Exercise set

Uncertainty Budget and Guard Banding Exercises

Solved uncertainty budget and guard banding exercises for Type A and Type B components, RSS combination, coverage, TUR and acceptance decisions.

Branch: Engineering Physics
Content: Exercise set
Updated: Jul 03, 2026
Revision: v1.0.1 · reviewed

These exercises practise uncertainty budgets as release evidence. The goal is to show how repeatability, resolution, reference uncertainty, environment, sensitivity coefficients and decision rules combine before a measurement can support acceptance or rejection.

Assume independent components unless an exercise states otherwise. Real uncertainty statements should also define distribution type, degrees of freedom when needed, coverage factor, measurand, measurement model, traceability chain, environmental state and decision rule.

How to Use These Exercises

For each problem, identify the uncertainty component, convert it to standard uncertainty, apply any sensitivity coefficient, combine components, expand with the stated coverage factor and then decide whether the guarded result supports the engineering decision.

Release Evidence Notes

An uncertainty budget is only useful when every term has a physical source, a distribution assumption and a connection to the measurement model. A list of numbers without reference certificates, repeatability data, environmental limits, resolution model, sensitivity coefficients and decision rule is not a defensible release record. For engineering work, the budget should state whether it supports calibration, inspection, commissioning, laboratory reporting, process control or regulatory evidence.

Guard banding should be chosen before results are known. If the rule is changed after seeing a borderline value, the measurement process becomes biased toward the desired decision. Conservative guard bands reduce false acceptance risk, but they may increase false rejects, rework, retest load and supplier disputes. The right rule depends on consequence, tolerance width, process capability, measurement capability and whether the item can be safely remeasured.

When the result is close to a limit, do not hide the decision inside a rounded number. Report the measured value, corrected value if any, combined standard uncertainty, coverage factor, expanded uncertainty, guard-band rule and disposition. If any dominant component is estimated from weak evidence, the release decision should say that explicitly.

Engineering Boundary Notes

These exercises use simplified uncertainty models and mostly independent components. They do not replace a full measurement model, degrees-of-freedom analysis, correlation review, method validation, interlaboratory comparison or accredited calibration procedure. A correct RSS calculation is not enough if the measurand, sensitivity coefficients or decision rule are wrong.

Common Release Mistakes

mixing expanded and standard uncertainty values in the same RSS budget;
assuming a rectangular, normal or triangular distribution without basis;
applying a guard band only after seeing a borderline result;
quoting TUR without saying whether uncertainty is expanded or standard;
rounding away a failed guarded decision near the specification limit;
omitting correlation, environmental sensitivity or resolution when it dominates the budget.

Scenario Map

Scenario	Exercises	Primary check	Engineering decision
Component conversion	1, 2, 3, 4, 5	Type A, rectangular, resolution, certificate and sensitivity terms	Put all components on a common standard-uncertainty basis.
Budget combination	6, 7, 8, 9	RSS, expanded uncertainty, relative uncertainty and dominant contributors	Decide which terms control measurement quality.
Guard bands and TUR	10, 11, 12, 13, 14	acceptance limits, TUR, false accept screen and correlation	Decide whether a measured value can be released.
Full release decisions	15, 16, 17, 18	environmental correction, budget table, borderline result and final gate	Accept, reject, hold or improve the measurement method.

Validation Package Checklist

measurand, measurement model and units are defined;
each uncertainty component has source, distribution and standard-uncertainty conversion;
sensitivity coefficients and correlations are reviewed;
combined standard uncertainty, coverage factor and expanded uncertainty are reported separately;
guard-band rule, TUR basis and acceptance limit are set before disposition;
dominant contributors and weak-evidence assumptions are named;
final decision states accept, reject, hold, remeasure or improve the method.

Exercise 1: Type A Standard Uncertainty

Five repeated readings have sample standard deviation:

s=0.030\ \text{mm}

Estimate the standard uncertainty of the mean.

Solution

For $n=5$ :

\displaystyle u_A=\frac{s}{\sqrt{n}}=\frac{0.030}{\sqrt{5}}=0.0134\ \text{mm}

Engineering Comment

The uncertainty of the mean decreases with repeated observations only when the readings are independent and the same measurand is stable.

Plausibility Check

The result is smaller than the individual-reading scatter, as expected for an average.

Exercise 2: Rectangular Type B Component

A temperature effect is bounded by $\pm 0.60^\circ\text{C}$ with no better distribution knowledge. Convert to standard uncertainty.

Solution

For a rectangular distribution:

\displaystyle u=\frac{a}{\sqrt{3}}=\frac{0.60}{\sqrt{3}}=0.346^\circ\text{C}

Engineering Comment

Bounds are not standard uncertainties. The distribution assumption must be stated because it changes the contribution.

Plausibility Check

The standard uncertainty should be lower than the half-width, and $0.346<0.60$ .

Exercise 3: Resolution Standard Uncertainty

A digital indicator has resolution $0.01\ \text{mm}$ . Estimate standard uncertainty from rounding.

Solution

Rounding half-width:

\displaystyle a=\frac{0.01}{2}=0.005\ \text{mm}

Standard uncertainty:

\displaystyle u_{res}=\frac{0.005}{\sqrt{3}}=0.00289\ \text{mm}

Engineering Comment

Resolution is often small but not always negligible. It can dominate when tolerances are tight or when the physical process is very repeatable.

Plausibility Check

The standard uncertainty is about one third of the half-count, which is expected for a rectangular model.

Exercise 4: Certificate Expanded Uncertainty

A reference standard certificate reports:

U=0.12\ \text{kPa},\quad k=2

Find standard uncertainty.

Solution

\displaystyle u=\frac{U}{k}=\frac{0.12}{2}=0.060\ \text{kPa}

Engineering Comment

Do not combine expanded uncertainty directly with standard uncertainty terms. Convert first.

Plausibility Check

For $k=2$ , standard uncertainty is half the expanded value.

Exercise 5: Sensitivity Coefficient

A length result depends on temperature correction:

\Delta L=\alpha L\Delta T

with $\alpha L=0.012\ \text{mm}/^\circ\text{C}$ and temperature standard uncertainty $u_T=0.5^\circ\text{C}$ . Find contribution to length uncertainty.

Solution

Sensitivity coefficient:

\displaystyle c_T=0.012\ \frac{\text{mm}}{^\circ\text{C}}

Contribution:

u_L=c_Tu_T=0.012(0.5)=0.006\ \text{mm}

Engineering Comment

Sensitivity coefficients convert uncertainty from input units to output units. Without them, the budget mixes incompatible quantities.

Plausibility Check

Half a degree at $0.012\ \text{mm}/^\circ\text{C}$ should contribute only a few micrometers.

Exercise 6: RSS Combined Standard Uncertainty

Three independent standard uncertainties are:

0.020,\quad 0.015,\quad 0.010\ \text{mm}

Find combined standard uncertainty.

Solution

u_c=\sqrt{0.020^2+0.015^2+0.010^2}

u_c=\sqrt{0.0004+0.000225+0.0001}=0.0269\ \text{mm}

Engineering Comment

RSS combination means the largest terms dominate. Adding many tiny terms will not matter if one component is much larger.

Plausibility Check

The result is larger than the largest component, but far less than the arithmetic sum $0.045\ \text{mm}$ .

Exercise 7: Expanded Uncertainty

Using $u_c=0.0269\ \text{mm}$ and $k=2$ , find expanded uncertainty.

Solution

U=ku_c=2(0.0269)=0.0538\ \text{mm}

Engineering Comment

The coverage factor should match the reporting convention and confidence requirement. It is not a safety factor for bad data.

Plausibility Check

For $k=2$ , expanded uncertainty should be twice the combined standard uncertainty.

Exercise 8: Relative Expanded Uncertainty

A flow measurement is:

Q=80.0\ \text{L/min},\quad U=1.6\ \text{L/min}

Find relative expanded uncertainty.

Solution

\displaystyle U_r=100\frac{1.6}{80.0}=2.0\%

Engineering Comment

Relative uncertainty is useful for comparing methods across ranges, but absolute uncertainty still matters near limits.

Plausibility Check

$1.6$ is one fiftieth of $80$ , so the result is $2\%$ .

Exercise 9: Dominant Contributor

An uncertainty budget has standard components:

0.040,\quad 0.012,\quad 0.010,\quad 0.006\ \text{mm}

Which component dominates variance?

Solution

Variances:

0.040^2=0.0016

The other variances are $0.000144$ , $0.000100$ and $0.000036$ . The $0.040\ \text{mm}$ component dominates.

Engineering Comment

Improve the dominant component first. Reducing small contributors may look tidy while barely changing the release uncertainty.

Plausibility Check

The largest standard uncertainty is more than three times the second largest, so it should dominate variance.

Exercise 10: Simple Guard Band

A specification limit is:

x\le 10.00\ \text{mm}

Expanded uncertainty is $U=0.06\ \text{mm}$ . Use a conservative acceptance limit:

x_{accept}\le 10.00-U

Solution

x_{accept}\le 10.00-0.06=9.94\ \text{mm}

Engineering Comment

Guard banding reduces false acceptance risk but increases false rejection or hold decisions.

Plausibility Check

The guarded acceptance limit must be inside the specification limit, and $9.94<10.00$ .

Exercise 11: Guarded Acceptance Decision

A measured value is $9.96\ \text{mm}$ for the limit in Exercise 10. Does it pass the conservative guard band?

Solution

The guarded acceptance limit is:

9.94\ \text{mm}

Since:

9.96>9.94

the result does not pass guarded acceptance, even though it is inside the nominal specification.

Engineering Comment

This is a classic hold condition. The part may be conforming, but the measurement evidence is not strong enough for conservative acceptance.

Plausibility Check

The value is close to the limit, so a guard-band failure is expected.

Exercise 12: Test Uncertainty Ratio

A tolerance half-width is $0.50\ \text{mm}$ and expanded measurement uncertainty is $0.10\ \text{mm}$ . Find TUR.

Solution

\displaystyle TUR=\frac{0.50}{0.10}=5

Engineering Comment

TUR does not replace a complete decision rule, but it is a useful capability screen.

Plausibility Check

The tolerance half-width is five times the expanded uncertainty.

Exercise 13: Borderline False Acceptance Screen

A part limit is $x\le 20.0\ \text{mm}$ . Measured value is $19.97\ \text{mm}$ and $U=0.08\ \text{mm}$ . Is nominal acceptance strong?

Solution

Distance to limit:

20.00-19.97=0.03\ \text{mm}

Since:

0.03<U=0.08

the nominal pass is weak and should be held or guard-banded.

Engineering Comment

The result is inside specification but too close to the limit relative to uncertainty.

Plausibility Check

The uncertainty interval can cross the limit, so a cautious decision is reasonable.

Exercise 14: Correlated Components Warning

Two uncertainty terms each equal $0.03\ \text{mm}$ . If they are fully correlated in the same direction, what is their combined contribution?

Solution

For fully correlated same-direction terms, add linearly:

u=0.03+0.03=0.06\ \text{mm}

If independent, RSS would give:

\sqrt{0.03^2+0.03^2}=0.042\ \text{mm}

Engineering Comment

Correlation matters when contributors share a source, such as the same temperature measurement, reference standard or correction model.

Plausibility Check

The correlated result is larger than the independent RSS result, as expected.

Exercise 15: Environmental Correction Uncertainty

A temperature correction is $-0.030\ \text{mm}$ . Its standard uncertainty is $0.012\ \text{mm}$ . A measured value is $50.080\ \text{mm}$ . Apply the correction.

Solution

Corrected value:

x_c=50.080-0.030=50.050\ \text{mm}

The uncertainty component remains:

u=0.012\ \text{mm}

Engineering Comment

Correction and uncertainty are different. Applying the correction does not remove uncertainty in the correction.

Plausibility Check

A negative correction reduces the reported value, while the uncertainty remains a positive magnitude.

Exercise 16: Budget Table Combination

A measurement has standard uncertainty components:

Source	Standard uncertainty
Repeatability	$0.018\ \text{mm}$
Resolution	$0.003\ \text{mm}$
Reference	$0.010\ \text{mm}$
Temperature	$0.012\ \text{mm}$

Find $u_c$ .

Solution

u_c=\sqrt{0.018^2+0.003^2+0.010^2+0.012^2}

u_c=\sqrt{0.000324+0.000009+0.000100+0.000144}=0.0240\ \text{mm}

Engineering Comment

Repeatability is the largest term, but temperature and reference are not negligible.

Plausibility Check

The result is a little larger than $0.018\ \text{mm}$ and well below the sum, which is expected.

Exercise 17: Acceptance with Expanded Uncertainty

A lower specification limit is $5.00\ \text{mm}$ . Measured value is $5.09\ \text{mm}$ and $U=0.06\ \text{mm}$ . Use conservative guard banding.

Solution

For a lower limit, guarded acceptance requires:

x\ge 5.00+U

x\ge 5.06\ \text{mm}

Since:

5.09\ge 5.06

the result passes guarded acceptance.

Engineering Comment

Guard band direction depends on whether the limit is upper, lower or two-sided.

Plausibility Check

The measured value is farther from the lower limit than the uncertainty band, so acceptance is consistent.

Exercise 18: Uncertainty Release Gate

A measurement method must support a $\pm 0.20\ \text{mm}$ tolerance. Evidence shows:

Check	Result	Gate
Combined standard uncertainty	$0.055\ \text{mm}$	$\le 0.050\ \text{mm}$
Coverage factor used	$2$	documented
TUR	$1.8$	$\ge 4.0$
Dominant contributor known	yes	yes
Guard-band rule	missing	required

Can the method be released?

Solution

Combined standard uncertainty fails:

0.055>0.050

TUR fails:

1.8<4.0

The guard-band rule is missing. The method should not be released for final acceptance.

Engineering Comment

The next action is not only more calculation. The method needs reduced uncertainty, a defined decision rule, or a wider tolerance assignment.

Plausibility Check

Multiple capability and decision-rule gates fail, so hold is the only defensible result.

REF