Uncertainties

Uncertainty in measurements indicates the range of possible values for a given quantity. There are several methods to calculate uncertainties in experimental measurements:

  1. Absolute Uncertainty

    • This is the uncertainty of a measurement expressed in the same units as the measurement itself. It is often given as ± some value.

    • Example: If a length is measured as 10.0 ± 0.2 cm, the absolute uncertainty is 0.2 cm.

  2. Relative Uncertainty

    • This is the ratio of the absolute uncertainty to the measured value, often expressed as a percentage.

    • Formula: Relative Uncertainty = (Absolute Uncertainty / Measured Value) × 100%

    • Example: For the measure 10.0 ± 0.2 cm, the relative uncertainty is (0.2 / 10.0) × 100% = 2%.

  3. Propagation of Uncertainty

    • This method calculates the uncertainty in a result derived from multiple measurements. The propagation rules depend on whether measurements are added, subtracted, multiplied, or divided.

    • Example: If you measure a length (L) of 10.0 ± 0.2 cm and a width (W) of 5.0 ± 0.1 cm to find the area (A = L × W), the total uncertainty is calculated using:

      • Absolute uncertainty in area = A × √((ΔL/L)² + (ΔW/W)²), where ΔL and ΔW are the uncertainties in length and width.

  4. Combined Uncertainty

    • This involves combining different sources of uncertainty (systematic and random) using statistical methods to obtain a single uncertainty value.

Example Question Based on an Experiment:

An experiment involves measuring the diameter of a cylindrical object. The measured diameter is 20.0 ± 0.3 mm.

If you need to calculate the volume of the cylinder, you will need to use the formula V = πr²h, where r is the radius and h is the height of the cylinder (measured to be 100 ± 1 mm). Calculate the absolute and relative uncertainties in the volume of the cylinder including the propagation of uncertainty from both diameter and height measurements.