Sig Figs, Weighing, and Averages

Off-topic conversations and chit-chat.

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This question has come up multiple times at work and I was wondering what everyone here thought...

Let's say I need to determine the average weight of 10 dosage units. Each individual dosage weighs about 3 grams. I weigh each of the 10 individually on a four decimal balance, and get an average of 3.0124 grams (by adding up the individual weights and dividing the total by 10, rounding to 4 decimals because that's the most precise measurement). My coworker, Steve, reviews my work and says that the guiding method states to weigh all 10 at once and report the average; he then repeats the experiment, putting all 10 on the balance at once; he gets a total of 30.1243 grams, divides by 10 to get 3.01243 grams as the average (because he had 6 significant figures to work with). Another coworker, Mildred, reviews Steve's work, but says the average is 3.0124 because one cannot add precision to the measurement and the balance is only precise to 4 decimals.

Our guiding SOP states to use the significant figures rules and not round until the final reportable value, but is quiet on this specific circumstance.

Who has the right weight, Steve or Mildred? What is the right number of significant figures?
1 All leading zeros;
2 Trailing zeros when they are merely placeholders to indicate the scale of the number (exact rules are explained at identifying significant figures); and
3 Spurious digits introduced, for example, by calculations carried out to greater precision than that of the original data, or measurements reported to a greater precision than the equipment supports.

Taken from rules on significant figures. Number 3 would say that Mildred is correct.

In all honesty though, as I was taught in school, especially my high school physics teacher, if you have a balance with 4 decimal places you can only have 3 significant figures past the decimal because the fourth is in doubt. You only use the fourth place to round the third. If the fourth place is 5, then you round up or down to the even number in the third place.
The past is there to guide us into the future, not to dwell in.
Ooooh, this is why I get so aggravated by the regulatory world. If the difference between Steve and Mildred's approaches actually matters, then neither is adequate. If the fourth decimal place after the point actually has any relevance to the decision on product safety (etc.) then the 4-figure balance isn't fit for purpose because it has no safety-factor; you need at least a 5-figure balance.
I get aggravated with a world that (correctly) makes rules, but then (incorrectly) spends so much time worrying about trivial differences in how they're interpreted that it doesn't have time to spend worrying about whether the rules were actually correct or useful in the first place. In this instance, for example, the rule of weighing 10 doses in a single measurement denies you any knowledge of the standard deviation, which is probably at least as important as the accuracy of average dose. I have limited brain-cells; I'd rather spend them thinking about whether my measurement is fit-for-purpose (what's the purpose anyway) rather than thinking about whether my measurement is fit-for-auditors. But I understand most analysts don't have the luxury, and have to do what they're told. Sad!
... but back to the true question, Mildred would argue her data aren't beyond the capacity of the instrument, because she's merely taking the exact figures the balance produced, and reporting them as they stand. Although she reports to a higher precision than the balance could achieve on a single sample, she's actually reporting the mass of 10 samples to exactly the precision of the raw data. She is right (though unhelpfully so).
(1) The method should have stated what precision was required,
(2) the expectation is that measurements will be made at full precision until the final step, at which they should be rounded to the method's required precision.
If this had been done, Mildred would have been correct, because (a) she followed the method and weighed 10 together, and (b) the act of weighing individually actually meant the balance was making 10 roundings in Steve's case, before the final step. But as I said, if there were any difference between Steve's and Mildred's answers sufficiently significant to affect the overall process, then I'd be very alarmed about the whole situation.
In all events, the method (the set requirements for the overall measurement) should set the precision, not the instrument used to achieve it. The dog wags the tail...
I would agree with Mildred (and others here). If a scale only calculated to 4 decimals, than the final answer can't include 5. I would also say that measuring each dose individually is incorrect if all you're after is the average. However, if I were concerned with absolute deviation from a prescribed weight, I would weigh each dose individually. If this were a drug, for instance, and a deviation greater than 2% was unacceptable in a sample of 10 doses, than individual weights would be necessary.

... actually I'd disagree that a measurement can never contain more precision than the device that measured it.

Assuming the rounded output from the measuring instrument is derived from something that has random errors, then the average of multiple measurements should converge on the true value of what's being measured (if the instrument is accurate).

The crude example: a coin. We can use a coin to measure the likelihood of it landing heads. The precision of a coin is zero decimal places: it lands heads (1) or tails (0). But if we toss the coin 1000 times, most of us would feel fairly confident in reporting the probability of it landing heads to at least one, if not two decimal places.
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