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Measurements · Module 5 of 10

Uncertainty Analysis

You can never know the error of a measurement, but you can estimate its likely size. Uncertainty analysis combines the error sources into one honest interval and carries it through to the final result.

01

Readiness check

This module estimates and combines uncertainties. Tick only what you can do closed-notes.

  • Square numbers and add them.
  • Take a square root of a sum.
  • Form a relative (fractional) quantity.
  • Recall that a result can depend on several measurements.
  • Recall a confidence interval from the last module.
0 or 1 weak itemsContinue with this module.
2 weak itemsRevisit confidence intervals in Module 4.
3 or more weak itemsRevisit uncertainty propagation in VVUQ, Module 7.
02

The core idea

Error is the unknowable difference from truth; uncertainty is our estimate of its probable range. Independent uncertainties combine by root-sum-square, not by adding. When a result is computed from measured inputs, their uncertainties propagate, with each input weighted by how strongly the result depends on it.

combine sources: u = √(u12 + u22 + ...)product: (uR/R)2 = Σ(ui/xi)2general: uR = √[Σ(∂R/∂xi · ui)2]

A single measurement has an error, the gap from the true value, which you can never actually know. What you can do is estimate its probable size, and that estimate is the uncertainty, reported as a plus-or-minus at a stated confidence. Uncertainty has contributions from many elemental sources: instrument resolution, calibration, random scatter, and more. These are combined not by simple addition, which would be pessimistic, but by root-sum-square, adding them in quadrature as if they were independent, so a 0.3 and a 0.4 combine to 0.5, not 0.7. The same quadrature idea governs propagation: when a result R is computed from measured quantities, each input's uncertainty contributes in proportion to how sensitively R depends on it, the partial derivative, and the contributions add in quadrature. For a pure product or quotient this simplifies beautifully: the relative uncertainties add in quadrature. So an area from a length good to 1 percent and a width good to 2 percent carries an uncertainty of the square root of the sum of their squares, about 2.2 percent. This is how a measured input's imperfection becomes the uncertainty of the final engineering number, the whole point of the analysis.

The skill works when: you combine independent uncertainties in quadrature and weight each input by its sensitivity.
The skill breaks down when: uncertainties are added linearly, or error and uncertainty are treated as the same thing.
The concept. Root-sum-square combination is Pythagoras for uncertainty: 0.3 and 0.4 combine to 0.5, less than their linear sum of 0.7.
03

The skills, taught in order

Five skills turn error sources into a reported uncertainty.

5.1 Error versus uncertainty

Error is the actual, unknowable difference from truth; uncertainty is the estimated bound on it. Uncertainties split into systematic, a fixed bias, and random, the scatter, and both must be accounted for.

5.2 Design-stage uncertainty

Before any data, you can estimate an instrument's uncertainty from its resolution and its stated accuracy, combined in quadrature. This design-stage figure decides whether an instrument is even worth using for a task.

5.3 Combining elemental uncertainties

Several independent sources combine by root-sum-square: square each, sum, and take the root. This gives a realistic total that neither ignores small sources nor pessimistically adds them all linearly.

SituationCombination rule
Independent sourcesu = √(Σ ui2)
Product or quotientrelative: √(Σ (ui/xi)2)
Sum or differenceabsolute: √(Σ ui2)

Quadrature everywhere: absolute uncertainties for sums, relative uncertainties for products.

5.4 Propagation through a result

When a result depends on several measurements, each contributes its uncertainty times the sensitivity of the result to that input, the partial derivative, and the contributions add in quadrature. For products, this reduces to combining relative uncertainties.

5.5 Reporting uncertainty

A result is stated as value plus or minus uncertainty at a confidence level, for example 95 percent. Without the interval and its confidence, a measured number cannot support a design decision.

Engineering connection: a flow rate computed from a measured pressure drop and pipe area inherits both their uncertainties in quadrature, which is what a specification's tolerance must accommodate.

04

Worked example 1: combining uncertainties

Two independent uncertainty sources contribute 0.3 and 0.4 (same units) to a reading. Find the combined uncertainty.

Figure 1. The two sources combine like the legs of a right triangle: 0.3 and 0.4 give a hypotenuse of 0.5.
  1. ProblemFind the combined uncertainty for the two sources in Figure 1.
  2. Given / findu1 = 0.3, u2 = 0.4, independent. Find the combined u.
  3. AssumptionsThe sources are independent and in the same units.
  4. Modelu = √(u12 + u22).
  5. Equationsu = √(0.32 + 0.42)= √(0.09 + 0.16)
  6. Solveu = √0.25 = 0.5.
  7. CheckThe result exceeds the larger source (0.4) but is well below the linear sum (0.7), as quadrature requires.
  8. ConclusionThe combined uncertainty is 0.5, showing that adding sources linearly would overstate it by 40 percent.
Result. Combined uncertainty 0.5.
05

Worked example 2: propagating to an area

An area is A = L × W with L = 100 ± 1 mm and W = 50 ± 1 mm. Find the area and its uncertainty.

Figure 2. For a product, the relative uncertainties add in quadrature: 1 percent and 2 percent give about 2.2 percent, so about 112 mm2.
  1. ProblemFind the area and its uncertainty in Figure 2.
  2. Given / findL = 100 ± 1 mm, W = 50 ± 1 mm. Find A and uA.
  3. AssumptionsL and W are independent; A = L × W.
  4. ModelFor a product, (uA/A)2 = (uL/L)2 + (uW/W)2.
  5. EquationsA = 100 × 50 = 5000 mm2uA/A = √(0.012 + 0.022)
  6. SolveuA/A = √0.0005 = 0.0224; uA = 0.0224 × 5000 = 112 mm2.
  7. CheckThe width's 2 percent dominates; the combined 2.2 percent is just above it, as quadrature with a smaller term predicts.
  8. ConclusionThe area is 5000 ± 112 mm2, the length and width uncertainties propagated correctly through the product.
Result. A = 5000 ± 112 mm2.
06

Misconceptions and diagnostics

MistakeSymptomDiagnostic questionCorrection
Adding uncertainties linearlyOverstated total"Did I add in quadrature?"Independent sources combine by root-sum-square.
Error equals uncertaintyClaiming to know the exact error"Do I know the true value?"Error is unknowable; uncertainty is its estimated bound.
Absolute where relative belongsWrong combination for a product"Product, or sum?"Products combine relative uncertainties.
Ignoring sensitivityWeak and strong inputs weighted equally"How much does R depend on this input?"Weight each by its partial derivative.
07

Practice ladder

Level 1 · Direct skill

Combine two independent uncertainties of 0.6 and 0.8 (same units).

Show answer

u = √(0.62 + 0.82) = √1.00 = 1.0.

Level 2 · Mixed concept

An instrument has a resolution uncertainty of ±0.5 and an accuracy uncertainty of ±1.2. Find the design-stage uncertainty.

Show answer

u = √(0.52 + 1.22) = √1.69 = 1.3.

Level 3 · Independent problem

Power is P = V × I with V = 10 ± 0.1 V and I = 2 ± 0.05 A. Find P and its uncertainty.

Show answer

P = 20 W; uP/P = √(0.012 + 0.0252) = 0.0269; uP = 0.0269 × 20 ≈ 0.54 W.

Transfer task | Real engineering

Density is ρ = m/V. Given the uncertainties in m and V, write the expression for the relative uncertainty in ρ and explain each term.

What good work looks like

Because ρ = m/V is a quotient, (uρ/ρ)2 = (um/m)2 + (uV/V)2: the relative uncertainties of mass and volume add in quadrature. A good answer identifies the quotient rule and notes that the larger relative term dominates.

08

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that I combined these uncertainties in quadrature, not linearly."
"Give me a product formula; I will propagate the relative uncertainties."
"What is the uncertainty?" Combine the sources yourself.
"Propagate this for me." Identify the rule and apply it yourself.

Portfolio task

Take a result you compute from two or more measurements and propagate the input uncertainties to a final ± value.

Must include: elemental uncertainties, a quadrature combination, and a reported value ± uncertainty.
09

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. Error versus uncertainty?

Error is the unknowable difference from truth; uncertainty is its estimated bound.

2. How do independent sources combine?

By root-sum-square, adding in quadrature.

3. Propagation through a product?

Relative uncertainties add in quadrature.

4. What weights each input in general?

Its sensitivity, the partial derivative of the result.

5. How is a result reported?

Value plus or minus uncertainty at a stated confidence.

TodayFinish this quiz and Levels 1 and 2 of the ladder.
+1 dayRe-derive a propagated uncertainty from scratch.
+3 daysReport one real result with propagated uncertainty.
+7 daysMove on to analog electrical measurements in Module 6.
+30 daysAttach a quadrature uncertainty to every computed result.
10

Textbook mapping

This module follows Figliola and Beasley, Theory and Design for Mechanical Measurements, 5th edition. Use these references to read further.

Topic in this moduleWhere to read more
Measurement errors and uncertaintyFigliola and Beasley, Section 5.2, Measurement Errors
Design-stage uncertaintyFigliola and Beasley, Section 5.3, Design-Stage Uncertainty Analysis
Uncertainty propagationFigliola and Beasley, Section 5.6, Uncertainty Analysis: Error Propagation

Section numbers refer to Figliola and Beasley, 5th edition. Any edition with the same chapter titles is equivalent for study.