Problem — Ranking instrument uncertainty

Problem

Four students measure the same length $L$ with different instruments. (A) Calliper, $20$ times, results agreeing within $0{,}05$ mm. (B) Ruler, $5$ times, spread $1$ mm. (C) Micrometer, $50$ times, spread $0{,}01$ mm. (D) Tape measure read “by eye”, $3$ times, spread $1$ mm. Rank the estimates of the statistical uncertainty on the mean from smallest to largest (recall $\sigma/\sqrt{N}$).

Solution

Criterion. The statistical uncertainty on the mean is not the spread $\sigma$ of a single measurement, but is given by $\sigma/\sqrt{N}$: it depends both on how precise the instrument is (small $\sigma$) and on how many times the measurement was repeated (large $N$). We compute $\sigma/\sqrt{N}$ for each, taking the spread as an estimate of $\sigma$.

$\begin{aligned} \text{(A)}\ & \frac{0{,}05}{\sqrt{20}} \approx 0{,}011\ \mathrm{mm} \\ \text{(B)}\ & \frac{1}{\sqrt{5}} \approx 0{,}45\ \mathrm{mm} \\ \text{(C)}\ & \frac{0{,}01}{\sqrt{50}} \approx 0{,}0014\ \mathrm{mm} \\ \text{(D)}\ & \frac{1}{\sqrt{3}} \approx 0{,}58\ \mathrm{mm} \end{aligned}$

Ascending order. $\text{C}\,(0{,}0014) \ <\ \text{A}\,(0{,}011) \ <\ \text{B}\,(0{,}45) \ <\ \text{D}\,(0{,}58)$

The micrometer (C) wins twice: it has the smallest $\sigma$ and was repeated the most times. The calliper (A) follows. The ruler (B) and the tape measure “by eye” (D) have the same spread for a single measurement ($1$ mm), so they are distinguished only by $N$: $B$ has $N=5$ against $D$‘s $N=3$, so $B$ is slightly better. To a first approximation one writes $\text{C} < \text{A} < \text{B} \approx \text{D}$, since with equal $\sigma$ and similar $N$ the two uncertainties are comparable; the exact calculation, however, gives $\text{B} \lesssim \text{D}$.

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

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Topics: Metodo sperimentale Concepts: Cifre significative e incertezza Skills: Ragionamento di ranking · Propagazione degli errori