# Significant figures when standard deviation is 0

For analytical chem lab, our group did a titration in which we got the same exact value every single time. As a result, our standard deviation was 0 and our instructor says to report answer using real rule of sig figs (our last significant figure should be the first digit in the uncertainty measurement) I am not sure what to do as my STDev doesn't have any significant digits?

edit: Molarity is the measurement in question!

• How many titrations? What readings did you get? How is burette marked? How many significant figures are you reading on burette? I wonder if you technique is bad, or if you have so cheap student grade burettes. – MaxW Nov 5 '15 at 7:56
• I'm one of the top research universities (hint public) for chemistry in the country, the burets are surely calibrated as this is an upper level chemistry course. We did 4 trials, Moles CaCO3 Volume EDTA added corrected for blank molarity EDTA 0.0009975 33.7 33.39 0.029874214 0.0009975 33.7 33.39 0.029874214 0.0009975 33.7 33.39 0.029874214 0.0009975 33.7 33.39 0.029874214 – camerobn Nov 5 '15 at 8:19
• You can be pretty sure that no matter where you go, how well funded or unfunded schools are, the burettes will be calibrated to something. That does not mean they are all accurate. The precision of the instrument should be printed on it (or you can find it in the manual). Your standard deviation cannot be smaller than the actual precision of the instrument. I would also like to encourage you to include your comment into the post with proper formatting. In the way it is presented now it is hardly comprehensible. – Martin - マーチン Nov 5 '15 at 9:30
• – Loong Nov 5 '15 at 9:47
• @camerobn - I think 33.7 were your burette volumes. Correct? Using 50 ml burette with 0.1 ml marks? – MaxW Nov 5 '15 at 15:46

Various sources of error are inherent in measurements. The so-called Type A evaluation, i.e. the statistical analysis of series of observations (e.g. calculation of the standard deviation and the standard error of the mean), can be used to evaluate the uncertainty arising from random effects.

However, this evaluation does not include the uncertainty arising from systematic effects. For example, if the concentration of the titrant (the solution of a known concentration that is added during the titration) has a small unknown error and the same titrant is used for all titrations, each of the individual results includes the same systematic error. Or, if the burette has a small unknown error and the same burette is used for all titrations, again each of the individual results includes the same systematic error. You can neither eliminate such errors nor reduce the caused uncertainty by making additional trials, and you cannot evaluate the caused uncertainty by using statistical analysis (i.e. by calculating the standard deviation and the standard error of the mean) of the results. The uncertainty arising from such systematic effects has to be evaluated by other means. The so-called Type B evaluation is usually based on scientific judgment using all of the relevant information available on the possible variability of the results. Finally, you can combine all calculated or assumed uncertainties by using the rules for error propagation.

For example, the uncertainty arising from the unknown error of the burette may be estimated using the manufacturer’s specifications as follows:
Usually, you estimate (interpolate) the measurement between the graduation divisions of the burette. You might be able to estimate the measurement to one tenth of the smallest division. However, 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 burettes are specified in the international standard ISO 385. Two classes of accuracy are specified: Class A or AS 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.

$$\textbf{Burettes, Classes A and AS} \\ \begin{array}{lll} \hline \text{Nominal capacity} & \text{Graduation divisions} & \text{Accuracy limits} \\ \mathrm{ml} & \mathrm{ml} & \pm\ \mathrm{ml} \\ \hline \hphantom{00}1 & 0.01 & 0.006 \\ \hphantom{00}2 & 0.01 & 0.01 \\ \hphantom{00}5 & 0.01 & 0.01 \\ \hphantom{00}5 & 0.02 & 0.01 \\ \hphantom{0}10 & 0.02 & 0.02 \\ \hphantom{0}10 & 0.05 & 0.03 \\ \hphantom{0}25 & 0.05 & 0.03 \\ \hphantom{0}25 & 0.10 & 0.05 \\ \hphantom{0}50 & 0.10 & 0.05 \\ 100 & 0.20 & 0.10 \\ \hline \end{array}$$

$$\textbf{Burettes, Class B}\\ \begin{array}{lll} \hline \text{Nominal capacity} & \text{Graduation divisions} & \text{Accuracy limits} \\ \mathrm{ml} & \mathrm{ml} & \pm\ \mathrm{ml} \\ \hline \hphantom{00}1 & 0.01 & 0.01 \\ \hphantom{00}2 & 0.01 & 0.02 \\ \hphantom{00}5 & 0.01 & 0.02 \\ \hphantom{00}5 & 0.02 & 0.02 \\ \hphantom{0}10 & 0.02 & 0.05 \\ \hphantom{0}10 & 0.05 & 0.05 \\ \hphantom{0}25 & 0.05 & 0.05 \\ \hphantom{0}25 & 0.10 & 0.10 \\ \hphantom{0}50 & 0.10 & 0.10 \\ 100 & 0.20 & 0.20 \\ \hline \end{array}$$

For example, a burette of Class AS with a nominal capacity of 50 ml has graduation divisions of 0.10 ml. By interpolation, you might be able to estimate the measurement to one tenth of the division, i.e. to 0.01 ml. However, the accuracy limits of the burette are ±0.05 ml (at any point on the scale).

• As I remember it the burettes were rated as "accuracy was at least" the quoted figure. – MaxW Nov 5 '15 at 15:45