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Standard Deviation Conceptual [intro. Stats]

  1. Feb 5, 2014 #1

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    Hello, PF!

    [My question pertains to a non-rigorous, undergraduate introductory Probability and Statistics course. I'm no math major, so please correct me if I've mishandled any terms or concepts as I try to express myself. I'm always eager to learn!]

    * * *​

    In a discussion of the standard deviation of a sample in relation to the 68-95-99.7 rule, the following "conceptual" example was given—or rather, made up on the spot—by our professor:

    Assume [itex]\bar{x}=50 \%[/itex] and [itex]s=20 \%[/itex] for test scores (in units of percent correct), and assume that the sample represents the normal distribution (symmetrical and bell-shaped) of a test where no test score range below [itex]0 \%[/itex] and none above [itex]100 \% [/itex] (sorry, fellas, no extra credit).

    It occurred to me that any score beyond [itex]2.5 [/itex] standard deviations would be a score of more than [itex]100 \%[/itex] or less than [itex]0 \%[/itex]. According to the three-sigma rule, this would still only encompass approximately [itex]98.7\%[/itex] of the scores meaning that approximately [itex]1.3\%[/itex] of the scores fall outside this possible range.

    My question:

    Is the above example even possible given the "parameters" (limits?—I can't find the right word) [itex]{0 \%}≤x_i≤{100 \%}[/itex]?

    And

    Extrapolating this question to the overall concept, can any standard deviation [itex]s[/itex] of a normal distribution ever exceed the possible range of data points/values within that distribution?

    My guess is that this was simple oversight and an error on the part of my professor.

    Thank you!
     
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  3. Feb 5, 2014 #2

    Stephen Tashi

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    A random variable with a bounded range (such as 0 to 100) is not actually normally distributed. However, a normal distribution can often be used to get approximate answers to questions about such a random variable. Many problems in textbooks expect students to make such an approximation. I'd call this type of approximation a tradition, not an oversight.
     
  4. Feb 5, 2014 #3

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    Ah, bounded was the word I was looking for! Thank you, Stephen.


    Would the example then be considered a Truncated normal distribution ?

    If this is the case, what would 2.5 or 3 standard deviations imply in relation to the three-sigma rule when the values simply cannot extend beyond the boundaries? Does the three-sigma/68-95-99.7 rule simply not apply to this (and other truncated distributions); i.e., this example would have a different set of probabilities in relation to the various standard deviations: [tex]Pr(\bar{x} - 2.5s ≤x≤ \bar{x} + 2.5s) = 1.00 \ ?[/tex]



    Thank you.
     
  5. Feb 6, 2014 #4

    Stephen Tashi

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    The example used the normal distribution as an approximation. If you want to make a different example, you could use a different distribution. A truncated normal distribution is but one example of what could be used.


    In general, a truncated normal distribution would have a different set of probability values for such an interval than a non-truncated normal distribution. (Keep in mind that a truncated normal distribution has a different "s" than the normal distribution that was truncated.)
     
  6. Feb 7, 2014 #5
    There is no such thing as a normal distribution in the real world. It is a mathematical ideal. Quite often there are deviations in the tails. Often it is close enough for jazz.
     
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