What is the Uncertainty in the Total Weight of a Collection of Similar Items?

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Discussion Overview

The discussion revolves around determining the uncertainty in the total weight of a collection of similar items weighed separately, with a focus on the implications of scale accuracy and the propagation of uncertainty in measurements. The scope includes theoretical considerations and practical implications related to measurement accuracy and error analysis.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant seeks to understand how to calculate the total weight uncertainty for N similar items, each weighed separately, given a scale accuracy of ±U units.
  • Another participant suggests using the propagation of uncertainty, proposing that the total uncertainty can be expressed as $$U_{total}=\sqrt{N} U$$, applicable when measurements have the same variance.
  • Some participants question the meaning of scale accuracy, discussing potential systematic errors and their impact on the proposed formula.
  • There is a discussion about the interpretation of measurement specifications, with one participant noting that reported specifications often refer to standard deviation rather than confidence intervals.
  • Another participant expresses uncertainty about the implications of systematic errors and whether they can still be bounded by the known accuracy of the scale.
  • One participant reflects on the complexity of measurement distributions and the relationship between precision and accuracy in the context of consumer-grade scales.

Areas of Agreement / Disagreement

Participants express differing views on the implications of scale accuracy and the applicability of the uncertainty propagation formula. There is no consensus on the interpretation of measurement specifications or the impact of systematic errors on the calculations.

Contextual Notes

Participants highlight the potential for systematic bias in measurements and the complexity of defining accuracy and precision in practical scenarios. The discussion does not resolve these complexities or provide definitive answers regarding the assumptions involved.

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TL;DR
Determine uncertainty in total weight of N different objects weighed separately (NOT homework)
Not a homework assignment. Just what must be a solved problem that I'm embarrassed I don't know how look up or figure out on my own.

I want to know the total weight of a collection of N items. They are all similar but of different, unknown weights. My scale has a known accuracy of ±U units. If I weigh 1 object I can say that it's weight is as measured, ±U. So what can I say about the uncertainty of the total weight of N objects?
EDIT: Each item weighed separately.
 
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How many items can you place on the scales at the one time?
 
JT Smith said:
TL;DR Summary: Determine uncertainty in total weight of N different objects weighed separately (NOT homework)

Not a homework assignment. Just what must be a solved problem that I'm embarrassed I don't know how look up or figure out on my own.

I want to know the total weight of a collection of N items. They are all similar but of different, unknown weights. My scale has a known accuracy of ±U units. If I weigh 1 object I can say that it's weight is as measured, ±U. So what can I say about the uncertainty of the total weight of N objects?
You can use the propagation of uncertainty to determine this. In this situation, the variance of the total weight is the sum of the variances. And since each measurement has the same variance we get $$U_{total}=\sqrt{N} U$$
 
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Baluncore said:
How many items can you place on the scales at the one time?

One at a time. Sorry, I left that important fact out of the description.
 
Dale said:
You can use the propagation of uncertainty to determine this. In this situation, the variance of the total weight is the sum of the variances. And since each measurement has the same variance we get $$U_{total}=\sqrt{N} U$$

Thank you, that's what I wanted to know.

I mistakenly thought that rule only applied when repeatedly measuring the same thing.
 
JT Smith said:
Thank you, that's what I wanted to know.

I mistakenly thought that rule only applied when repeatedly measuring the same thing.
You are welcome. It applies any time you are adding up a bunch of measurements each with the same uncertainty.
 
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JT Smith said:
My scale has a known accuracy of ±U units. I
What does that mean?

If it always reads a half-gram high, this formula will not be right.
 
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Vanadium 50 said:
What does that mean?

If it always reads a half-gram high, this formula will not be right.
If it's always a half-gram high, it doesn't have "a known accuracy of ±U", so the formula does not apply.
 
Vanadium 50 said:
What does that mean?

If it always reads a half-gram high, this formula will not be right.

What does it mean when a scale has a specification of a certain precision? I take it to mean that, ideally at least, that a large number of measurements would generate a symmetric distribution, around the mean value, that has a width corresponding to the precision. And by width I mean a large percentage of the measurements (95%?) would fall within. Maybe this isn't exactly right but I think I have the basic idea.

In general it's probably a very complicated question. I just wanted a basic approach to extending the uncertainty because the worst-case scenario, where every measurement is at maximum uncertainty in the same direction, isn't very realistic -- provided the error is truly random. If there is some sort of systematic bias then it's a different question.
 
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JT Smith said:
In general it's probably a very complicated question.
That is correct. That is why I wanted you to think about it.

If you take two meter sticks and place them side by side, they may differ by 1 or even 2 mm. Obviously, they can't both be right. So the idea "distribution around the mean value: is non-trivial. Your single number represents two concepts - "how wide is this distribution" and 'how well is it centered around the true value".
 
  • #11
JT Smith said:
And by width I mean a large percentage of the measurements (95%?) would fall within. Maybe this isn't exactly right but I think I have the basic idea.
Usually the reported specification would be one standard deviation. Hopefully the documentation is clear, eg by using the term “standard uncertainty” or “standard deviation”. But if it is ambiguous then assume it is a standard deviation rather than a 95% confidence interval.
 
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  • #12
Squizzie said:
If it's always a half-gram high, it doesn't have "a known accuracy of ±U", so the formula does not apply.
I disagree. An unknown systematic error may still be known to be bounded by ##\pm U##.
 
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  • #13
Vanadium 50 said:
If you take two meter sticks and place them side by side, they may differ by 1 or even 2 mm. Obviously, they can't both be right. So the idea "distribution around the mean value: is non-trivial. Your single number represents two concepts - "how wide is this distribution" and 'how well is it centered around the true value".

That's why I said "accurate" instead of "precise". I wanted to convey the message that the scale was reporting a range of values centered around the true value.

I don't know what is standard for specifications. I don't own any expensive laboratory balances. The consumer grade scales I have used usually just specify precision and linearity. I suppose I could carefully test my scale and figure out the confidence level. A real scale may be less precise at higher weights, which is why I said that the items in question were of similar weight.

But I never meant for this to go very deep. My question was simple and has been answered. That's not to say you guys can't take it further though.
 
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