Using significant figures when measuring with an instrument with marks other than powers of ten

Question

If an instrument does have markings that are powers of ten, then finding the precision of the instrument is easy. For example, if a graduated cylinder had markings for every $\pu{0.1 mL}$, or every $\pu{mL}$, or every $\pu{L}$, then it is easy to tell that those measurements will be to the hundredth mL, to the tenth of a $\pu{mL}$, and to the tenth of a $\pu{L}$, respectively. These would be valid measurements with the correct number of significant figures.

See this image, for example:

These markings are to the tenths and ones. This is simple.

However, what if that graduated cylinder went up by something like $\pu{5 mL}$? Would its precision be to the tens or to the ones (in other words, will the measurements be to the $\pu{0.1 mL}$ or to the $\pu{1 mL}$)? What would be the valid measurement?

Answer 1

I think the rule of thumb for the "reading error" being $\pm\,0.1$ of the smallest division applies here, so for your case, it'd be $5\;\mathrm{mL}\pm\, 0.5\,\mathrm{mL}$. See this page, entitled "Math Skills - Scientific Notation", from Texas A&M. Specifically, it is noted that

A rule of thumb: read the volume to 1/10 or 0.1 of the smallest division. (This rule applies to any measurement.) This means that the error in reading (called the reading error) is 1/10 or 0.1 of the smallest division on the glassware. If you are less sure of yourself, you can read to 1/5 or 0.2 of the smallest division.

Answer 2

Usually, you estimate (interpolate) the measurement between the graduation divisions. You should be able to estimate the measurement to one tenth of the smallest division. However, you cannot indefinitely improve the accuracy (the significant figures) of the measurement by interpolation. Volumetric instruments, such as graduated cylinders, volumetric flasks, bulb pipettes, graduated pipettes, and burettes, – like any other measuring instrument – have a limited accuracy. The dimensions, graduating divisions, and accuracy limits of volumetric instruments are standardized. For example, the requirements of measuring cylinders are specified in the international standard ISO 4788. Two classes of accuracy are specified: Class A for the higher grade and Class B for the lower grade. The nominal capacity, graduation divisions, and accuracy limits are shown in the following tables.

$$ \textbf{Graduated measuring cylinders with spouted neck, tall form, Class A} \\ \begin{array}{lll} \hline \text{Nominal capacity} & \text{Graduation divisions} & \text{Accuracy limits} \\ \mathrm{ml} & \mathrm{ml} & \pm\ \mathrm{ml} \\ \hline 5 & 0.1 & 0.05 \\ 10 & 0.2 & 0.1 \\ 25 & 0.5 & 0.25 \\ 50 & 1 & 0.5 \\ 100 & 1 & 0.5 \\ 250 & 2 & 1 \\ 500 & 5 & 2.5 \\ 1000 & 10 & 5 \\ 2000 & 20 & 10 \\ \hline \end{array}$$

$$ \textbf{Graduated measuring cylinders with spouted neck, tall form, Class B}\\ \begin{array}{lll} \hline \text{Nominal capacity} & \text{Graduation divisions} & \text{Accuracy limits} \\ \mathrm{ml} & \mathrm{ml} & \pm\ \mathrm{ml} \\ \hline 5 & 0.1 & 0.1 \\ 10 & 0.2 & 0.2 \\ 25 & 0.5 & 0.5 \\ 50 & 1 & 1 \\ 100 & 1 & 1 \\ 250 & 2 & 2 \\ 500 & 5 & 5 \\ 1000 & 10 & 10 \\ 2000 & 20 & 20 \\ \hline \end{array}$$

For example, a measuring cylinder of Class A with a nominal capacity of 500 ml has graduation divisions of 5 ml. By interpolation, you might be able to estimate the measurement to one tenth of the division, i.e. to 0.5 ml. However, the accuracy limits of the measuring cylinder are ±2.5 ml (at any point on the scale).

Various sources of error are inherent in calibration and use. When the greatest possible accuracy is desired, the measuring cylinder should be used as closely as possible to the manner in which it has been calibrated. Most importantly, the capacity of the measuring cylinder varies with change of temperature. The particular temperature at which a measuring cylinder is intended to contain its nominal capacity is the reference temperature of the measuring cylinder. Usually, the reference temperature is 20 °C.

Note that 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. To set the meniscus precisely, fill the cylinder with the relevant liquid by means of a plastic tube to a distance of a few millimetres above the selected graduation line, so that the walls of the measuring cylinder considerably above the graduation line are not wetted. Wait 2 min to allow liquid in the cylinder to drain. Then withdraw the surplus of liquid by means of a tube. Careful swaying may be necessary to refresh the meniscus shape.