Consider the scenario where you have a measurement system which records at a resolution of 0.001 but has an accuracy of +/- 0.01. Then, with the following measurement:
8.925
I would report this measurement to three significant figures because the last digit should have the same order of magnitude as the uncertainty (+/- 0.01):
8.93 +/- 0.01
Now, consider that I collect 100 measurements in the same manner so that I end up with a spread. I take the average of the 100 measurements, and I want to specify a tolerance for the expected result.
Are there any rules about how much resolution I can reasonably specify with limits for the averaged measurement?
For example, it's obvious that I can be at least as precise as 9.00 +/- 0.01 based on the resolution/accuracy of the individual measurements themselves.
Consider that my averaged measurement is 9.014. With the criteria above this would round-down to 9.01 and would be considered acceptable, even though the averaged measurement was 0.004 above the limit. I could increase the resolution of the limit to account for that, e.g. 9.000 +/-0.01, giving an upper limit of 9.010. This would create a failure for the measurement of 9.014. I could go further and start worrying about the next decimal place along, and so on.
MY QUESTION BOILS DOWN TO:
Are there any rules which govern how precise it is reasonable to be with a collection of individual measurements which are averaged?
UPDATE:
I have been doing some experiments averaging measurements to two decimal places and then changing the allowed tolerance.
I have found that when the number of decimal places in the tolerance value is the same or less than the measurements, then the averaged raw value, averaged upper limit and averaged lower limit all have the same digits in the remaining decimal places. This is best explained by showing the tables:
You can see that the digits "9825" occur in all the all the averaged measurements in decimal places greater than that of the measurement values (2). This number occurs regardless of the value of the tolerance, it only matters that it has the same or less number of decimal places as the measurements. I must admit, I found this surprising.
Something slightly different happens if I increase the number of decimal places in the tolerance so that it is greater than the decimal places in the measurement:
Here you can see that there is a common value of 825 in the averaged values. The 9 that was previously common is no longer common.
There is clearly something I don't understand here. My original question still stands but I thought this extra information was worth adding to it.