PHS 104A | MEASUREMENT UNCERTAINTY & GRAPH ANALYSIS | 2026–2027
PHS 104A Measurement
Uncertainty & Graph Analysis
Comprehensive Review 2026–2027
110 Original Practice Questions • Worked Solutions • Answer Key
Independent educational practice resource; not an official PHS 104A assessment. Institution-specific
syllabus alignment is not verified. Numerical uncertainty problems state the propagation convention
used. Calculations are based on idealized illustrative datasets.
Formula & Methods Reference
• Mean: ̄ x = Σx/n. Sample standard deviation: s = √[Σ(x−x̄)²/(n−1)]. Standard error: SEM = s/√n.
• Percent error: |measured−accepted|/|accepted| × 100%. Percentage uncertainty: (absolute
uncertainty/|value|) × 100%.
• Slope: m = (y₂−y₁)/(x₂−x₁); intercept: b = y−mx; residual: observed y−predicted y.
• Maximum-error convention: for addition/subtraction, absolute uncertainties add; for
multiplication/division, fractional uncertainties add. This is distinct from root-sum-square propagation
for independent random uncertainties.
• Graph quality: labeled axes and units, sensible scales, error-bar definitions, appropriate fit and
residual checks.
Independent practice resource • Page 1
,PHS 104A | MEASUREMENT UNCERTAINTY & GRAPH ANALYSIS | 2026–2027
Uncertainty and percent uncertainty
Question 1. A measured length is (12.0 ± 0.2) cm. What is its percentage uncertainty (two
decimals)?
A. 0.20%
B. 6000.00%
C. 3.33%
D. 1.67%
Correct Answer: D. 1.67%
Worked Solution / Detailed Rationale: Percentage uncertainty = (absolute uncertainty / measured
value) × 100 = (0.2/12.0) × 100 = 1.67%.
Why the Other Options Are Incorrect: A (0.20%) does not match the required calculation or
measurement principle. B (6000.00%) does not match the required calculation or measurement
principle. C (3.33%) does not match the required calculation or measurement principle.
Question 2. A measured length is (15.0 ± 0.3) cm. What is its percentage uncertainty (two
decimals)?
A. 5000.00%
B. 4.00%
C. 2.00%
D. 0.30%
Correct Answer: C. 2.00%
Worked Solution / Detailed Rationale: Percentage uncertainty = (absolute uncertainty / measured
value) × 100 = (0.3/15.0) × 100 = 2.00%.
Why the Other Options Are Incorrect: A (5000.00%) does not match the required calculation or
measurement principle. B (4.00%) does not match the required calculation or measurement
principle. D (0.30%) does not match the required calculation or measurement principle.
Question 3. A measured length is (18.0 ± 0.4) cm. What is its percentage uncertainty (two
decimals)?
A. 4.44%
B. 2.22%
C. 0.40%
D. 4500.00%
Independent practice resource • Page 2
,PHS 104A | MEASUREMENT UNCERTAINTY & GRAPH ANALYSIS | 2026–2027
Correct Answer: B. 2.22%
Worked Solution / Detailed Rationale: Percentage uncertainty = (absolute uncertainty / measured
value) × 100 = (0.4/18.0) × 100 = 2.22%.
Why the Other Options Are Incorrect: A (4.44%) does not match the required calculation or
measurement principle. C (0.40%) does not match the required calculation or measurement
principle. D (4500.00%) does not match the required calculation or measurement principle.
Question 4. A measured length is (21.0 ± 0.5) cm. What is its percentage uncertainty (two
decimals)?
A. 2.38%
B. 0.50%
C. 4200.00%
D. 4.76%
Correct Answer: A. 2.38%
Worked Solution / Detailed Rationale: Percentage uncertainty = (absolute uncertainty / measured
value) × 100 = (0.5/21.0) × 100 = 2.38%.
Why the Other Options Are Incorrect: B (0.50%) does not match the required calculation or
measurement principle. C (4200.00%) does not match the required calculation or measurement
principle. D (4.76%) does not match the required calculation or measurement principle.
Question 5. A measured length is (24.0 ± 0.6) cm. What is its percentage uncertainty (two
decimals)?
A. 0.60%
B. 4000.00%
C. 5.00%
D. 2.50%
Correct Answer: D. 2.50%
Worked Solution / Detailed Rationale: Percentage uncertainty = (absolute uncertainty / measured
value) × 100 = (0.6/24.0) × 100 = 2.50%.
Why the Other Options Are Incorrect: A (0.60%) does not match the required calculation or
measurement principle. B (4000.00%) does not match the required calculation or measurement
principle. C (5.00%) does not match the required calculation or measurement principle.
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, PHS 104A | MEASUREMENT UNCERTAINTY & GRAPH ANALYSIS | 2026–2027
Question 6. A measured length is (27.0 ± 0.2) cm. What is its percentage uncertainty (two
decimals)?
A. 13500.00%
B. 1.48%
C. 0.74%
D. 0.20%
Correct Answer: C. 0.74%
Worked Solution / Detailed Rationale: Percentage uncertainty = (absolute uncertainty / measured
value) × 100 = (0.2/27.0) × 100 = 0.74%.
Why the Other Options Are Incorrect: A (13500.00%) does not match the required calculation or
measurement principle. B (1.48%) does not match the required calculation or measurement
principle. D (0.20%) does not match the required calculation or measurement principle.
Question 7. A measured length is (30.0 ± 0.3) cm. What is its percentage uncertainty (two
decimals)?
A. 2.00%
B. 1.00%
C. 0.30%
D. 10000.00%
Correct Answer: B. 1.00%
Worked Solution / Detailed Rationale: Percentage uncertainty = (absolute uncertainty / measured
value) × 100 = (0.3/30.0) × 100 = 1.00%.
Why the Other Options Are Incorrect: A (2.00%) does not match the required calculation or
measurement principle. C (0.30%) does not match the required calculation or measurement
principle. D (10000.00%) does not match the required calculation or measurement principle.
Question 8. A measured length is (33.0 ± 0.4) cm. What is its percentage uncertainty (two
decimals)?
A. 1.21%
B. 0.40%
C. 8250.00%
D. 2.42%
Correct Answer: A. 1.21%
Independent practice resource • Page 4
PHS 104A Measurement
Uncertainty & Graph Analysis
Comprehensive Review 2026–2027
110 Original Practice Questions • Worked Solutions • Answer Key
Independent educational practice resource; not an official PHS 104A assessment. Institution-specific
syllabus alignment is not verified. Numerical uncertainty problems state the propagation convention
used. Calculations are based on idealized illustrative datasets.
Formula & Methods Reference
• Mean: ̄ x = Σx/n. Sample standard deviation: s = √[Σ(x−x̄)²/(n−1)]. Standard error: SEM = s/√n.
• Percent error: |measured−accepted|/|accepted| × 100%. Percentage uncertainty: (absolute
uncertainty/|value|) × 100%.
• Slope: m = (y₂−y₁)/(x₂−x₁); intercept: b = y−mx; residual: observed y−predicted y.
• Maximum-error convention: for addition/subtraction, absolute uncertainties add; for
multiplication/division, fractional uncertainties add. This is distinct from root-sum-square propagation
for independent random uncertainties.
• Graph quality: labeled axes and units, sensible scales, error-bar definitions, appropriate fit and
residual checks.
Independent practice resource • Page 1
,PHS 104A | MEASUREMENT UNCERTAINTY & GRAPH ANALYSIS | 2026–2027
Uncertainty and percent uncertainty
Question 1. A measured length is (12.0 ± 0.2) cm. What is its percentage uncertainty (two
decimals)?
A. 0.20%
B. 6000.00%
C. 3.33%
D. 1.67%
Correct Answer: D. 1.67%
Worked Solution / Detailed Rationale: Percentage uncertainty = (absolute uncertainty / measured
value) × 100 = (0.2/12.0) × 100 = 1.67%.
Why the Other Options Are Incorrect: A (0.20%) does not match the required calculation or
measurement principle. B (6000.00%) does not match the required calculation or measurement
principle. C (3.33%) does not match the required calculation or measurement principle.
Question 2. A measured length is (15.0 ± 0.3) cm. What is its percentage uncertainty (two
decimals)?
A. 5000.00%
B. 4.00%
C. 2.00%
D. 0.30%
Correct Answer: C. 2.00%
Worked Solution / Detailed Rationale: Percentage uncertainty = (absolute uncertainty / measured
value) × 100 = (0.3/15.0) × 100 = 2.00%.
Why the Other Options Are Incorrect: A (5000.00%) does not match the required calculation or
measurement principle. B (4.00%) does not match the required calculation or measurement
principle. D (0.30%) does not match the required calculation or measurement principle.
Question 3. A measured length is (18.0 ± 0.4) cm. What is its percentage uncertainty (two
decimals)?
A. 4.44%
B. 2.22%
C. 0.40%
D. 4500.00%
Independent practice resource • Page 2
,PHS 104A | MEASUREMENT UNCERTAINTY & GRAPH ANALYSIS | 2026–2027
Correct Answer: B. 2.22%
Worked Solution / Detailed Rationale: Percentage uncertainty = (absolute uncertainty / measured
value) × 100 = (0.4/18.0) × 100 = 2.22%.
Why the Other Options Are Incorrect: A (4.44%) does not match the required calculation or
measurement principle. C (0.40%) does not match the required calculation or measurement
principle. D (4500.00%) does not match the required calculation or measurement principle.
Question 4. A measured length is (21.0 ± 0.5) cm. What is its percentage uncertainty (two
decimals)?
A. 2.38%
B. 0.50%
C. 4200.00%
D. 4.76%
Correct Answer: A. 2.38%
Worked Solution / Detailed Rationale: Percentage uncertainty = (absolute uncertainty / measured
value) × 100 = (0.5/21.0) × 100 = 2.38%.
Why the Other Options Are Incorrect: B (0.50%) does not match the required calculation or
measurement principle. C (4200.00%) does not match the required calculation or measurement
principle. D (4.76%) does not match the required calculation or measurement principle.
Question 5. A measured length is (24.0 ± 0.6) cm. What is its percentage uncertainty (two
decimals)?
A. 0.60%
B. 4000.00%
C. 5.00%
D. 2.50%
Correct Answer: D. 2.50%
Worked Solution / Detailed Rationale: Percentage uncertainty = (absolute uncertainty / measured
value) × 100 = (0.6/24.0) × 100 = 2.50%.
Why the Other Options Are Incorrect: A (0.60%) does not match the required calculation or
measurement principle. B (4000.00%) does not match the required calculation or measurement
principle. C (5.00%) does not match the required calculation or measurement principle.
Independent practice resource • Page 3
, PHS 104A | MEASUREMENT UNCERTAINTY & GRAPH ANALYSIS | 2026–2027
Question 6. A measured length is (27.0 ± 0.2) cm. What is its percentage uncertainty (two
decimals)?
A. 13500.00%
B. 1.48%
C. 0.74%
D. 0.20%
Correct Answer: C. 0.74%
Worked Solution / Detailed Rationale: Percentage uncertainty = (absolute uncertainty / measured
value) × 100 = (0.2/27.0) × 100 = 0.74%.
Why the Other Options Are Incorrect: A (13500.00%) does not match the required calculation or
measurement principle. B (1.48%) does not match the required calculation or measurement
principle. D (0.20%) does not match the required calculation or measurement principle.
Question 7. A measured length is (30.0 ± 0.3) cm. What is its percentage uncertainty (two
decimals)?
A. 2.00%
B. 1.00%
C. 0.30%
D. 10000.00%
Correct Answer: B. 1.00%
Worked Solution / Detailed Rationale: Percentage uncertainty = (absolute uncertainty / measured
value) × 100 = (0.3/30.0) × 100 = 1.00%.
Why the Other Options Are Incorrect: A (2.00%) does not match the required calculation or
measurement principle. C (0.30%) does not match the required calculation or measurement
principle. D (10000.00%) does not match the required calculation or measurement principle.
Question 8. A measured length is (33.0 ± 0.4) cm. What is its percentage uncertainty (two
decimals)?
A. 1.21%
B. 0.40%
C. 8250.00%
D. 2.42%
Correct Answer: A. 1.21%
Independent practice resource • Page 4
