincingly
Every input uncertainty still needs evidence from measurements, calibration, literature or justified assumptions.
139. Parents can use uncertainty language productively
Instead of “the child is a 70-mark student,” say recent independent performance is around a range under comparable tasks, with uncertainty from task mix and day-to-day variation.
140. Tutors can separate measurement uncertainty from learning change
A two-mark change may be ordinary task variation. A repeated ten-mark improvement across unseen papers is stronger evidence of real change.
141. Examination prediction should include ranges
Practice performance does not produce one guaranteed score. The final examination includes new questions, timing variation and normal performance noise.
142. Independent-attempt task 1: rectangle Monte Carlo
Choose plausible distributions for length and width, sample 100 pairs and calculate areas. Compare the output spread with a simple linear approximation.
143. Independent-attempt task 2: correlation challenge
Propagate two positively correlated inputs once assuming independence and once including correlation. Explain why the answers differ.
144. Independent-attempt task 3: uncertainty budget
List five uncertainty sources for a speed experiment and rank which one contributes most to final speed uncertainty.
145. Independent-attempt task 4: nonlinear threshold
Create a model where output switches at a threshold. Explain why derivative-based propagation can fail near the switch.
146. Independent-attempt task 5: decision relevance
Compare two cases with identical measurement uncertainty but different distance from a safety threshold. Explain why the decision consequence differs.
147. Diagnostic error: uncertainty equals decimal places
Repair by identifying physical, calibration, model and sampling sources.
148. Diagnostic error: all uncertainties added directly
Repair by using the measurement model and appropriate variance/covariance combination.
149. Diagnostic error: independence assumed by default
Repair by asking which inputs share calibration, environment, baseline or preprocessing.
150. Diagnostic error: linear propagation used for strong nonlinearity
Repair with simulation or a more appropriate analytical method.
151. Diagnostic error: standard uncertainty reported as guaranteed bound
Repair by stating the distributional interpretation and coverage convention.
152. Diagnostic error: Monte Carlo sample size confused with evidence size
Repair by remembering simulation draws reproduce the uncertainty model; they do not create new empirical information.
153. Diagnostic error: model discrepancy omitted
Parameter uncertainty is tiny, so the prediction is called certain. Repair by checking whether the model structure itself is validated.
154. Diagnostic error: uncertainty propagated beyond the model boundary
Some simulations enter a new regime where the equation is invalid. Repair with regime-aware models or restrict claims.
155. The independence test
Give a learner a derived quantity with three uncertain inputs. Can they identify dependence, choose a propagation method, find the dominant contributor and explain what the final interval means? That is transferable uncertainty reasoning.
156. The evidence boundary
An uncertainty interval is conditional on the measurand definition, model, input uncertainty descriptions and dependence assumptions. It quantifies uncertainty under that framework; it does not guarantee reality lies inside because every model can be incomplete.
157. A compact uncertainty-propagation checklist
- What output or measurand is being calculated?
- What measurement model links inputs to output?
- Which inputs are uncertain?
- How was each input uncertainty evaluated?
- What probability distribution or interval represents each input?
- Which inputs are correlated?
- What sensitivity coefficients apply?
- Is first-order linear propagation adequate?
- Does nonlinearity or a boundary require Monte Carlo?
- What is the combined standard uncertainty?
- Is the output distribution symmetric or skewed?
- What coverage interval is reported?
- Which input dominates the uncertainty budget?
- Does model discrepancy matter?
- How does uncertainty affect the decision?
- What additional measurement would reduce decision-relevant uncertainty most?
158. Frequently asked questions
What is uncertainty propagation?
It is the process of calculating how uncertainty in input quantities affects uncertainty in a derived output through a measurement model or scientific calculation.
What is error propagation?
The phrase is commonly used for similar calculations, though modern measurement science distinguishes uncertainty from unknowable measurement error more carefully.
What is a sensitivity coefficient?
It describes how strongly the output changes for a small change in one input, usually represented by a partial derivative in a smooth model.
When should Monte Carlo propagation be used?
It is especially useful for nonlinear models, asymmetric distributions, complex dependence and cases where linear approximation is poor.
Why does covariance matter?
Because inputs that move together do not contribute uncertainty independently; their shared movement can increase or reduce output variance.
How does uncertainty propagation help PSLE Science?
The mathematics is advanced, but the core habit is accessible: derived answers inherit uncertainty from the measurements used to calculate them.
How does it deepen in Secondary Science?
Students can connect significant figures, measurement error, sensitivity, covariance, probability distributions, simulation and decision thresholds.
159. Continue the Science Education Systems series
- How Scientific Uncertainty Works
- How Scientific Sensitivity Analysis Works
- How Scientific Identifiability Works
Conclusion: Uncertainty does not disappear when numbers enter an equation
Maya sees the measured inputs.
Jia Jun writes the measurement model.
Hana checks which uncertainties move together.
Ethan asks whether the final uncertainty changes the decision.
Science needs all four.
Define the measurand.
map the inputs.
quantify their uncertainty.
preserve covariance.
propagate through the real model.
show the output interval.
Then improve the measurement that matters most rather than pretending the calculation made uncertainty vanish.
