# WHat is the uncertainty in a metre rule?

1. May 30, 2012

### 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 ?

2. May 30, 2012

### K^2

Re: WHat is the uncertainty in a metre rule??

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]

Last edited: May 31, 2012
3. May 31, 2012

### Studiot

Re: WHat is the uncertainty in a metre rule??

Do you not think it should be

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

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.

Last edited: May 31, 2012
4. May 31, 2012

### Staff: Mentor

Re: WHat is the uncertainty in a metre rule??

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.

5. May 31, 2012

### Studiot

Re: WHat is the uncertainty in a metre rule??

This is a different sort of error.

6. May 31, 2012

### Staff: Mentor

Re: WHat is the uncertainty in a metre rule??

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

7. May 31, 2012

### Studiot

Re: WHat is the uncertainty in a metre rule??

Don't you think that's pushing it?

8. May 31, 2012

### K^2

Re: WHat is the uncertainty in a metre rule??

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.

9. May 31, 2012

### Studiot

Re: WHat is the uncertainty in a metre rule??

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

10. May 31, 2012

### Staff: Mentor

Re: WHat is the uncertainty in a metre rule??

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).

11. May 31, 2012

### Studiot

Re: WHat is the uncertainty in a metre rule??

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.

12. May 31, 2012

### truesearch

Re: WHat is the uncertainty in a metre rule??

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.

13. May 31, 2012

### bahamagreen

Re: WHat is the uncertainty in a metre rule??

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...

14. May 31, 2012

### Dickfore

Re: WHat is the uncertainty in a metre rule??

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.

15. May 31, 2012

### Studiot

Re: WHat is the uncertainty in a metre rule??

Wow that's a good ruler?

16. May 31, 2012

### Staff: Mentor

Re: WHat is the uncertainty in a metre rule??

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.

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.

17. May 31, 2012

### truesearch

Re: WHat is the uncertainty in a metre rule??

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 relise 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.

18. May 31, 2012

### banerjeerupak

Re: WHat is the uncertainty in a metre rule??

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.

19. May 31, 2012

### truesearch

Re: WHat is the uncertainty in a metre rule??

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

20. May 31, 2012

### Studiot

Re: WHat is the uncertainty in a metre rule??

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

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.

Last edited: May 31, 2012