Question
How is the precision of a calculated result related to the precision of the measurements used in the calculation?
The result's precision is the average of the precision of the measurements used.
The result is only as precise as the most precise measurement used in the calculation.
The result is only as precise as the least precise measurement used in the calculation.
The result's precision is the average of the precision of the measurements used.
The result is only as precise as the most precise measurement used in the calculation.
The result is only as precise as the least precise measurement used in the calculation.
Answers
GPT-5 mini
The third statement is correct: the result is limited by the least precise measurement used.
Reasoning and caveats:
- Rule-of-thumb (significant-figures rules): when multiplying or dividing, the result should have the same number of significant figures as the factor with the fewest; when adding or subtracting, align decimal places and keep the least precise decimal place.
- Example (addition): 12.11 + 0.3 = 12.4 (keep 1 decimal place).
- Example (multiplication): 4.56 × 1.4 = 6.4 (keep 2 significant figures).
- More rigorously, use uncertainty propagation: combine measurement uncertainties (usually in quadrature). In many practical cases the largest uncertainty dominates the propagated uncertainty, so the least-precise input limits the output precision.
- Exceptions: averaging many independent measurements can improve precision (uncertainty falls roughly as 1/√N), and correlated errors or particular functional dependencies can change which input dominates.
Reasoning and caveats:
- Rule-of-thumb (significant-figures rules): when multiplying or dividing, the result should have the same number of significant figures as the factor with the fewest; when adding or subtracting, align decimal places and keep the least precise decimal place.
- Example (addition): 12.11 + 0.3 = 12.4 (keep 1 decimal place).
- Example (multiplication): 4.56 × 1.4 = 6.4 (keep 2 significant figures).
- More rigorously, use uncertainty propagation: combine measurement uncertainties (usually in quadrature). In many practical cases the largest uncertainty dominates the propagated uncertainty, so the least-precise input limits the output precision.
- Exceptions: averaging many independent measurements can improve precision (uncertainty falls roughly as 1/√N), and correlated errors or particular functional dependencies can change which input dominates.