# The uncertainty of a metre ruler?

I have been taught that the uncertainty in the measurement of a metre ruler is +-1 mm. However , I was also taught that the uncertainty is half of the smallest division in the measuring instrument. So, it should be that the uncertainty of the measurement of a metre ruler is +-0.5 mm( its smallest division is 1 mm) ?

• Nothing is certain with uncertainties. Dec 30, 2015 at 16:30
• @ja72 - that is certainly true. Oh wait... Dec 30, 2015 at 16:41
• Dec 30, 2015 at 16:43
• I'm pretty sure "uncertainty" isn't the term you're talking about. The uncertainty in a measurement may be caused by the person using the ruler (and other things), in addition to the ruler itself. All else being the same, I could make 100 measurements with that ruler, and you could then make 100 measurements of the same thing with the same ruler, and the uncertainty could be different in both cases. Dec 30, 2015 at 20:20
• @ja72 - the truth is that Epimenides was right - he just didn't have the theoretical framework to understand that Cretans exist in a Schroedinger's-cat-like quantum state where neither truth or untruth can be determined without actually observing the Cretan - which just annoys them, because they don't like being stared at. (I suppose this begs the question of "How many quantum Cretans can dance of the head of a quantum pin?", but I ain't goin' there! :-) Dec 31, 2015 at 2:54

First - the size of the smallest division on a meter ruler need not be one mm. I have a ruler that only goes down to half cm divisions, and I have one that gives half mm divisions.

Second - a ruler may not be accurate to the nearest division. Wooden rulers in particular will grow and shrink with humidity, they can become bent, and they may have been poorly constructed to begin with. Metal rulers tend to be better in this regard.

Third - your ability to align the ruler with the thing you are measuring. Parallax error can come into play (more so for thicker rulers), as well as a "zero" error: does the end of the ruler really correspond to zero? Is the end fully straight, or worn? Is the ruler accurately aligned with the direction of the thing you are measuring?

Example of two rulers that don't agree on "zero" (by about 1.2 mm) - note also the effect of parallax, where the line of 1" aligns exactly, but the 0.5" and 1.5" lines seem to be shifted; this is due to the relatively close distance of the camera to the ruler, and the magnifying effect this has on the metal ruler relative to the wooden one behind it:

All these factors matter in determining the error of your measurement. But if you are just interested in quoting the number you read off your ruler (assuming it is marked in mm) and you thought the nearest value was 345 mm, then you would have to ask yourself - could it have been 346? If your measurement was "almost half way" between two values the answer is clearly "yes", and you can see you would be wrong to say +- 0.5 mm; this is why one would commonly quote the error due to the device as one unit of the least significant measurement. But note that other factors may contribute additional error.

• +1. Re "if… quoting the number you read… (assuming it is marked in mm) and you thought the nearest value was 345 mm, then… ask… could it have been 346? If your measurement was 'almost half way' between two values the answer is clearly 'yes', and you can see you would be wrong to say +- 0.5 mm": Right, but that's only because you're choosing imprecision. If instead you say exactly what you think you see (e.g. "345.4 mm" even though the lines are 1 mm apart), you won't be off by more than 0.5 mm (for this reason). But you need to drop a digit after e.g. multiplying that measurement. Dec 30, 2015 at 18:49
• @msh210 while it is possible to estimate subdivisions, you have to be careful about interpretation of the error: was the ruler intended to be that accurate? In other words is the scale sufficiently linear to allow that interpolation? Usually the answer is "no". 1 mm on 1 m is 0.1% which is actually rather good... Dec 30, 2015 at 19:06
• Careful: you're mixing accuracy and precision there. A ruler which has 0.1mm gradations will give you an answer with precision of 0.05mm but the entire ruler might only be accurate to 20% if it was built badly. Dec 30, 2015 at 21:02
• @CarlWitthoft I understand that. My point was that a ruler with 0.1 mm graduations is likely to be no more accurate than 0.1 mm - and possibly less. Which is why I am advocating against trying to estimate subdivisions. Dec 30, 2015 at 21:41
• Fair enough!ONEONEONE (needed more chars to be able to post) Dec 30, 2015 at 22:18

Indeed, the uncertainty associated with each reading on your meter rule would carry an uncertainty of 0.5mm. However, in order to measure the length of anything, you would really need to make two readings: one at each end of the object under measurement. Even if you start from the "zero" mark on your rule, it is really a reading of 0.0 cm by you, so it too carries some undertainty. Then I am sure your physics class on uncertainty would have taught you how to combine uncertainties: in this case because you subtract two quantities, their absolute uncertainties add - Hence you get an uncertainty of 1mm.

The above argument is the theoretical explanation for the apparent inconsistency you raised in your question (i.e. whole division or half division to quote as uncertainty?) Meaning: this probably would be what most marking schemes of physics exams expect from you when they ask about it.

In reality of course, things are quite complicated, and I completely agree with the points raised by Floris. You should certainly take into account those practical considerations, if you are concerned about what is the real uncertainty of your actual measurement in reality in an experiment, not what you should write on the answer script of your exams.

• This was such a beautiful reply. Most people just think they are taking only one measurement, but actually as you pointed out so well, they are taking two readings when making a measurement. Apr 30 at 17:18