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I was studying calculating power and sample sizes and i came across this paragraph:

Calculating Power for comparing two proportions has the same idea as with comparing means, except that no standard deviation estimate is necessary (as the standard deviation of a proportion is a function of the proportion itself)

what does the sentence between the parenthesis mean?

Elmahy
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1 Answers1

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The mean of a proportion is p, then the variance is $p(1-p)$. The standard deviation is then the square root.

This clearly shows what is meant by a "function of itself". Once you have the proportion you also have the variance.

Using the sample analogy principle it can, with ease, be shown that sample variance (estimate) is: $\hat{p}(1-\hat{p})$.

The version I had (pre edit) was divided by n. This is interesting if you wish to do testing, or calculate a confidence interval

Repmat
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    I believe the quote refers to the variance of a sample proportion; the variance of a sample proportion is $p(1-p)/n$. – Glen_b Jan 12 '16 at 13:23
  • Yeah I suppose, still same principle – Repmat Jan 12 '16 at 13:30
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    The variance of a sample proportion is $p(1-p)/n$, which depends on $p$, not $\hat{p}$. The version with $\hat{p}$ is simply a sample estimate of the "true" variance of the estimate. If you're going to keep "the variance is $p(1-p)$" in your first sentence you'll need to clarify what you intend it to be the variance *of*. – Glen_b Jan 12 '16 at 14:51
  • are you sure that it can be shown that the sample variance of a proportion is $\frac{\hat{p} (1-\hat{p})}{n}$ ? With $\hat{p}$ the estimated proportion ? I think I must agree with @Glen_b (+1) –  Jan 12 '16 at 17:11
  • The relation given is not a completely general relation for any kind of distribution for proportions, but it is the relation for a binomial. Another interesting case is the beta distribution (see e.g. http://stats.stackexchange.com/questions/12232/calculating-the-parameters-of-a-beta-distribution-using-the-mean-and-variance). More positively, there must be a qualitative dependence of variance on the mean for proportions, as the variance will be zero if the mean proportion is 0 or 1 but will be positive otherwise, apart from degenerate cases. – Nick Cox Jan 12 '16 at 19:04