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Non-applicability of uniform distribution formulae?

  1. Oct 20, 2012 #1
    I have tried finding a table's length using two methods. Both should be applicable, as using a ruler implies uniform distribution, does it not? The first method involved calculating the mean length from a set of measurements, finding the variance, and substituting it in the formula for the total uncertainty, namely sqrt((variance/N) + (ruler's resolution)^2).
    The second method involved using the formulae for uniform distribution to find the mean and deviation.
    The results were different. I am 100% certain the first method is correct. But why would the uniform distribution formulae not be applicable in this case?
     
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  3. Oct 20, 2012 #2

    HallsofIvy

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    No, "using a ruler" does NOT imply uniform distribution. The error in measurement will be normally distributed around 0.
     
  4. Oct 20, 2012 #3
    I wondered about that, but is that really true? Why would it be normally distributed?

    I thought peripatein meant that the error will be uniformly distributed between 0 and the size of the smallest gradation in the ruler (or something like that), but I don't know if that is true.
     
  5. Oct 20, 2012 #4
    Are you more likely to be slightly off, or to be off by the entire length of that "smallest gradation"? Surely one would be more noticeable, and so more easily correctible than the other.
     
  6. Oct 20, 2012 #5

    mfb

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    For a specific table with a specific length? I would not expect that.

    The uncertainty depends on the way you record your readings.
    The variance in your data sample might contain a part of the reading uncertainty.
     
  7. Oct 20, 2012 #6

    haruspex

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    It's not clear why there are multiple readings. If it's just a question of rounding to nearest on the marked scale, it would be a uniform distribution (in the sense of samples being the errors in measurements of many different tables), but all measurements between the same two points on the same table should produce the same answer.
    If measurements between the same two points are producing different answers then there is something else going on. E.g. the positions of the reader's eye might introduce parallax error. In this case, for a given table, you would see something closer to a normal distribution (the variance depending on the inherent rounding error); taken over multiple tables this distribution will be combined with the uniform one, rounding off the shoulders.
    Or are multiple measurements taken along the same table at different offsets from the side?
     
  8. Oct 20, 2012 #7
    So under what conditions is uniform distribution applicable? What are its requirements, aside equal probability in the range [a, b]? I found something related to minimum bounding limits, but am uncertain whether it actually means that my range, say [a, b], must also be the minimum and maximum values the parameter could get.
    Please explain.
     
  9. Oct 20, 2012 #8

    haruspex

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    I'll find it much easier to explain if you can first clarify the measurement process. Do you make essentially the same measurement multiple times? Measure the same table's length along different paths? Or..?
     
  10. Oct 20, 2012 #9
    Haruspex,
    (A) The question I am asked to provide an answer for is general - when in the lab would uniform distribution may be applied/used? Under what conditions and in what type of measurements?

    (B) Regarding the table's length, it is measured five times using a ruler of a given resolution.

    May you kindly provide a clear explanation to both A and B?
     
  11. Oct 20, 2012 #10

    mfb

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    Digital displays of measurement tools are a common example. The measured value can be anything which gets rounded to the value you see.

    At the same position, with the same person? I would not expect different values in that case, and assign ~half of the scale as uncertainty (and do not care about the distribution) if that is true. Otherwise, there is something odd going on.
     
  12. Oct 20, 2012 #11

    haruspex

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    As a general answer, I'd say "when the possible error is known to lie in a specific range, with no particular reason to favour any part of that range". As an example, rounding error in a measurement sounds a reasonable answer, but need to be careful what this means. If you carry out exactly the same measurement multiple times there's no reason why the rounding would be any different, so you'd expect to keep getting the same answer. Any variation in the answer would be due to or influenced by other random elements which could have a different distribution. The sense in which the measurement error is uniform is that if you were to conduct the same measurement process on many different samples (with a wide variety of different lengths - rather wider than the error range) and could determine the error in each such measurement you would expect to see a uniform distribution.
    OK, but is each measurement between the same two points, or along parallel lines along the length? Either way, any variation in reading will be a result of the measurement resolution combined with another source of variation: eye parallax, variation in actual length across the table, variation in orientation of the line of measurement... Each of these may have a different distribution, and I doubt any of them would be uniform.
    The shape of distribution of measurements for a given table would depend on these influences; the resolution error would affect only its parameters - the closer the actual length to a gradation length the less the variance in the measurements.
    Oh, and of course the set of errors for a given actual entity would be discrete, not continuous.
     
  13. Oct 20, 2012 #12

    chiro

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    Hey peripatein.

    Although error terms do not have to be Normally distributed, you might want to consider how likely it is to get an error of say +30 as opposed to +3 or +0.3 from the true result.

    Following a uniform distribution for the errors means that getting an error in (+10,+11) is the same as in (+0,+1) and this is just not consistent with how errors actually come in and how measurements are taken.

    It makes a lot more sense to get most of the errors around the true value where they have some kind of decay from that true value.

    Also if you want to understand this statistically, you may want to consider the distribution of the sample mean with respect to the population mean and you'll find that under the Central Limit Theorem, regardless of the true population distribution if you deal with I.I.D samples then the distribution of the sample mean will have a normal distribution with specific parameters.

    So if you are looking at the true value of the measurement as the population mean then the distribution of this real value will be a normal distribution in th asymptotic limit (again look at the CLT).
     
  14. Oct 20, 2012 #13

    haruspex

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    Depends. If you measure a whole series of different objects, varying over a wide range of actual lengths, and most of the error comes from the fact that you are rounding to the nearest cm, then it will be pretty much a uniform distribution.
    Although this doesn't match the actual measurement procedure in the OP (namely, making the 'same' measurement multiple times), it would serve the OP's true objective: finding an example of a uniform distribution that might arise in a lab context.
     
  15. Oct 20, 2012 #14

    chiro

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    I see what you are saying, but it just wouldn't make sense to have extreme values be the same as values closer to the true value with regards to probability.

    Maybe you could have some kind of "stair-case" effect with regard to the marks on your measuring device (like your ruler or some other metric instrument), but it just wouldn't make sense to have a true uniform distribution if you did allow non-zero probability past the resolution of your instrument (so if your resolution is 0.1 and you got non-zero probability for say +1.0, then I would expect this is a lot lower then the +- 0.1 region for the residual).
     
  16. Oct 21, 2012 #15

    haruspex

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    No, it wouldn't go past that. If you round to the nearest cm then the error has uniform distribution over -5mm to +5mm. If you also allow, say, a little parallax error then the corners would be rounded off slightly, maybe uniform -4mm to +4mm and falling off to zero outside ±6mm.
     
  17. Oct 21, 2012 #16
    I thank you sincerely for all your replies!
     
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