# Measuring instruments and significant figures

To start with, I am familiar with the rules that we have for sig-figs. I have two chemistry books by different authors (One by McMurry and the other one by Nivaldo J Tro).

I was doing some casual readig, I am familiar with sig-figs (or that's what I thought).

According to the first book, let's say I measure the thickness of the cover of a hardbound version of a book with a ruler that is accurate up to the nearest millimeter, and I find that it is 3±1 mm thick, or 0.3±0.1 cm thick. The book says I can claim just one significant figures when I use a ruler because it is accurate up to 0.1 cm, or 1 mm. The book clearly wants to say that I can't claim more than one sig fig if I use the ruler. For example, I can't say that the cover is about 3.2 mm thick, or 3.4 mm thick because the limitations of the ruler, it is not accurate up to 0.1 cm, or 1 mm. On the other hand, if I use an instrument that is accurate up to 0.01 mm (Vernier caliper, for example), I find that the book is 3.25±0.01 mm thick. It means I can claim three sig figs if I use a Vernier caliper because it is accurate up to 0.01 mm. Out of these three sig figs (3 2 5), 3 and 2 are certain digits and 5 is uncertain.

The other book, however, says that if I use a ruler (accurate up to the nearest mm) to measure the cover of the book, I can be sure the the cover is at least 3 mm thick, and then I can mentally divide the little space for a mm on the ruler into 10 parts and I can extrapolate that the cover is about 3.2 mm thick (or 3.3 mm, whatever). Now I can say that the cover is about 3.2 mm thick, out of which 3 is the certain digit and 2 is uncertain. That's what the book says.

In other words, there are two sig-figs according to the 2nd book, when we used a ruler to measure the thickness of the cover of a hardbound book. On the other hand, the first book says that we can claim only one sig-fig if we use the same ruler that is accurate up to the nearest mm.

I am confused, which one is correct? From what I understand about significant figures, I would think that the first book is correct in saying that we can't have two sig-figs if we use a ruler that can accurately measure up to the nearest mm. But the second book says that we can have two sig-figs.

Please help me with it

• @Poutnik In large samples, where error of standard deviation estimation is small, there can be overlap of 2 digits. Isn't it why we should say it is 3±1 mm, and not 3.2±0.1 mm? Unless I did not understand what you said. Can you please elaborate? – π times e May 7 '20 at 11:09
• Is this a site about measuring books now? – Mithoron May 7 '20 at 16:58
• @Mithoron not books, just covers – π times e May 7 '20 at 18:52
• If you're an old timer like me you used a slide rule and interpolating between lines was common. You also have to (well at least you should...) interpolate when using a burette. You'd also run into that problem when reading a mercury barometer which would probably have a vernier scale which would help greatly. So the second book is right. – MaxW May 23 '20 at 19:04

## 1 Answer

Last significant digit is for the most of common measurements the first unsure digit. The eye+ruler can estimate $$\pu{0.1 mm}$$ , so it is $$\pu{ 3.2 +- 0.1 mm}$$, not $$\pu{3 +- 1 mm}$$.

There is big difference between e.g some 1st generation digital voltmeter, displaying integer values of voltage, and analogue voltmeter with integer voltage marks, where you can estimate the position between marks. The 1st book speaks about the former, the 2nd book about the latter. The common sense must tell you you know better than the thickness is within $$\pu{ 2-4 mm}$$.

In large samples, where error of standard deviation estimation is small, there can be overlap of 2 digits. That applies e.g. to massive and very accurate parallel measumurent of physical values. Like $$1.12345678 \pm 0.00000034$$, by convention written as $$1.12345678(34)$$.

• Thanks. So in some situations, I need to see if I can extrapolate another digit if it gives me more useful (accurate) information? – π times e May 7 '20 at 18:51
• It is done always when discrete marks are plotted along continuous scale. – Poutnik May 7 '20 at 18:58