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I understand that a common convention is to report the uncertainty of an analogue instrument as plus or minus half of the smallest scale division.

By this logic, a measuring cylinder with $\pu{0.5 mL}$ graduations has an uncertainty of $\pu{\pm 0.25 mL}$.

Because the measurement should also be reported to the same number of decimal places as the uncertainty, however, this suggests that any measurements with this measuring cylinder should be reported to two decimal places (e.g., $\pu{(17.85 \pm 0.25) mL}$).

Would this not be an example of false precision? Reporting a volume measurement with 2 decimal places seems excessive for a measuring cylinder. Would it not make more sense to report an uncertainty of $\pu{0.3 mL}$ (0.25 rounded to 1 d.p.) and a measurement with only one decimal place (e.g. $\pu{(17.9 \pm 0.3) mL}$)?

Before you answer, I understand that the rules for uncertainty are not set in stone and depend on the experimenter's judgement to some extent. However, I'd love to know which approach would be more conventional in this situation. Thanks so much!

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    $\begingroup$ Related: chemistry.stackexchange.com/a/37788/103369 $\endgroup$
    – Loong
    Jan 15, 2022 at 15:09
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    $\begingroup$ It depends on the quality of the graduate cylinder. It should say something like 25:0.5mL ±0.4 mL on the top of the cylinder (and a temperature). If it doesn't, all bets are off. $\endgroup$
    – Karsten
    Jan 15, 2022 at 15:25
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    $\begingroup$ Welcome to the site. Note, chemical information may be advantageously formatted using on ChemSE with mhchem. Take moment to familiarize with this. You are encouraged to use it in the body of questions, answers, and comments. Because it is something special not all web browsers understand well, do not use it in the title of questions or answers. Here especially: the \pu{} syntax, which allows you to type recordings without unwanted line breaks between the floating number, and the unit of the property. $\endgroup$
    – Buttonwood
    Jan 15, 2022 at 17:02

1 Answer 1

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common convention is to report the uncertainty of an analogue instrument as plus or minus half of the smallest scale division

Yes, this is merely a poor convention propagated by general chemistry texts.

Actually, this does not reflect analytical measurement standard deviation, and this is not the way volume uncertainty should be expressed. The correct method to determine uncertainty is called volumetric calibration. Take for example a $\pu{100 mL}$ cylinder. You would measure, for example, $\pu{60 mL}$ of water or any other solvent of choice. The liquid would be transferred to a weighing vessel and the exact mass would be measured. The process would be repeated at least 10 times. The volume would be calculated based on the density of the liquid. The volume delivered by the graduated cylinder would be calculated as the average and standard deviation.

If we look up industry standards, such as ASTM (American Society of Testing and Materials, e.g., ASTM E1272-02(2019), reference), it says that a $\pu{100 mL}$ class A is supposed to have a tolerance (std. dev.) of $\pu{\pm 0.5 mL}$ but a different $\pu{100 mL}$ class B cylinder will have a tolerance of $\pu{\pm 1 mL}$. Same graduations, different prices! Class A cylinder is way more expensive. It is all about how precisely, the cylinder was manufactured under controlled conditions.

One can immediately see fallacies associated with this general chem rule.

A reader brought another useful point about TC (to contain) or TD (to deliver) glassware. ASTM mentions both TC and TD type cylinders and both types exist. Still in reality one might use a TC graduated cylinder to deliver. In order to determine the actual volume delivered by a TC/TD graduated cylinder, calibration is the way to go to determine how much actual volume is being delivered. To be pedantic, if a higher capacity balance is available, the cylinder could be measured directly on the balance (true TC measurement).

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  • $\begingroup$ Measuring cylinders are calibrated ‘to contain’ (In); i.e. the contained (not the delivered) quantity of liquid corresponds to the capacity printed on the measuring cylinders. Thus, the suggested method of transferring the liquid to a weighing vessel and measuring the exact mass would not work as intended. $\endgroup$
    – Loong
    Jan 15, 2022 at 16:28
  • $\begingroup$ I added a paragraph on TC/TD glassware. This was a useful point. $\endgroup$
    – AChem
    Jan 15, 2022 at 16:38
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    $\begingroup$ @M.Farooq Hint: for chemistry.se, mhchem has a nifty \pu{} syntax. An input like $\pu{(13 \pm 0.5) mL}$ coalesces into $\pu{(13 \pm 0.5) mL}$ with much less danger that a line feed will separate a floating number from its following units. Have a look at the examples in the lower section of the documenting page. (No, it is not -- and indeed does not intend to be -- a replacement for siunitx for LaTeX2e.) $\endgroup$
    – Buttonwood
    Jan 15, 2022 at 17:11

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