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    Ogre in the Playground
     
    BardGuy

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    Default statistics question on normal distribution & quartiles

    In the standard normal distribution (mean 0, standard deviation of 1), would we expect 25% of the data points to fall in each the quartiles (that is, below 25th percentile, 25th-49th, 50th-74th, and 75th-100th)?

    Would changing the standard deviation change what we expect?
    I'm thinking yes, but not sure how.

    Would changing the mean change what we expect?
    I'm thinking so, since all the data elements would shift. Do we need to know to mean to know how the standard deviation changing matters (if it does)?

    For the purposes of this question, assume the data is perfectly normal. That is, there is no deviation from normality due to being a random sample.

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    SwashbucklerGuy

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    Default Re: statistics question on normal distribution & quartiles

    I'm not entirely sure what you're asking, so I'm sorry if this response isn't helpful. Please elaborate and correct me if I've misread you. I think you're asking if changing the standard deviation will change the population of the quartiles (i.e. if changing the SD will make some quartiles hold more or less than 25% of the data).

    In the standard normal distribution (mean 0, standard deviation of 1), would we expect 25% of the data points to fall in each the quartiles (that is, below 25th percentile, 25th-49th, 50th-74th, and 75th-100th)?
    This is true, but "In the standard normal distribution" part is redundant. By definition, 25% of the data falls in each quartile. That's what the word "quartile" means. For a normal distribution, changing the standard deviation will change where precisely the quartiles lie, but it can't change the actual population of the quartiles, since that's established by definition.

    ETA: same goes for the mean; no possible change to the data set will change the proportional population of each quartile.
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    Lord Torath's Avatar

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    Default Re: statistics question on normal distribution & quartiles

    Changing the mean would shift the entire dataset, and would move the percentile breaks along with it. The average value in each quartile would change, but you'd still get 25% in the first quartile, 25% in the second, etc.
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    Ogre in the Playground
     
    BardGuy

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    Default Re: statistics question on normal distribution & quartiles

    Thank you.
    And, yes, I realize what I'm asking is fairly redundant.

    Just to be sure: regardless of what the standard deviation is, 25% of the values from a normal distribution would fall into the 1st quartile because that's what a first quartile is.

    I was trying to answer a question about if data likely came from a normal distribution with a certain mean and standard deviation, by comparing where the data fell in that distribution's quartiles. I thought the expected values were 25% in each quartile (since that's what the word means), but I was confused since that made it seem like knowing the mean and standard deviation was just extra info not needed for the problem.

    But I think part of it might be to check if we know we don't need the mean and standard deviation (sorta a trick question.)

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    Dwarf in the Playground
     
    MindFlayer

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    Default Re: statistics question on normal distribution & quartiles

    What the mean and standard deviation of the hypothesized normal distribution do for you is determine where its quartiles are. If you know the locations of the quartiles of a standard normal distribution, you can calculate the locations of the quartiles of any other normal distribution (with any mean and standard deviation).

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    Dwarf in the Playground
     
    SwashbucklerGuy

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    Default Re: statistics question on normal distribution & quartiles

    Quote Originally Posted by JeenLeen View Post
    Thank you.
    And, yes, I realize what I'm asking is fairly redundant.

    Just to be sure: regardless of what the standard deviation is, 25% of the values from a normal distribution would fall into the 1st quartile because that's what a first quartile is.

    I was trying to answer a question about if data likely came from a normal distribution with a certain mean and standard deviation, by comparing where the data fell in that distribution's quartiles. I thought the expected values were 25% in each quartile (since that's what the word means), but I was confused since that made it seem like knowing the mean and standard deviation was just extra info not needed for the problem.

    But I think part of it might be to check if we know we don't need the mean and standard deviation (sorta a trick question.)
    Sorry, I don't mean to be insulting when I say "redundant." I meant some of the information was redundant, not that you are being redundant.

    Where precisely the quartiles fall is well defined for any standard distribution. If you've studied Z scores*, the quartiles boundaries of a standard distribution will always fall at Z = -0.67, Z = 0 (the mean), and Z = 0.67. If that's where your quartiles are, then we'd say that your data is consistent with a standard distribution.

    If you don't know where your quartiles are, just that each quartile holds 25% of the population, then you really can't say anything about the data set without further analysis. The quartiles holding 25% doesn't tell you anything.


    * Z = (x - xmean)/SD
    Quote Originally Posted by crayzz
    That a given person is known for his sex appeal does not mean that he is only known for his sex appeal.
    Quote Originally Posted by jere7my
    For instance, I am also known for my humility.

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    Ogre in the Playground
     
    BardGuy

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    Default Re: statistics question on normal distribution & quartiles

    No offense taken, and thanks for the additional info.

    In my particular instance, I don't need to know where the quartiles are, since I'm just told the # of data points that fall into each quartile of the given normal distribution, to figure out if those data points come from that normal distribution.
    I mainly wanted to confirm that, in a normal distribution, 25% of the data points would be expected to be in each quartile, so in a sense I was confirming the definition of quartile.

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    Default Re: statistics question on normal distribution & quartiles

    For a true Normal distribution:
    The 25th percentile is the mean minus 0.6745 times the standard deviation (μ 0.6745 σ).
    The 75th percentile is the mean plus 0.6745 times the standard deviation (μ + 0.6745 σ).

    So for a standard Normal distribution, the 25th percentile is -0.6745, and the 75th percentile is 0.6745. Any time the mean and the standard deviation change, the 25th and 75th percentiles also change, exactly as those formulas indicate.

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