Problem — Significant figures at the supermarket

Problem

Significant figures at the supermarket. On the label of a packet of pasta you read “net weight $500\;\text{g}\,\pm\,1\%$”. (a) How many significant figures does the value have? (b) Between which extremes can the real mass range? (c) If you buy 6 packets, what is the absolute uncertainty on the total mass? And the relative one? (d) Why does the relative uncertainty decrease?

Solution

Step 1 — Significant figures (a). The value $500$ read with an uncertainty of $1\%$ ($\approx 5\;\text{g}$) already has the units digit uncertain: so the significant digits are $5$, $0$, $0$, i.e.

$\ev{3\ \text{significant figures}}$

Step 2 — Range of variation (b). The absolute uncertainty is $\Delta m = 1\%\cdot 500\;\text{g} = 5\;\text{g}$. The real mass can thus vary between

$\ev{m \in [\,500-5;\ 500+5\,] = [\,495;\ 505\,]\;\text{g}}$

Step 3 — 6 packets, absolute uncertainty (c). The nominal total mass is $M = 6\cdot 500 = 3000\;\text{g}$. The uncertainties of the six packets are independent and random (one packet may be over, another under): they do not add linearly but in quadrature:

$\Delta M = \sqrt{\Delta m_1^2 + \dots + \Delta m_6^2} = \sqrt{6}\cdot 5\;\text{g} = 12{,}25\;\text{g} \approx 12{,}2\;\text{g}$

The relative uncertainty on the total is

$\frac{\Delta M}{M} = \frac{12{,}2}{3000} \approx 0{,}41\%$

(If instead all the packets were off in the same direction — a systematic error — the uncertainty would add linearly, $6\cdot 5 = 30\;\text{g}$, i.e. again $1\%$.)

Step 4 — Why the relative uncertainty decreases (d). With $N$ identical packets, the absolute uncertainty grows as $\sqrt{N}$ ($\Delta M = \sqrt{N}\,\Delta m$) while the total mass grows as $N$ ($M = N m$). The ratio therefore decreases:

$\frac{\Delta M}{M} = \frac{\sqrt{N}\,\Delta m}{N m} = \frac{1}{\sqrt{N}}\cdot\frac{\Delta m}{m}$

$\ev{\dfrac{\Delta M}{M} = \dfrac{1}{\sqrt{N}}\,\dfrac{\Delta m}{m}}$

It is the same reason why averaging several measurements reduces the random error by a factor $1/\sqrt{N}$: independent errors partly cancel out.

Linked notes

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