Understanding Uncertainty in Measurement
Uncertainty in a scientific measurement quantifies the inherent doubt about the exact true value. It arises from limitations of measuring instruments, environmental conditions, or human factors, and it is crucial for communicating the reliability and precision of experimental data. Expressing uncertainty demonstrates a thorough understanding of experimental limitations, distinguishing a raw reading from a scientifically rigorous result.
Absolute vs. Relative Uncertainty
There are two primary ways to express uncertainty: absolute and relative. Absolute uncertainty is given in the same units as the measurement itself (e.g., 20.0 ± 0.1 cm), providing a direct range. Relative uncertainty, however, expresses the uncertainty as a fraction or percentage of the measured value (e.g., 0.1/20.0 = 0.005 or 0.5%). Relative uncertainty is particularly useful for comparing the precision of measurements of vastly different magnitudes.
Reporting with Significant Figures
When reporting a measurement with its uncertainty, the number of significant figures is paramount. The uncertainty itself should generally be stated to one significant figure. Subsequently, the measured value must be rounded so that its last significant figure aligns with the decimal place of the uncertainty. For example, a measurement of 12.345 cm with an uncertainty of ±0.02 cm would be correctly reported as 12.35 ± 0.02 cm, ensuring consistency in precision.
Propagation of Uncertainty in Calculations
When multiple measured quantities, each with its own uncertainty, are combined in a calculation (e.g., addition, subtraction, multiplication, division), their individual uncertainties must be 'propagated' to determine the overall uncertainty of the final result. Specific mathematical rules apply for combining these uncertainties, ensuring that the computed outcome accurately reflects the cumulative doubt from all component measurements.