Measurement Uncertainty and Propagation

Overview

This lecture covers measurement uncertainties, methods of determining and expressing uncertainty, significant figures, and rules for propagating uncertainty when combining measurements.

Measurement and Uncertainty

• Every measurement has uncertainty; perfect accuracy is impossible.
• Uncertainty is the range within which the true value is likely to be found.
• Always report measurements with their uncertainty using the format Q ± ΔQ, where Q is the value and ΔQ is the uncertainty.

Determining Uncertainty

• Uncertainty is the degree of doubt in a measurement and represents measurement reliability.
• Least count: smallest value measurable by an instrument; for digital devices, uncertainty equals least count; for analog, it’s usually half the least count.
• Range method: For repeated measures, uncertainty = (highest value - lowest value) ÷ 2.

Expressing Uncertainty

• Measurement is recorded as value ± uncertainty (with units).
• Decimal places in uncertainty should match the instrument’s precision.
• Significant figures are determined by uncertainty.

Types of Uncertainty

• Absolute uncertainty: Actual size of uncertainty, e.g., 20.1 ± 0.05 cm.
• Fractional uncertainty: Ratio of uncertainty to measured value, e.g., 0.05/20.1.
• Percent uncertainty: Fractional uncertainty × 100, e.g., 0.25%.

Propagating Uncertainty in Calculations

• Addition/Subtraction: Add absolute uncertainties.
• Multiplication/Division: Add relative uncertainties (uncertainty/value).
• Powers: Multiply relative uncertainty by the power.
• Constant Multiplication: Multiply both value and absolute uncertainty by the constant.

Worked Examples

• For a rectangle: perimeter = 2(L+W), uncertainties added, then multiplied by 2.
• Area: multiply uncertainties as relative fractions, then apply to area.
• Density: propagate uncertainty through both mass and volume using relative uncertainties and the rules above.

Key Terms & Definitions

• Uncertainty — Range in which the true value is likely to lie.
• Least Count — Smallest value an instrument can measure.
• Absolute Uncertainty — Numerical value of measurement doubt.
• Fractional Uncertainty — Absolute uncertainty divided by measured value.
• Percent Uncertainty — Fractional uncertainty × 100.
• Significant Figures — Digits in a measurement meaningful to its precision.

Action Items / Next Steps

• Practice determining uncertainty for different measuring devices.
• Apply propagation rules to sample measurements.
• Complete assigned concept builder problems on uncertainty calculations.