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.

 

[Dickfore]

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

No, the uncertainty is one half of the length of the interval, in this case (3-2)/2 = 0.5.

 

[truesearch]

sorry studiot...that is not an error or uncertainty in measurement.

 

[Studiot]

Truesearch

I have no idea what you mean.

Dickfore.

If the result could be reported as 2 or 3 that is an uncertainty of 1.

 

[truesearch]

I am concerned when I read in post 6 (a PF Mentor) that measurements can be made to within +/- 0.1mm using a mm scale.

 

[Dickfore]

Dickfore.

If the result could be reported as 2 or 3 that is an uncertainty of 1.

http://en.wikipedia.org/wiki/Plus-minus_sign#Precision_indication

 

[mfb]

I am concerned when I read in post 6 (a PF Mentor) that measurements can be made to within +/- 0.1mm using a mm scale.

As a standard deviation, 0.2 is easy to achieve and 0.1 might be possible (all in mm).

As an example for 0.2 standard deviation, this means that 0.3 is usually (~70%) read somewhere between 0.1 and 0.5. It is sufficient to see that 0.3 is smaller than 0.5, but not close to 0, to do this.
0.5 should usually be read somewhere between 0.3 and 0.7 - which is everything not close to a mark on the scale.

If you want to give some "upper bound" for the error, you should use larger values, of course. But an upper bound is not always well-defined (apart from digital displays). And if you want to use the marks on the scale, you should format your numbers like 0.3^{+0.7}_{-0.3}mm

 

[Studiot]

Yes indeed using ± notation.

But if you are limited to scale divisions you cannot report half divisions, your statement has to be 2 or 3, not 2.5 ± 0.5.
That implies you have to report 0,1,2,3,4 etc.

So a report that the thickness is 2 implies that it is between 1 and 3 ie 2 ±1

Which is what I said.

Incidentally you need to revise your statement on end standards.

 

[Dickfore]

As an example for 0.2 standard deviation, this means that 0.3 is usually (~70%) read somewhere between 0.1 and 0.5. It is sufficient to see that 0.3 is smaller than 0.5, but not close to 0, to do this.

I guess you meant 0.2 in the bolded number. Nevertheless, this is only true for a normal distribution. When we measure with a coarse scale such that we always get the length to be between the same two divisions, the error is not of statistical nature, and uncertainty has a different meaning from standard deviation. For a uniform distribution with ends a, and b, the standard deviation is:
<br /> \sigma_{U} = \frac{b - a}{2 \sqrt{3}}<br />
Notice that 2 \sqrt{3} \approx 3.5[/tex]<br /> <br /> <blockquote data-attributes="" data-quote="Studiot" data-source="post: 3936920" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> Studiot said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Yes indeed using ± notation.<br /> <br /> But if you are limited to scale divisions you cannot report half divisions, your statement has to be 2 or 3, not 2.5 ± 0.5.<br /> That implies you have to report 0,1,2,3,4 etc.<br /> <br /> So a report that the thickness is 2 implies that it is between 1 and 3 ie 2 ±1<br /> <br /> Which is what I said.<br /> <br /> Incidentally you need to revise your statement on end standards. </div> </div> </blockquote><br /> You can report half divisions, as it is customary in experimental physics.<br /> <br /> As for end standards, we are not doing calibration of etalons. We are measuring the length of an object. Thus, we are free to slide the scale so that the left end coincides <b>exactly</b> with one of the ruler&#039;s divisions. Then, there is uncertainty in reading off only the right end.<br /> <br /> There might be systematic errors due to the bad calibration of the rulers divisions, but that&#039;s another point.

 

[Studiot]

As for end standards, we are not doing calibration of etalons. We are measuring the length of an object. Thus, we are free to slide the scale so that the left end coincides exactly with one of the ruler's divisions. Then, there is uncertainty in reading off only the right end.

This is a fundamental error. The process of sliding still constitutes a 'reading' or alignment error. It is not as accurate as the engineering process of aligning an end standard ruler, even with a comparator microscope which engineers also use.

And BTW why mention etalons?

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

Actually there is but you require a draftsman's scale rule with diagonal scales. Have you heard of these?
However I have not seen one as long as 1 metre.

 

[truesearch]

Last few comments are missing the point and rely on unsubstantiated assumptions.
Trying to see something between divisions which is not there!
Assuming that there is some uniform scale within the division
Assuming there is no distortion.
The example of digital instruments should serve as a clue... There is no way to 'eyeball' how close the last digit is to the one above or the one below. +/-1 max is a safe, sensible, objective bet.

 

[Dickfore]

This is a fundamental error. The process of sliding still constitutes a 'reading' or alignment error.

Even if it does consists of an error, the error is of the order of the width of the mark, and not of the order of half the distance between two marks. I don't know about your rulers, but the marks on mine are pretty thin.

 

[truesearch]

I have used draughts mans scales ...I have also used verniers

 

[K^2]

truesearch, error is not the absolute maximum or minimum you can be wrong by. It's the average error. So it behaves like a random walk. For a random walk of N steps of distance L each, the average distance you travel is L*Sqrt(N). So average error behaves like RMS.

In practice, the actual value of how much you are off by will be normally distributed. The quoted error is the standard deviation of that distribution.

 

[Studiot]

Even if it does consists of an error, the error is of the order of the width of the mark, and not of the order of half the distance between two marks. I don't know about your rulers, but the marks on mine are pretty thin.

So why can't you 'read' the other end to the same precision?

As a matter of inerest how do you guarantee that the aligned 'zero' stays put while you read the other end?

I know how the navy does it for a traverse tape and how an engineering workshop does it for an engineering endstop rule and similarly how a drapers shop does it for a drapers endstop rule. Why do you think they do it this way with an end stop rather than your way?

 

[Studiot]

I have used draughts mans scales

So you know they are commonly calibrated in 1/100 inch or 0.1mm?

 

[Dickfore]

So why can't you 'read' the other end to the same precision?

Because not all lengths in Nature are integer multiples of the divisions of our scale.

As a matter of inerest how do you guarantee that the aligned 'zero' stays put while you read the other end?

You don't. But, you are describing sources of error that are order of magnitude smaller than the precision of the scale of the measuring instrument.

In fact, if the sources you allude to start giving comparable contributions, then it means your measuring scale is so precise that you do not get the same result as you repeat the measurement. In other words, you start getting statistical errors.

I know how the navy does it for a traverse tape and how an engineering workshop does it for an engineering endstop rule and similarly how a drapers shop does it for a drapers endstop rule. Why do you think they do it this way with an end stop rather than your way?

Probably to account for the fact that in these cases the object being measured is violently moved during the measuring process. This, on the other hand, happens rarely in a Physics Lab.

 

[Studiot]

Dickfore all you are proving is that different people in different 'laboratories' use different techniques and thereby achieve (slightly) different results by going their own way.

The whole object of calibration and standardisation is so that anyone anywhere can achieve the same result under the same conditions. This involves standardisation of measurement technique as well as tools in order to remove 'operator bias'. Measurement against a common stop end is one such standard.

If a laboratory develops its own special techniques it needs to report these as part of the results.
I once worked in such a laboratory measuring the lengths of bricks, accurate to less than a 10thou using the lab's specially developed technique.
But we never pretended it was 'standard' or that the method should be widely adopted.

 

[Dickfore]

The whole object of calibration and standardisation is so that anyone anywhere can achieve the same result under the same conditions.

Provided that the results are reported to same precision! What you are describing in the previous posts is comparing the result of a measurement done by a school ruler, to that of a micrometer screw gauge.

Let us say that (by the method I described), I get a length measurement of 3.5±0.5 mm.

Then, you come with your fancy equipment and get a result 3.329±0.007 mm (you had to repeat the experiment several times because you noticed that every time you get a different reading with your fine equipment. Then, you took the mean, and you found the standard deviation of the mean, and you took a 95% confidence interval for the mean.)

Does that make my measurement "wrong"?

 

[jtbell]

I am concerned when I read in post 6 (a PF Mentor) that measurements can be made to within +/- 0.1mm using a mm scale.

Ultimately, the proper ± figure for scale-reading uncertainty depends on the person who is making the measurement and the instrument that he is using, and that person has to make a judgment about this.

I feel confident in assigning ±0.1 mm when using a metal scale with finely-engraved lines, in a way that eliminates or minimizes parallax error due to the thickness of the scale. It helps that I'm rather nearsighted so I can get my eye about 10 cm from the scale if I take off my eyeglasses. If I'm using a typical plastic ruler with relatively thick mm-lines, I might use ±0.2 mm. If I'm using a thick meter stick and can't lay it edgewise on the object being measured so that I have to sight across the thickness of the meter stick, I might use ±0.5 mm or even ±1.0 mm.

 

[Studiot]

Not at all to do with statistics.

But everything to do with technique which contains inherent sources of error v technique which avoids these.

When you place your test piece and ruler against the stop end you have a guaranteed square and reproducible 'zero'.

When you estimate the alignment of two lines along a sight line that may or may not be square and hold the ruler and testpiece at some random (albeit small) angle to each other you have a recipe for variability of measurement. Notice I said 'sight line'. Two operators will align the pieces slightly differently by sight. They cannot do this with a stop end.


Edit
JTbell has just described the visual alignment issue to a T whilst I was posting.