A chemist weighs a crucible (±0.0002 g), adds a sample, then weighs again (±0.0002 g), and subtracts to find the sample mass. What is the uncertainty of the mass difference?
A±0.0004 g — absolute uncertainties simply add
B±0.0002 g — only one weighing contributes to the difference
C±0.00028 g — absolute uncertainties add in quadrature
D±0.0001 g — the average of the two uncertainties
For subtraction, absolute uncertainties add in quadrature: √(0.0002² + 0.0002²) = √(0.00000004 + 0.00000004) = √0.00000008 ≈ 0.00028 g. The common mistake (option A) is to add uncertainties directly; this overestimates the combined uncertainty because the errors are statistically independent and partially cancel on average. The quadrature rule is not a convention — it follows from the mathematical behavior of independent random variables.
Question 2 Multiple Choice
A final concentration C is computed as C = n/V, where n has a relative uncertainty of 1.0% and V has a relative uncertainty of 0.1%. What is the approximate combined relative uncertainty of C?
A±1.1% — relative uncertainties add directly for multiplication and division
B±0.9% — subtract the smaller uncertainty from the larger
C±0.1% — only the smaller source matters since it is more precise
D±1.005% — relative uncertainties add in quadrature, and the larger source dominates
For multiplication and division, relative uncertainties add in quadrature: √(1.0² + 0.1²) = √(1.0 + 0.01) ≈ √1.01 ≈ 1.005%. The 0.1% contribution barely changes the total because quadrature addition is dominated by the largest term. This is the key practical insight: when one uncertainty source is much larger than others, improving the smaller sources is essentially wasted effort. Option A (direct addition) overestimates; option C underestimates by ignoring the dominant source.
Question 3 True / False
When a ±1% uncertainty source is combined with a ±0.1% source, the combined uncertainty is approximately ±1.1%, since both contributions are meaningful.
TTrue
FFalse
Answer: False
Quadrature addition gives √(1.0² + 0.1²) ≈ 1.005%, not 1.1%. The ±0.1% contribution adds only 0.5% to the total, not 10%. This is the practical message of quadrature rules: smaller sources of uncertainty become negligible once a larger source dominates. Improving the 0.1% step would have virtually no effect on the combined uncertainty. The effort should go to reducing the 1.0% source first.
Question 4 True / False
An uncertainty budget that identifies one measurement step as the dominant contributor reveals where effort to improve the method will be most effective.
TTrue
FFalse
Answer: True
This is precisely the diagnostic value of the uncertainty budget. Because uncertainties combine in quadrature, the largest source dominates the total — and reducing smaller sources while the dominant source remains unchanged barely improves the overall uncertainty. If the volumetric step contributes ±0.5% and the weighing step contributes ±0.01%, investing in a more precise balance is wasted until the volumetric step is addressed. The budget turns uncertainty propagation from a reporting requirement into a rational guide for method improvement.
Question 5 Short Answer
Why do uncertainties combine in quadrature (root-sum-of-squares) rather than adding directly? What practical consequence does this have for identifying the limiting step in an analysis?
Think about your answer, then reveal below.
Model answer: Quadrature addition follows from the statistical independence of measurement errors — if errors in two measurements are uncorrelated, the variance of their sum or difference is the sum of their variances, and standard deviation (uncertainty) is the square root of variance. Because each term is squared before adding, the largest uncertainty dominates the total; smaller uncertainties contribute negligibly once a larger source is present. The practical consequence is that the dominant uncertainty source determines overall quality, so improvement efforts should focus there — improving other steps has diminishing returns.
This is why the uncertainty budget is a tool for rational resource allocation, not just bookkeeping. A chemist who improves balance precision from ±0.0002 g to ±0.00001 g while still using a ±0.5% volumetric flask will see essentially no improvement in the final uncertainty — the flask dominates. Only by identifying and attacking the largest contributor can overall precision be meaningfully improved.