WHat is the uncertainty in a metre rule?

[mutineer123]

WHat is the uncertainty in a metre rule??

For a single value is it 1 mm or is it 1/2mm(half the smallest division) ? And what about measuring something like a length of a stick (we need to take 2 readings, and deduct them like 15-0=15), then is the uncertainty 1+1=2mm or is it .5+.5=1mm ?

 

[K^2]

The rule is half the smallest division. So if your ruler has 1mm divisions, then the error is 0.5mm. [strike]I believe, the errors do add. So it does sound like 0.5mm+0.5mm = 1mm is the correct answer there, but I'm less certain about that.[/strike]

 

[Studiot]

So it does sound like 0.5mm+0.5mm = 1mm is the correct answer there, but I'm less certain about that.

Do you not think it should be


$$\sqrt {{{\left( {0.5} \right)}^2} + {{\left( {0.5} \right)}^2}} = 0.7mm$$

For a single value is it 1 mm or is it 1/2mm(half the smallest division) ? And what about measuring something like a length of a stick (we need to take 2 readings, and deduct them like 15-0=15), then is the uncertainty 1+1=2mm or is it .5+.5=1mm ?

That rather depends upon your ruler.

If it is a school type ruler which does not have zero at the end of the ruler then yes you have two measurements as above.

If it is an engineer's rule with zero flush ground to one end then there is only one comparison to account for.

 

[mfb]

I would not expect that the ruler has an accuracy of .5mm over the full range of a meter. While it is possible to read the values with an even higher accuracy, the scale itself might be wrong by 1-2 mm.

 

[Studiot]

I would not expect that the ruler has an accuracy of .5mm over the full range of a meter. While it is possible to read the values with an even higher accuracy, the scale itself might be wrong by 1-2 mm.

This is a different sort of error.
The OP asked about reading errors.

 

[jtbell]

If the ruler or meter stick is marked off in mm, you should be able to estimate the reading to ±0.1 mm.

 

[Studiot]

If the ruler or meter stick is marked off in mm, you should be able to estimate the reading to ±0.1 mm.

Don't you think that's pushing it?

 

[K^2]

Do you not think it should be


$$\sqrt {{{\left( {0.5} \right)}^2} + {{\left( {0.5} \right)}^2}} = 0.7mm$$

Hmm...

Δ²(X+Y) = <(X+Y)²> - <X+Y>² = <X²> + <2XY> + <Y²> - (<X>+<Y>)² = <X²> - <X>² + <Y²> - <Y>² + 2<XY> - 2<X><Y> = Δ²(X) + Δ²(Y)

(<XY>-<X><Y>=Cov(X,Y)=0 for independent variables.)

So yes, you are right. I probably should have done this to begin with to check myself. Apologies for sloppy reply.

 

[Studiot]

At least mutineer will be very well advised after all this discussion.

 

[mfb]

This is a different sort of error.
The OP asked about reading errors.

I cannot see this in the first post. The basic question is "WHat is the uncertainty in a metre rule??", and possible errors in the scale are clearly part of this uncertainty (and should be discussed, even if the conclusion is that they can be neglected).

 

[Studiot]

I didn't say that there were no other (potential) sources of error or that they were insignificant.

If you feel that they need discussing fire away and advise mutineer what they are and what to do about them.

 

[truesearch]

Anyone quoting measurements using a mm scale to +/- 0.1mm will not be believed.
This sort of accuracy can only be approached with a vernier scale.

 

[bahamagreen]

Setting the "0" end as one of the ends of the measurement is incorrect. A measurement of length must have two values both of which have a limit to their precision. Setting the "0" end as one measurement implies one of your measures has perfect precision, which it does not.

The ruler should be placed casually so both readings fall randomly within the interior of the ruler (so both readings are greater than 0). Then the measures are read.
This gives you two proper measured values, each of which will have significant digits the last of which is an estimate... (an estimate of how many tenths of the previous digit), then you take the difference between the two measures to find that length.

The precision of the measurement is only as good as your estimated last significant digit. If the ruler is marked in steps of 0.001mm and you are using your eyes to read it, your last digit will be the one where you reach the limit of what you can see, so you have to estimate. If you are making the reading with a magnifying glass you may get another significant digit, with a microscope you might be justified in getting additional significant digits... the scale of the ruler and the resolution of your view is what determines the precision of your measurement, not the ruler's markings alone...

 

[Dickfore]

For a single value is it 1 mm or is it 1/2mm(half the smallest division) ? And what about measuring something like a length of a stick (we need to take 2 readings, and deduct them like 15-0=15), then is the uncertainty 1+1=2mm or is it .5+.5=1mm ?

Neither. You align the left end with exactly the zero of the meter scale. There is only uncertainty with the right end, which does not necessarily fall onto a division of the meter scale.

The uncertainty in an analog scale is equal to half the smallest division of the scale. If your meter scale has divisions of 1 mm, then the uncertainty is 0.5 mm.

 

[Studiot]

If the ruler is marked in steps of 0.001mm

Wow that's a good ruler?

 

[jtbell]

If the ruler or meter stick is marked off in mm, you should be able to estimate the reading to ±0.1 mm.

Don't you think that's pushing it?

You're supposed to "push it" a bit when you're reading a scale. That's what the reading uncertainty is for. I can't eyeball the difference between 0.5 and 0.6 mm reliably, but I can definitely tell the difference between 0.5 and 0.7 mm.

(Of course I'm referring only to the "scale-reading" uncertainty, which is random, and not any inaccuracy in the scale itself (e.g. due to expansion or contraction or mis-calibration), which is systematic.

Anyone quoting measurements using a mm scale to +/- 0.1mm will not be believed. This sort of accuracy can only be approached with a vernier scale.

With a vernier scale you can get the 0.1 "exactly." I'd assign an uncertainty of no greater than ±0.05. With a good vernier scale you can even tell when you're halfway between two 0.1 divisions, and use that to reduce the reading uncertainty still further, say to ±0.025.

 

[truesearch]

When using a measuring a scale you are not advised/supposed to "push it a bit".
You are better off, and more credible, if you recognise the limitations of the scale. If the divisions are 1mm then anything between divisions is a guess... maybe an educated guess because we would all say which half of the division we are guessing in.
It is then a completely subjective judgement as to who has the best eyeball.
It is recommended to start with the assumption that the error is +/- 1 scale division
(+/- 1mm on a mm scale) or +/- 1 in the last digit of a digital scale.
When you realize what measurements are used for, being able to measure length greater than 100mm to within 1mm represents better than 1% uncertainty... In science this would be considered an excellent degree of accuracy... there is no necessity to measure better than 1% in the vast majority of cases.
Make life easy for yourself... +/- 1 division is all you need.

 

[banerjeerupak]

This method of uncertainty calculation is correct, but it holds for calculating the uncertainty when using different rulers (sensors in general). In this case, the maximum uncertainty is 1mm. This is because in the first reading you could be off by -0.5 mm and in the second reading it could be off by +0.5 mm.

 

[truesearch]

agree with you banerjeerupak... this is more like a treatment of observational error

 

[Studiot]

The uncertainty in an analog scale is equal to half the smallest division of the scale. If your meter scale has divisions of 1 mm, then the uncertainty is 0.5 mm

This I agree with as it conforms to standards/calibration lab practice.

Setting the "0" end as one of the ends of the measurement is incorrect. A measurement of length must have two values both of which have a limit to their precision. Setting the "0" end as one measurement implies one of your measures has perfect precision, which it does not.

The ruler should be placed casually so both readings fall randomly within the interior of the ruler (so both readings are greater than 0). Then the measures are read.
This gives you two proper measured values, each of which will have significant digits the last of which is an estimate... (an estimate of how many tenths of the previous digit), then you take the difference between the two measures to find that length.

The precision of the measurement is only as good as your estimated last significant digit. If the ruler is marked in steps of 0.001mm and you are using your eyes to read it, your last digit will be the one where you reach the limit of what you can see, so you have to estimate. If you are making the reading with a magnifying glass you may get another significant digit, with a microscope you might be justified in getting additional significant digits... the scale of the ruler and the resolution of your view is what determines the precision of your measurement, not the ruler's markings alone...

Neither. You align the left end with exactly the zero of the meter scale. There is only uncertainty with the right end, which does not necessarily fall onto a division of the meter scale.

This is just wrong, because both opposing statements are incomplete and provide a false impression of linear measurement.

I suggest you get hold of a good (engineering) metrology textbook.

There are two types of standards identified.

End standards.
Length standards used in standards and equipment calibration labs are end standards.

Line standards which you are referring to.

A cheap ruler from a toyshop has only line standards.

An engineering workshop or drawing office standard ruler has one end standad (zero) and one line standard - the scale.
I have both types.

 

[truesearch]

Given a specimen of thin sheet material, known to be between 3 and 5mm thick...what would you, as a scientist, quote its thickness using a steel, engineering rule with smallest scale divisions of 1mm?
What would you give as the uncertainty that would be acceptable in the world of science communication.

 

[Studiot]

Given a specimen of thin sheet material, known to be between 3 and 5mm thick...what would you, as a scientist, quote its thickness using a steel, engineering rule with smallest scale divisions of 1mm?
What would you give as the uncertainty that would be acceptable in the world of science communication

Let us say that you measured your sheet or block as 3mm or 30mm or 300mm, with an uncertainty of half a millimetre (0.50mm).

This means that the expected value falls between 2.50 and 3.49 mm , 29.5 and 30.49 or 299.5 and 300.49.

As you say this is a range of 1mm , but we do not call this an uncertainty of 1mm.

This is a similar situation to the RMS value of voltage in electricity, which has a peak and peak to peak value of twice the peak.

 

[Dickfore]

Given a specimen of thin sheet material, known to be between 3 and 5mm thick...what would you, as a scientist, quote its thickness using a steel, engineering rule with smallest scale divisions of 1mm?
What would you give as the uncertainty that would be acceptable in the world of science communication.

The thing is, you can do better than the information that the thickness is between 3-5 mm by measuring with such a ruler.

 

[truesearch]

I do not see the analogy with rms...peak...this is a mathematical relationship not a matter of uncertainty. If you could expand on this I would be interested.
I think that what I wrote does mean an uncertainty in measurement of +/- 1mm
(+/- 0.5mm if you want to split hairs).
I work by the guidline of +/- 1 scale division... I have no choice... an exam board, no less, requires this of students.

 

[truesearch]

please explain how Dickfore

 

[mfb]

When using a measuring a scale you are not advised/supposed to "push it a bit".
You are better off, and more credible, if you recognise the limitations of the scale. If the divisions are 1mm then anything between divisions is a guess... maybe an educated guess because we would all say which half of the division we are guessing in.

You can determine the quality of your educated guess. Something close to 1/2 half of the scale would be equivalent to random guessing between the marks. As you do not guess randomly, your uncertainty is smaller. An uncertainty of 1 scale would provide an upper limit for random guessing (and not a typical deviation).

When you realize what measurements are used for, being able to measure length greater than 100mm to within 1mm represents better than 1% uncertainty... In science this would be considered an excellent degree of accuracy... there is no necessity to measure better than 1% in the vast majority of cases.[/QUOTE]
That really depends on the measurement. If you want to build any high-tech product with moving parts, 1% is way too much. If you try to dig tunnels with 1% accuracy, everything longer than 100m is a mess. And I think the length of the LHC tunnel (27km) is known with an uncertainty of some micrometers.

 

[Dickfore]

By measuring it! If the sheet is smooth enough, then there are the following values that you might get:
3.0, 3.5, 4.0, 4.5, 5.0, each with an uncertainty ±0.5 mm.

You may ask how we can estimate 3.5, 4.5 mm on a ruler with 1 mm divisions. You can judge if the length is closest to 3.0, to 4.0, or is closest to the midpoint 3.5 mm, for example. This is how you choose between these three values.

Then, when reporting 3.5 ± 0.5 mm, for example, it means that you expect the length to be between 3.0 mm and 4.0 mm. This is an interval that is half as wide than your initial assumption for the length! Thus, you did better.

 

[truesearch]

That is brilliant Dickfore... I did hint that if splitting hairs was important I would go with +/- 0.5mm but there is no indication in what you have given that it could ever be better than that. For me it is a matter of no importance +/-1mm or +/-0.5mm is the same principle. I will stick with +/- 1 scale division.
There is no way anyone could say +/- 0.1mm

PS I won't do it, but it would not be difficult to make up a string of numbers that would show an answer between 4.0 and 5 mm or any other number I wanted to be important... I will stick with +/- 1 scale division. It would be difficult to come up with a string of measurements that would take us out ofthat range.

 

[Studiot]

If you could expand on this I would be interested.

Happily.

The standard mains voltage in Europe is 230 volts. This has a peak voltage of 325 volts and a peak to peak value of 650 volts. This means that the voltage varies 325 volts above and below the zero ± if you like.

But you can never ever apply or measure 635 volts across a pair of terminals
The most you can obtain is 325.

This is because the ± means plus or minus not plus and minus. It can never be both at once.

Now apply this to the measurement process.

1) You should be able to decide whether the measurement point is between 1 & 2, 2 & 3 or 3 & 4.

Let us say it is between 2 and 3. This is an uncertainty of 1

So to the left and right of your measurement point you have scale markings 2 and 3.

2) You should be able to estimate whether the measurement point is closer to 2, 2.5 or 3. This cuts the uncertainty in half.

So if you can read to the nearest division (2 or 3) the uncertainty is 1

If you can read to the nearest half division the uncertainty is 0.5

Does this help?

 

[Dickfore]

There is no way anyone could say +/- 0.1mm

Of course! If that was the case, we might as well have divided our scale on marks 0.1 mm apart!

The "1/2 divisor rule" is trying to extract as much as possible from an analog scale.

For digital instruments however, the uncertainty is ±1 of the smallest displayed digit.