Absolute and Relative Uncertainty

               

Resolution is the smallest change an instrument can detect. For example, on a standard measuring tape with 1 cm markings, the resolution is 1 cm.

By definition, resolution is the smallest measurable increment — the tiniest difference that can affect a reading. Trying to measure a keyhole’s diameter with such a tape will yield the same result each time, even after 100 attempts, because the resolution is too low.

Examples:

🔸If the instrument steps by 1 unit, any value between 6.5 and 7.5 is read as 7.

    🔸 If it steps by 2 units, anything between 7 and 9 is read as 8.

     We discussed measurement uncertainty in another post. It should be expressed as:

                                                          (X ± ∆X) unit

     where X is the best estimate and ∆X the associated uncertainty. This means future results likely lie           between:

                                                  (X - ∆X) and (X + ∆X)

     In single measurements, the instrument’s resolution is key. Suppose a newborn measures 50 cm. If a       ruler graduated in 1 cm was used, we can report:

                                                        (50.0 ± 0.5) cm

      This clearly states an uncertainty of ±0.5 cm.

                               

      Assuming a properly calibrated instrument, this is a Type B uncertainty — not based on statistics,            but on the instrument’s resolution.

       Now for mass: if a digital scale has 10 g increments (0.01 kg), and reads 3.54 kg, then:

                                                   (3.540 ± 0.005) kg

       In X ± ∆X, ∆X is the absolute uncertainty. Relative uncertainty is:

                                          relative uncertainty = ∆X / X

      As a percentage:

                                 relative uncertainty (%) = (∆X / X) × 100

     Example: A speedometer with 2 km/h steps shows 60 km/h.

                             - Absolute uncertainty = 1 km/h
                             - Relative = (1 / 60) × 100 = 1.67%

    Relative uncertainty is unitless and allows comparisons across quantities.

    Another example: Which measurement has more relative uncertainty — length or mass of a newborn?

                         - Length: (0.5 / 50.0) × 100 = 1.00%
                         - Mass: (0.005 / 3.540) × 100 ≈ 0.14%

    Thus, the length has greater relative uncertainty.

    These definitions apply to both Type B and Type A (statistical) uncertainties.

Exercises

          1. A child’s height is measured as 80 cm. What is the uncertainty?
          2. A thermometer with 2°C steps reads 38°C. What is the uncertainty?

Answers

          1. (80 ± 0.5) cm → 0.625%
          2. (38 ± 1) °C → 2.63%