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Suppose that I have a vector X of 50 numbers between 1 and 10. Then I calculate the variance of X. Let us say it turns out to be 8, that is, VAR(X) is equal to 8.

Is it meaningful to ask the statistical significance of this result? If yes, how can I calculate it?

Note: Although Matlab calculates statistical significance of correlation coefficient, it does not calculate the statistical significance of variance. This led me to think that may be it is not meaningful to ask the statistical significance of variance. However, it seems to me a meaningful question because for a list of uniformly distributed random numbers with a certain range some variances are quite unlikely.

Thanks. Ahmet

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    $\begingroup$ I think that rather than a hypothesis testing problem, this is an estimation problem, unless you have an explicit hypothesis about the variance of the random variable (not to be confounded with the sample variance). $\endgroup$
    – user10525
    Commented May 12, 2012 at 11:36

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Statistical significance is always measured in relation to a hypothesis. In the case of the correlation coefficient, Matlab reports significance (a p-value) in relation to the hypothesis that the correlation is zero (and it uses the assumption of normality - not uniformity - for the calculations). A low p-value is interpreted as evidence against the hypothesis.

In this case, you can talk about significance if you have a hypothesis (eg $Var(X)$>5). Note that the hypothesis is a statement about the underlying population (a random variable) and not about the sample.

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  • $\begingroup$ I think this is probably the right answer as Mans T has formulated this about the variance of a population from which the vector X is a sample. I find like both Mans T and procrastinator that a population and a sampling mechanism are not specified. The question is posed by saying "Is it meaningful to say that this is a significant result?" That is vague because Var(X) is not related to an inference problem and nothing is said about whether or not the components of X are independent. $\endgroup$
    – Michael R. Chernick
    Commented May 12, 2012 at 20:19
  • $\begingroup$ Man T assumes these things and if that is done the fact thatthe sample variance is 8 can be used to test a hypothesis about the population variance or obtain a confidence interval for it. $\endgroup$
    – Michael R. Chernick
    Commented May 12, 2012 at 20:20

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