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Math Help - Rounding off results / significant figures

  1. #1
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    Rounding off results / significant figures

    Hi!

    Finally decided to seek for online help after repeatedly banging my head to the wall with an annoying problem. It's a fault in my thinking but I just can't seem to be able to shake it off.

    So, I know what decimals and sig figs are and I'm very familiar with the general rounding rules of +/-/*/: -calculations with measured values. It goes brilliantly as long as I don't think about it. But sometime ago I stopped to think about the difference of accuracy and precision and there I got lost. See, in my native language those two are rarely separated when it comes to the rounding rules. The same word is basically used for both although they are very different concepts. Anyhow, I'm fine with precision - no problem there. But does anyone have a magic trick to make my brain understand accuracy? I'll elaborate:

    1) A = 12,5 cm * 0,7 cm = 8,75 cm2 9 cm2 , because 0,7 cm has less sig figs as 12,5 cm
    The precision of values is the same, but why is 0,7 cm considered to be the less accurate value so that the sig fig rounding rule can be applied? Even my teacher friends were baffled by my problem, because they'd just accepted the rule without ever thinking.

    2) A = 0,123 m * 1200 m = 147,6 m2 150 m2 , because 1200 m has less sig figs as 0,123 m
    Again I fail to understand the accuracy aspect. What makes 1200 m the less accurate value? If you answer me, that it's because of the sig figs, I'll just carry on banging my head to the wall...

    Cheers in advance,
    F


    PS: Pardon my non-fluent English.
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  2. #2
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    Re: Rounding off results / significant figures

    AFAIK, accuracy and precision have to do with measurement.

    From Kirkup's Experimental Methods book, which we used in undergrad physics:

    Accurate - close to the true value but, unless given, the uncertainty could be of any magnitude.
    Precise - value has a small uncertainty, but this does not mean that it is close to the true value.
    Accurate and Precise - close to true value and with a small uncertainty.

    In part 1) of your question, 0.7 has less significant figures than 12.5. Its accuracy would be determined by whether or not its measured value (0.7) is in fact close to its true value. The sigfigs inform the precision of your statement of the answer, not the accuracy of the original measurement or your finding ( i.e you state the answer to the same number of figures as the least precise measurement made.)
    So I could measure the length of a pendulum with a laser and time its swing with an atomic clock (i suppose!), which would be very precise, but end up finding g = 10.23456 \pm 0.00001 \text{ ms} ^{-2} which is not very accurate.

    I hope this is useful!
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  3. #3
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    Re: Rounding off results / significant figures

    Saying that your answer is, say, 7.135 implies that you believe the "true" result is between 7.1345 and 7.1355 while stating your answer as only 7.13 would imply that you believe the "true" result lies between 7.125 and 7.135. The first is more "precise" than the second since it covers a smaller region. However, that doesn't tell you that the answer is correct! "Accuracy" describes how close your answer is to the actual "correct" answer to the problem.

    An example I sometimes use for the difference between "precision" and "accuracy" is this: the Boy Scout requirements for the "archery" merit badge require that five shots be put into an area within a given radius- they do NOT require that the shots be close to the center of the target. That is, they are requiring "precision", not "accuracy".
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