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Hypothesis testing for std. deviation (SD)

  1. Dec 24, 2013 #1

    What I know: In a hypothesis test for the mean, we compare a sample mean with a hypothetical sampling distribution of means. And depending on how far it is away from the mean of the sampling distribution, we attribute it the probability of getting that value purely by chance.

    What I don't understand - In a hypothesis test for the SD, why don't we compare the sample SD with a sampling distribution of SDs? (instead, I have seen people using the the Chi sq. distribution)

    As per the applet on the following web page, even the sampling distribution of SDs appears normally distributed around the population SD value. So why is it not used?
    [/PLAIN] [Broken]
    http://www.stat.tamu.edu/~west/ph/sampledist.html[/URL] [Broken]

    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Dec 28, 2013 #2

    Stephen Tashi

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    If you did that, what hypothesis would you be testing?

    It isn't clear what hypothesis you are testing. What specific problem are you talking about that was solved using the Chi square distribution?

    How did you use that applet to create a histogram of sample standard deviations?
    Last edited by a moderator: May 6, 2017
  4. Dec 29, 2013 #3
    Thanks Stephen,
    I am talking about the hypothesis test used to verify the standard deviation of a population.
    Here is one example.

    My question is why don't we use a sampling distribution of standard deviations (SD) to compare the sample SD, as we do for other statistics like mean or median. Why do we need to use the Chi Sq. statistic?
  5. Dec 30, 2013 #4

    Stephen Tashi

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    If f(x) is a 1-to-1 function and X is a random variable then the probability of the event X < v is the same as the probability of the event f(X) < f(v). If you are doing a hypothesis test about X and the distribution of f(X) is easiest to use then its simpler to formulate the test as a hypothesis about f(X). The sample standard deviation and the sample variance are related by a 1-to-1 function, so , when it's simpler, we can use the sample variance to test hypotheses about the sample standard deviation.

    This PDF suggests that when sampling from normal populations we might also use the distribution of the standard deviation directly:

    Different families of population distributions can have different families of sampling distributions for their sample variances and sample standard deviations. Normally distributed populations have chi-squared distributions for their sample variances. Other families of distributions may have sample variances that are not chi-squared. (So one cannot speak of "the" distribution of the sample standard deviation as if there was some type of distribution for it that applies to all situations.)
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