Very roughly - uncertainty of average is approximately equal to standard deviation of measurements divided by square root of number of measurements.

If the uncertainties are the same and you want to estimate the uncertainty of the total mass you can use propagation of uncertainties. In your case that would be the uncertainty of your measurements times the square root of the number of measurements. So if you measured 100 samples and each have an error of 0.1 g the uncertainty of your total mass at 1 sigma would be sqrt(100) x 0.1 = 1 gram.

I thank both of you for your answer. Here is a follow-up question. Would I be correct in now saying if I take a weight by difference or if I take a weight after zeroing the balance or taring the balance, there would have to be a propagation of uncertainty done for the final weight determined, ie, total weight -initial weight (or tare to zero) each with the same uncertainty as my uncertainty budget calculated. So the total uncertainty for each weight determination is square root of 2 times the uncertainty?

Well in fact your situation is more complex now because you have to deal with the net values which are given by differencies. So if say you have a gross G and tare T measurements, each of which is affected by the uncertainties u_G and u_T, the uncertaintiy associated to the net count N, which can be called u_N, is given by propagating the uncertainties on the gross and tare values, that is the square root of the squares of the uncertainties: u_N = sqrt(u_G^2 +u_T^2). If the 2 uncertainties are the same then yes you have u_N = sqrt(2) x u, where u=u_G=u_T.

Thank you. I am getting a hand on it. Somewhat fun to use it. Now I am wondering how you feel about reporting the uncertainty of a measurement with or without the error of measurement. I think both should be reported but at least the error of measurement would give more information as to the "found weight". The uncertainty expands this range. If you only use the uncertainty you know what range of values are possible based upon method but you know nothing about the accuracy of the measurement. Am I wrong?

A first consideration: try to avoid mixing the concepts of error and uncertainties. I will not go into detail but what you are interested, to start with, is the uncertainty associated with some estimates, that is the variance or dispersion associated with your data. In fact, in the GUM (the reference guide for the expression of uncertainties in measurements) uncertainties are basically standard deviations (square root of the variance). I think that if you want to seriously understand uncertainty propagation/analysis it is worth to spend some time reading the GUM. So, and this is the second point, when you want to summarize some measurements what you should give is your best estimate of some characteristic (say the mean mass of your samples) but also its standard deviation. You are right when you say that an analysis of this type allows you to evaluate the precision of your method (low dispersion of data indicates a good precision) but it says little about the accuracy. To do an example, you measure 10 samples that have similar weights (the real true weight is unknown). Your balance gives to you very close results and you think that everything is fine. For some reason however the balance was wrongly calibrated and there is a systematic shift of 5 grams in every single sample. So, your experiment will be very precise but will not be accurate, meaning that your results are close to each other but far away from the true (unknow) weight. Keep in mind however that the evaluation of accuracy is complex and is often system/method dependent. To conclude, at the beginning you might be confused by the terminology : systematic errors, random errors, bias, uncertainties of types A and B, and so on, are often used with different meanings. My advice is, try to understand your system first and then study the uncertainty associated with it using the GUM as a reference.

To come back to Yuri's comment. If you would like to estimate the uncertainty of the mean from a sample (like your 10 mass measurements) it is often used the so called "standard error of the mean SE" which is the ratio between the standard deviation SD and the square root of the number of samples: SE = SD / sqrt(N). This comes from the fact that if you take a new sample the mean can differ from the one calculated using the previous sample. In fact, if you take many many samples you can build a distribution for the mean (like an histogram). This is a good way to estimate the true (unknown) mean. Of course, the higher the number of samples the higher the precision of your estimation.

All good points. Some background-----I am an independent forensic scientist with a MS degree in Physical Chemistry and 10 yrs experience as a bench chemist at a State Crime Lab. The issues I deal with involve, generally, a total weight. I have now read the summary to the Guide twice and the am in the third reading of the GUM. The use of uncertainty by Crime Lab Analysts began several years ago without explanation and has been used in place of error of measurement (EM). Originally, the UM has been implied to be a representation of error which I know is wrong. In a recent case, I finally was given enough information to lead to the Summary of GUM and then to the GUM. I am now at the point, with yours and Yuri answers getting a good understanding of the use and misuse of the uncertainty. Accuracy is never fully known, but there are now logbooks used to track, at least, a monthly check against standard weights and can support some aspect of accuracy. I realize error of measurement goes towards precision and that accuracy and precision are apples and oranges.

I am now more convinced that the reports should show EM and UM or show a combination of EM with the UM added at the low end of the range and the high end of the range. Forensic chemistry results are based upon scientific protocols, analytical analysis protocols, and forensic protocols. It is my job to challenge and raise issues when I feel any of these protocols are not followed or are improperly being followed. This morning I realized that the uncertainty applies to all measurements and not just the resulting reported value. This prompted the first question and led to the second question. I now am more comfortable with my understanding of these "simple" conclusions and I am glad that you seem to be agreeing with me that both the EM, (showing precision), the balance logbook (implying some level of accuracy) and the UM (increasing the range in which the reported value may lie) should be required in a report. This set of Q&A's have been of great assistance to me and I am very grateful for the time and assistance you have given me. If you think I need more clarification to the above conclusions I have expressed, I am open to suggestions and directions.

Your job seems very interesting. I am glad my answers have helped. Do not hesitate to write back in case of further needs. Biagio.

And it sounds like you have a fascinating career.CERN wow. I will you let you know how things go in these cases. It may another year though at the rate these cases are going