Overview
This lecture explains the concept of measurement uncertainty, how to report it, methods for determining it, and rules for calculating with uncertainty in derived quantities.
Measurement and Uncertainty
- No measurement can be assumed perfectly accurate or to be the true value.
- Uncertainty is always present and must be reported with every measurement.
- Uncertainty refers to the range of possible values where the true value lies.
- Measured value is reported as Q ± ΔQ, where Q is the measurement and ΔQ is the uncertainty.
- Uncertainty expresses confidence in the measurement and is essential for meaningful data.
Determining Uncertainty
- Uncertainty can be estimated using the least count or the standard deviation (range method).
- Least count is the smallest value a measuring device can reliably indicate (resolution).
- For analog devices, uncertainty is usually half the least count.
- For digital devices, uncertainty is equal to the least count.
- Record measurements as: quantity ± uncertainty, including units.
Significant Figures and Uncertainty
- Uncertainty determines the number of significant figures in a related measurement.
- Measurements should be reported with the same decimal places as the uncertainty.
Types of Uncertainty
- Absolute Uncertainty: The actual size of the uncertainty (e.g., ±0.05 cm).
- Relative Uncertainty: Fraction of uncertainty to the measured value (ΔQ/Q).
- Percent Uncertainty: Relative uncertainty expressed as a percentage.
Calculating with Uncertainty (Error Propagation)
- Addition/Subtraction: Add absolute uncertainties (ΔC = ΔA + ΔB).
- Multiplication/Division: Add relative uncertainties (ΔC/C = ΔA/A + ΔB/B).
- Powers: Multiply relative uncertainty by the power (ΔQ/Q = n × (ΔA/A)).
- Constants: Multiply both the value and absolute uncertainty by the constant.
Example Problems
- For perimeter: Add the lengths and widths, add uncertainties, then multiply by constant if needed.
- For area: Multiply lengths, add relative uncertainties, then multiply by value to get absolute uncertainty.
- For volume (cube): Raise the side to the power, multiply relative uncertainty by 3.
- For density: Divide mass by volume, then sum relative uncertainties and multiply by calculated density value.
Key Terms & Definitions
- Uncertainty — Range within which the true value may be found.
- Least Count — Smallest value measurable by a device.
- Absolute Uncertainty — Actual error value in the same unit as the measurement.
- Relative Uncertainty — Ratio of absolute uncertainty to measured value.
- Percent Uncertainty — Relative uncertainty times 100.
- Significant Figures — Digits in a number that are reliable and necessary.
Action Items / Next Steps
- Practice determining and expressing uncertainty using both least count and range methods.
- Complete assigned concept builders covering rectangle perimeter, area, and density with uncertainty.
- Read up on significant figures and their relationship to measurement uncertainty.