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Use of σ in quoting measurement accuracy

  1. Oct 2, 2008 #1

    cepheid

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    I know that σ is the symbol typically used for standard deviation, but what does the use of σ mean in these contexts?

    ex. 1: "[our fancy new instrument] ... allows for the identification of > 500 sources at greater than 10-sigma"

    ex. 2: "These estimates assume ... a 1σ polarization uncertainty P = 1%.

    Please note that if the symbol doesn't show up for you, it is supposed to be the letter sigma.
     
  2. jcsd
  3. Oct 2, 2008 #2

    jtbell

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    It's the standard deviation of a hypothetical ensemble of measurements of the same quantity, which differ only in random experimental errors, and are assumed to be distributed according to a Gaussian probability distribution.

    An interval of [itex]\pm \sigma[/itex] around the ideal mean value contains about 63% of the hypothetical ensemble, and [itex]\pm 5 \sigma[/itex] gets you up to about 95%, if I remember correctly.
     
  4. Oct 2, 2008 #3

    brewnog

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    Indeed, so in terms of manufacturing defects, one sigma corresponds to the equivalent of 690,000 parts per million failures, four sigma 6,210 ppm, and six sigma 3.4 ppm.
     
  5. Oct 2, 2008 #4

    stewartcs

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    It's more like 99.99994% for [itex]\pm 5 \sigma[/itex].

    http://en.wikipedia.org/wiki/68-95-99.7_rule

    CS
     
  6. Oct 2, 2008 #5

    cepheid

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    I'm still not sure I understand. How do you construct this hypothetical ensemble? And are you saying that each measurement in the ensemble is a single number that differs from the others only *due to* random experimental errors? Most importantly, HOW does this apply to the examples above? I have another one that says:

    " [The instrument] has detected > 80 sources per square degree (at 5-sigma)." What does this mean? Or does this have nothing to do with the specific number 80 and more to do with the definition of what a source is as compared to the background (this is in the context of submillimetre astronomy)?
     
  7. Oct 2, 2008 #6

    stewartcs

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    In this example I believe the statement means that your fancy new instrument will detect source outliers up to 10 sigma from the mean (i.e. sources with a z-score of [itex]\pm 10[/itex]). Hard to say without knowing more about the application though.

    CS
     
  8. Oct 2, 2008 #7

    stewartcs

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    I would interpret that as meaning that instrument detected >80 sources per square degree that were within [itex] \pm 5 \sigma [/itex] of the mean. In other words the detected value, y, was within 5 sigma of the mean value. This y value was detected >80 times.

    CS
     
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