Problem — Ranking of relative uncertainties

Problem

Four experimental results: (A) $m = (52{,}3 \pm 0{,}4)\;\text{g}$; (B) $m = (103{,}1 \pm 1{,}2)\;\text{g}$; (C) $m = (9{,}87 \pm 0{,}05)\;\text{g}$; (D) $m = (1{,}034 \pm 0{,}003)\;\text{g}$. Order the relative uncertainties from smallest to largest.

Solution

Step 1 — Definition. The relative uncertainty is the ratio between the absolute uncertainty and the measured value, $\dfrac{\Delta m}{m}$; it is dimensionless and is often expressed as a percentage. What matters is not the mass value itself, but the percentage “quality” of the measurement.

Step 2 — Calculation for each case.

$\begin{aligned} \text{(A)}\quad &\frac{0{,}4}{52{,}3} = 0{,}0076 = 0{,}76\% \\ \text{(B)}\quad &\frac{1{,}2}{103{,}1} = 0{,}0116 = 1{,}16\% \\ \text{(C)}\quad &\frac{0{,}05}{9{,}87} = 0{,}0051 = 0{,}51\% \\ \text{(D)}\quad &\frac{0{,}003}{1{,}034} = 0{,}0029 = 0{,}29\% \end{aligned}$

Step 3 — Ascending order. Comparing the percentages: $0{,}29\% < 0{,}51\% < 0{,}76\% < 1{,}16\%$, i.e.

$\ev{\text{D} < \text{C} < \text{A} < \text{B}}$

Remark. Measurement B, despite having the largest mass, is the least precise in relative terms; D, the smallest mass, is the most precise. The absolute value of the mass does not determine the relative precision: it is the ratio between error and value that does.

Linked notes

Topics: Grandezze fisiche e misura Concepts: Cifre significative e incertezza Skills: Propagazione degli errori · Uso delle cifre significative