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In my lab we have to calculate uncertainties in measuring devices and we are given a document explaining different uncertainties for different tools (rulers, digital stopwatches, etc.) I'm just having a difficult time understanding what the uncertainty for a measuring tape is. In the document it is explained as "the uncertainty for an analogue device is half of the smallest graduation". Does this mean on a measuring tape is cm? Thank you.

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  • $\begingroup$ The trouble is we weren't given an actual measuring tape. We were just given values. For example, one of the values was 14.7919 m. My confusion is that I'm not sure what kind of measuring tape is in meters and if the smallest graduations would be 0.1 cm. $\endgroup$ Sep 23 '20 at 15:07
  • $\begingroup$ Many of the questions and the answer are commenting on accuracy... that is not your question as I understand it. The question is about precision. If you did everything else right there would still be an uncertainty in your measurement which your document defines as half the smallest graduation. Do all of the example numbers have 4 digits to the right of the decimal? $\endgroup$
    – DrFalcon
    Sep 23 '20 at 21:25
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Most meter sticks have millimeter markings, so your guideline would suggest an uncertainty of half a millimeter.

There's actually a technique for getting a factor of ten better than the smallest division, which I learned in high school. Suppose you're measuring a location between analog marks labeled 3 and 4:

.      v  .
3         4

You can judge by eye that the v is clearly more than halfway along. Maybe it's not clear whether it's at or less than three-quarters of the way along, but it's probably not more than three-quarters of the way along. So call it $3.7\pm0.1$. If that seems too confident, call it $3.7\pm0.2$. I think you would agree that $4.0\pm0.5$ (your text's guidance) is unnecessarily cautious.

You've asked for "what the uncertainty is," and here I am talking to you about judgement, clarity, probability, confidence, and caution. Some of my students get upset when I do this. But the entire point of an uncertainty analysis is to permit a mathematical analysis of our subjective confidence in our result.

For instance, if you look at a typical wooden meter stick, the millimeter markings might be half a millimeter wide (that is, the smallest gradiation on the ruler is 50% ink). It's a lot less plausible that you could measure to a tenth of a millimeter if you're also trying to decide whether to use the front, middle, or back of your millimeter mark as a reference. Sometimes you want to use the end of the meter stick as your zero, but you discover that long use has worn the wood away from the zero end. Or that there's some brass ferrule of unknown thickness attached to the end to prevent such wear. Or sometimes (this one is fun) you'll take two "identical" meter sticks, touch the measuring surfaces against each other, and discover that the two sets of millimeter markings make a kind of moiré pattern, because not all of the millimeters on the sticks are the same width. In these cases you have to use your judgement about how much you trust the measurement you're making.

A great thing about statistics is that we know how repeated measurements should vary if we've been estimating our uncertainties correctly, which gives us the confidence to state whether a result is "wrong" because of mistakes we know we might have made, or because of new effects.

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