Measurement Uncertainty

Probability and Statistics · FE Reference Handbook section

Probability and Statistics

5 formulas

10 exam-style examples

~55 min

All Probability and Statistics lectures

Learning objectives

What you must be able to do before leaving this section.

This chapter section covers Measurement Uncertainty within Probability and Statistics. Read it the way you would read a textbook chapter: the theory first so the relations mean something, then every equation with its use and its trap, then 10 fully worked examples with the arithmetic shown line by line, and finally a self-check you should be able to answer without notes.

Explain, in your own words, what measurement uncertainty describes physically and when it applies.
State every one of the 5 relations the handbook lists here and name each symbol with its unit.
Select the correct relation from the wording of an exam stem within 20 seconds.
Carry a complete solution from givens to a "most nearly" answer with the correct unit.
Recognise the distractors generated by the unit trap: probabilities are dimensionless and must land in [0, 1].

Lecture

Why this section exists. Measurement Uncertainty is the part of Probability and Statistics that lets you connect a sample of measurements from a construction or materials process to a number you can defend. Before any equation is useful you must be able to picture the physical situation it describes; the schematic below is that picture.

How the theory is built. The handbook prints results, not derivations. Each relation in this section comes from one governing principle applied to the idealised system: state the principle, impose the stated assumptions, and the printed equation follows. Knowing which assumption each relation rests on is what lets you reject a wrong answer choice in seconds.

How it is examined. Items from this page are written as one distribution or one counting rule, then a single probability or interval. Roughly two thirds are direct substitution, one third require one intermediate quantity from a neighbouring relation, and a small number are conceptual — testing whether you know the assumption, not the arithmetic.

The habit that earns the points. Unit discipline. probabilities are dimensionless and must land in [0, 1]. Every relation below is dimensionally consistent only when that rule is honoured, and the distractor set is deliberately built from candidates who ignored it. Write the unit next to every number you substitute, every time.

How to study this page. Read the theory, then cover the formula cards and try to reproduce each relation from its description. Then work the examples with the solution hidden, revealing one line at a time. Finish with the self-check questions; if you cannot answer one, return to the matching formula card.

Three engineers in hard hats and safety vests reviewing drawings on a truck tailgate. — Photo 1. Where this shows up in practice: measurement uncertainty.
Capstone Studio instructional photograph

Probability and Statistics — Measurement Uncertainty: reference schematic for orienting the symbols used in this section.

Theory, developed

Read this before the equations — it is what makes them memorable.

The physical situation

Every item from this section describes a sample of measurements from a construction or materials process. Sketch it before you compute — a labelled sketch with the givens on it converts a wordy stem into a solvable problem and exposes the quantity the examiner left out on purpose.

The governing principle

The 5 relations on this page are consequences of one principle applied to that idealised system. Identify which quantity is conserved, balanced, or defined, and the correct equation follows without memorisation.

Assumptions and limits of validity

Each printed relation carries silent assumptions — linearity, steady state, uniformity, small deformation, or standard conditions, depending on the subject. Conceptual exam items are written by violating exactly one of these, so read the sentence above the equation as carefully as the equation itself.

Solution procedure you should automate

1) Read the last sentence of the stem to identify the requested quantity. 2) Locate the relation on this page whose left-hand side is that quantity. 3) Tabulate the givens with units and mark the missing symbol. 4) If a symbol is missing, find the one relation that produces it. 5) Rearrange symbolically, substitute once, evaluate, and round only at the end.

Notation used in this section

y	Quantity produced by "y = f(x1, x2,…, xn)" — read its definition and unit from the handbook line directly above the equation.
vy	Quantity produced by "vy = e o 2 e o 2 e o 2" — read its definition and unit from the handbook line directly above the equation.

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

Measurement uncertainty is defined as: A quantitative estimate of the range of values about the reported or measured value in
which the true value is believed to lie. [Source: ISO JCGM 200:2012, definition 2.26]
Given a desired state or measurement y, which is a function of different measured or available states xi:
2 2 2
2f 2f 2f
2x1 v x1 + 2x2 v x2 + g + 2xn v xn
This represents 95% of the area under a Normal probability distribution and is often called 2 sigma.

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1

Measurement error, residual and combined uncertainty — Measurement Uncertainty

A distance with an accepted value of 97.50 ft is observed as 98.90 ft. The instrument contributes a standard uncertainty of ±0.18 ft and the target centring ±0.16 ft. Report the measurement error, the residual, and the combined standard uncertainty.

Given

True value = 97.50 ft
Observation = 98.90 ft
σ₁ = 0.18 ft, σ₂ = 0.16 ft

Find

Error, residual and combined uncertainty

Start with the thinking

Error is observation minus true value; the residual is the same magnitude with the sign convention of the adjustment.
Independent uncertainties combine in quadrature, never by simple addition.

Step-by-step solution

Formula
$e = o b ser v e d - t r u e$
Substituting
$e = 98.90 - 97.50 = 1.400 f t$
Relative error — 1.4359%
Formula
$u_{c} = (σ_{1}^{2} + σ_{2}^{2})$
Substituting
$u_{c} = (0.1 8^{2} + 0.1 6^{2}) = 0.2408 f t$
95% expanded uncertainty
$U = 2 u_{c} = 0.4817 f t$

Answer: Error = 1.400 ft, combined uncertainty = ±0.241 ft

Why the other options are there

±0.340 ft (uncertainties added directly)
-1.400 ft (sign reversed)

Reference: FE Reference Handbook — Probability and Statistics → Measurement Uncertainty

Example 2

Measurement error, residual and combined uncertainty — Measurement Uncertainty (2)

A distance with an accepted value of 62.00 ft is observed as 62.60 ft. The instrument contributes a standard uncertainty of ±0.12 ft and the target centring ±0.13 ft. Report the measurement error, the residual, and the combined standard uncertainty.

Given

True value = 62.00 ft
Observation = 62.60 ft
σ₁ = 0.12 ft, σ₂ = 0.13 ft

Find

Error, residual and combined uncertainty

Start with the thinking

Error is observation minus true value; the residual is the same magnitude with the sign convention of the adjustment.
Independent uncertainties combine in quadrature, never by simple addition.

Step-by-step solution

Formula
$e = o b ser v e d - t r u e$
Substituting
$e = 62.60 - 62.00 = 0.600 f t$
Relative error — 0.9677%
Formula
$u_{c} = (σ_{1}^{2} + σ_{2}^{2})$
Substituting
$u_{c} = (0.1 2^{2} + 0.1 3^{2}) = 0.1769 f t$
95% expanded uncertainty
$U = 2 u_{c} = 0.3538 f t$

Answer: Error = 0.600 ft, combined uncertainty = ±0.177 ft

Why the other options are there

±0.250 ft (uncertainties added directly)
-0.600 ft (sign reversed)

Reference: FE Reference Handbook — Probability and Statistics → Measurement Uncertainty

Example 3

Measurement error, residual and combined uncertainty — Measurement Uncertainty (3)

A distance with an accepted value of 187.0 ft is observed as 188.5 ft. The instrument contributes a standard uncertainty of ±0.03 ft and the target centring ±0.13 ft. Report the measurement error, the residual, and the combined standard uncertainty.

Given

True value = 187.0 ft
Observation = 188.5 ft
σ₁ = 0.03 ft, σ₂ = 0.13 ft

Find

Error, residual and combined uncertainty

Start with the thinking

Error is observation minus true value; the residual is the same magnitude with the sign convention of the adjustment.
Independent uncertainties combine in quadrature, never by simple addition.

Step-by-step solution

Formula
$e = o b ser v e d - t r u e$
Substituting
$e = 188.5 - 187.0 = 1.500 f t$
Relative error — 0.8021%
Formula
$u_{c} = (σ_{1}^{2} + σ_{2}^{2})$
Substituting
$u_{c} = (0.0 3^{2} + 0.1 3^{2}) = 0.1334 f t$
95% expanded uncertainty
$U = 2 u_{c} = 0.2668 f t$

Answer: Error = 1.500 ft, combined uncertainty = ±0.133 ft

Why the other options are there

±0.160 ft (uncertainties added directly)
-1.500 ft (sign reversed)

Reference: FE Reference Handbook — Probability and Statistics → Measurement Uncertainty

Example 4

Measurement error, residual and combined uncertainty — Measurement Uncertainty (4)

A distance with an accepted value of 177.5 ft is observed as 175.9 ft. The instrument contributes a standard uncertainty of ±0.02 ft and the target centring ±0.07 ft. Report the measurement error, the residual, and the combined standard uncertainty.

Given

True value = 177.5 ft
Observation = 175.9 ft
σ₁ = 0.02 ft, σ₂ = 0.07 ft

Find

Error, residual and combined uncertainty

Start with the thinking

Error is observation minus true value; the residual is the same magnitude with the sign convention of the adjustment.
Independent uncertainties combine in quadrature, never by simple addition.

Step-by-step solution

Formula
$e = o b ser v e d - t r u e$
Substituting
$e = 175.9 - 177.5 = - 1.600 f t$
Relative error — 0.9014%
Formula
$u_{c} = (σ_{1}^{2} + σ_{2}^{2})$
Substituting
$u_{c} = (0.0 2^{2} + 0.0 7^{2}) = 0.0728 f t$
95% expanded uncertainty
$U = 2 u_{c} = 0.1456 f t$

Answer: Error = -1.600 ft, combined uncertainty = ±0.073 ft

Why the other options are there

±0.090 ft (uncertainties added directly)
1.600 ft (sign reversed)

Reference: FE Reference Handbook — Probability and Statistics → Measurement Uncertainty

Example 5

Measurement error, residual and combined uncertainty — Measurement Uncertainty (5)

A distance with an accepted value of 111.0 ft is observed as 111.5 ft. The instrument contributes a standard uncertainty of ±0.04 ft and the target centring ±0.14 ft. Report the measurement error, the residual, and the combined standard uncertainty.

Given

True value = 111.0 ft
Observation = 111.5 ft
σ₁ = 0.04 ft, σ₂ = 0.14 ft

Find

Error, residual and combined uncertainty

Start with the thinking

Error is observation minus true value; the residual is the same magnitude with the sign convention of the adjustment.
Independent uncertainties combine in quadrature, never by simple addition.

Step-by-step solution

Formula
$e = o b ser v e d - t r u e$
Substituting
$e = 111.5 - 111.0 = 0.500 f t$
Relative error — 0.4505%
Formula
$u_{c} = (σ_{1}^{2} + σ_{2}^{2})$
Substituting
$u_{c} = (0.0 4^{2} + 0.1 4^{2}) = 0.1456 f t$
95% expanded uncertainty
$U = 2 u_{c} = 0.2912 f t$

Answer: Error = 0.500 ft, combined uncertainty = ±0.146 ft

Why the other options are there

±0.180 ft (uncertainties added directly)
-0.500 ft (sign reversed)

Reference: FE Reference Handbook — Probability and Statistics → Measurement Uncertainty

Example 6

Measurement error, residual and combined uncertainty — Measurement Uncertainty (6)

A distance with an accepted value of 129.0 ft is observed as 131.3 ft. The instrument contributes a standard uncertainty of ±0.16 ft and the target centring ±0.08 ft. Report the measurement error, the residual, and the combined standard uncertainty.

Given

True value = 129.0 ft
Observation = 131.3 ft
σ₁ = 0.16 ft, σ₂ = 0.08 ft

Find

Error, residual and combined uncertainty

Start with the thinking

Error is observation minus true value; the residual is the same magnitude with the sign convention of the adjustment.
Independent uncertainties combine in quadrature, never by simple addition.

Step-by-step solution

Formula
$e = o b ser v e d - t r u e$
Substituting
$e = 131.3 - 129.0 = 2.300 f t$
Relative error — 1.7829%
Formula
$u_{c} = (σ_{1}^{2} + σ_{2}^{2})$
Substituting
$u_{c} = (0.1 6^{2} + 0.0 8^{2}) = 0.1789 f t$
95% expanded uncertainty
$U = 2 u_{c} = 0.3578 f t$

Answer: Error = 2.300 ft, combined uncertainty = ±0.179 ft

Why the other options are there

±0.240 ft (uncertainties added directly)
-2.300 ft (sign reversed)

Reference: FE Reference Handbook — Probability and Statistics → Measurement Uncertainty

Example 7

Measurement error, residual and combined uncertainty — Measurement Uncertainty (7)

A distance with an accepted value of 123.0 ft is observed as 123.6 ft. The instrument contributes a standard uncertainty of ±0.17 ft and the target centring ±0.10 ft. Report the measurement error, the residual, and the combined standard uncertainty.

Given

True value = 123.0 ft
Observation = 123.6 ft
σ₁ = 0.17 ft, σ₂ = 0.10 ft

Find

Error, residual and combined uncertainty

Start with the thinking

Error is observation minus true value; the residual is the same magnitude with the sign convention of the adjustment.
Independent uncertainties combine in quadrature, never by simple addition.

Step-by-step solution

Formula
$e = o b ser v e d - t r u e$
Substituting
$e = 123.6 - 123.0 = 0.600 f t$
Relative error — 0.4878%
Formula
$u_{c} = (σ_{1}^{2} + σ_{2}^{2})$
Substituting
$u_{c} = (0.1 7^{2} + 0.1 0^{2}) = 0.1972 f t$
95% expanded uncertainty
$U = 2 u_{c} = 0.3945 f t$

Answer: Error = 0.600 ft, combined uncertainty = ±0.197 ft

Why the other options are there

±0.270 ft (uncertainties added directly)
-0.600 ft (sign reversed)

Reference: FE Reference Handbook — Probability and Statistics → Measurement Uncertainty

Example 8

Measurement error, residual and combined uncertainty — Measurement Uncertainty (8)

A distance with an accepted value of 167.5 ft is observed as 165.9 ft. The instrument contributes a standard uncertainty of ±0.04 ft and the target centring ±0.18 ft. Report the measurement error, the residual, and the combined standard uncertainty.

Given

True value = 167.5 ft
Observation = 165.9 ft
σ₁ = 0.04 ft, σ₂ = 0.18 ft

Find

Error, residual and combined uncertainty

Start with the thinking

Error is observation minus true value; the residual is the same magnitude with the sign convention of the adjustment.
Independent uncertainties combine in quadrature, never by simple addition.

Step-by-step solution

Formula
$e = o b ser v e d - t r u e$
Substituting
$e = 165.9 - 167.5 = - 1.600 f t$
Relative error — 0.9552%
Formula
$u_{c} = (σ_{1}^{2} + σ_{2}^{2})$
Substituting
$u_{c} = (0.0 4^{2} + 0.1 8^{2}) = 0.1844 f t$
95% expanded uncertainty
$U = 2 u_{c} = 0.3688 f t$

Answer: Error = -1.600 ft, combined uncertainty = ±0.184 ft

Why the other options are there

±0.220 ft (uncertainties added directly)
1.600 ft (sign reversed)

Reference: FE Reference Handbook — Probability and Statistics → Measurement Uncertainty

Example 9

Measurement error, residual and combined uncertainty — Measurement Uncertainty (9)

A distance with an accepted value of 62.50 ft is observed as 60.70 ft. The instrument contributes a standard uncertainty of ±0.04 ft and the target centring ±0.11 ft. Report the measurement error, the residual, and the combined standard uncertainty.

Given

True value = 62.50 ft
Observation = 60.70 ft
σ₁ = 0.04 ft, σ₂ = 0.11 ft

Find

Error, residual and combined uncertainty

Start with the thinking

Error is observation minus true value; the residual is the same magnitude with the sign convention of the adjustment.
Independent uncertainties combine in quadrature, never by simple addition.

Step-by-step solution

Formula
$e = o b ser v e d - t r u e$
Substituting
$e = 60.70 - 62.50 = - 1.800 f t$
Relative error — 2.8800%
Formula
$u_{c} = (σ_{1}^{2} + σ_{2}^{2})$
Substituting
$u_{c} = (0.0 4^{2} + 0.1 1^{2}) = 0.1170 f t$
95% expanded uncertainty
$U = 2 u_{c} = 0.2341 f t$

Answer: Error = -1.800 ft, combined uncertainty = ±0.117 ft

Why the other options are there

±0.150 ft (uncertainties added directly)
1.800 ft (sign reversed)

Reference: FE Reference Handbook — Probability and Statistics → Measurement Uncertainty

Example 10

Measurement error, residual and combined uncertainty — Measurement Uncertainty (10)

A distance with an accepted value of 135.0 ft is observed as 137.2 ft. The instrument contributes a standard uncertainty of ±0.20 ft and the target centring ±0.20 ft. Report the measurement error, the residual, and the combined standard uncertainty.

Given

True value = 135.0 ft
Observation = 137.2 ft
σ₁ = 0.20 ft, σ₂ = 0.20 ft

Find

Error, residual and combined uncertainty

Start with the thinking

Error is observation minus true value; the residual is the same magnitude with the sign convention of the adjustment.
Independent uncertainties combine in quadrature, never by simple addition.

Step-by-step solution

Formula
$e = o b ser v e d - t r u e$
Substituting
$e = 137.2 - 135.0 = 2.200 f t$
Relative error — 1.6296%
Formula
$u_{c} = (σ_{1}^{2} + σ_{2}^{2})$
Substituting
$u_{c} = (0.2 0^{2} + 0.2 0^{2}) = 0.2828 f t$
95% expanded uncertainty
$U = 2 u_{c} = 0.5657 f t$

Answer: Error = 2.200 ft, combined uncertainty = ±0.283 ft

Why the other options are there

±0.400 ft (uncertainties added directly)
-2.200 ft (sign reversed)

Reference: FE Reference Handbook — Probability and Statistics → Measurement Uncertainty

Self-check

Answer these without notes before moving on.

Without looking, state the relation on this page whose left-hand side is the quantity most often requested, and name every symbol in it.
Which assumption, if violated, makes the main relation of this section invalid?
Given a sample of measurements from a construction or materials process, what is the first quantity you would compute, and why that one first?
Which unit conversion in this subject most often produces a wrong answer choice, and what is its numerical factor?
Rework Example 1 above from the givens alone, without reading the solution lines.

Chapter summary

Measurement Uncertainty contains 5 relations; you must be able to find this page in under 15 seconds.
Exam style: one distribution or one counting rule, then a single probability or interval.
Unit rule: probabilities are dimensionless and must land in [0, 1].
Work the 10 examples until the solution path, not the answer, is automatic.

Common traps in this section

probabilities are dimensionless and must land in [0, 1]
Answering the intermediate quantity instead of the quantity requested.
Rounding intermediate values before the final step.
Using a relation from an adjacent handbook section that shares a symbol.
Skipping the sketch — most lost points on this page start with a misread geometry.