Source for sample size formula

Steve Simon


This page is currently being updated from the earlier version of my website. Sorry that it is not yet fully available.

Hello, I am looking at your page on sample size calculation, and I'm curious as to where you got the equation shown there:

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I can't seem to find that exact form in Cohen's book, not does it appear anywhere else that I've looked. Would you happen to know its original source?

I'm away from all my books for the time being, so I can only speculate. If you let the two standard deviations be equal, then the formula simplifies somewhat.

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I can't say for sure what Cohen's formula is, but I suspect that it assumes both variances are equal.

Some formulas will place the common standard deviation in the denominator of this equation, which yields

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The quantity

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is sometimes called the effect size, and is often called Cohen's d (notice the change from upper case to lower case).

Some formulas will substitute a t distribution for the z distribution although this requires iteration, as the degrees of freedom are dependent on the sample size. This is a slightly better approximation, but the best answer will come from the non-central t-distribution. You would have to rely on tables or software for any power or sample size calculation involving the non-central t-distribution..

As far as a source, I suspect you would find this formula in many textbooks. Perhaps Rosner would be a good source.

If there's a different formula from a definitive source, I would not be at all offended if you used it instead.

You can find an earlier version of this page on my original website.