What is Measurand, error, uncertainty? — Physics

Definition 1.15 University Physics — Year 1 · Chapter 1 — Units, Dimensions, and Measurement

The measurand is the quantity one intends to measure. Repeating a measurement gives scattered values: the measurement is variable. The error of one result is its unknown difference to the true value; the standard uncertainty $u(x)$ of a result $x$ is the standard deviation of the values one could reasonably attribute to the measurand — a quantity one can evaluate. Its sources: the operator (reaction time, parallax), the environment (temperature, vibrations), the instrument (resolution, calibration), the method (a model that is only approximately true).

Examples

Example 1.19 (The density of a cylinder)

A cylinder: $m = 47.32\,\mathrm{g}$ ( $u = 0.01\,\mathrm{g}$ ), $d = 12.00\,\mathrm{mm}$ ( $u = 0.02\,\mathrm{mm}$ ), $h = 50.2\,\mathrm{mm}$ ( $u = 0.1\,\mathrm{mm}$ ). $\rho = 4m/(\pi d^2 h)$ , so the relative uncertainty squared adds $(0.01/47.32)^2 = 4.5 \times 10^{-8}$ , $(2 \times 0.02/12.00)^2 = 1.1 \times 10^{-5}$ and $(0.1/50.2)^2 = 4.0 \times 10^{-6}$ : the diameter dominates, because it is squared and measured to $0.17\%$ . $u(\rho)/\rho = \sqrt{1.5 \times 10^{-5}} = 0.39\%$ ; $\rho = 8334\,\mathrm{kg}/\mathrm{m}^{3}$ , so $u(\rho) = 33\,\mathrm{kg}/\mathrm{m}^{3}$ : $\rho = (8.33 \pm 0.03)\times10^3\ \mathrm{kg}/\mathrm{m}^{3}$ — brass.

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