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I was able to calculate the sample size using two different effect sizes 0.3 and 0.5 and each gave me different sample sizes. 0.3 gave me 82 and 0.5 gave me 26. I have 45 students in that gradelevel. Does this mean that I am able to use these number of students. What exactly is effect size and why does it affect the sample size. Also, does anyone know how I should write this into my proposal. Thank you very much to Stephan Kolassa and Dandar, their answer to my first question helped me get here.

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Why are you calculating sample size if you only have 45 students? Why not use all 45 students and calculate the power of your test? –  AdamO Jan 22 '13 at 15:18
Because my professor said I have to. I told him I am using all the students, he said I have to perform a power analysis to determine if the number of students I have will be enough for the study. All the help I could get will be appreciated. –  Fran Jan 22 '13 at 17:20
Are you sure he understands your problem statement correctly? Usually if you have an available sample size, and are able to assume some effect size (measurable difference in outcomes compared treated to control individuals), you can directly calculate the power of the t-test and report that. –  AdamO Jan 22 '13 at 18:43
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1 Answer

In your described case, effect size is a statistical quantity explaining how big the differences between your two groups is relative to the amount of variability in the sample overall. There are two types of effect size usually used in relationship to t-tests are of one of two general families. One is Cohen's d (and includes Hedge's g) and the other is r (the Pearson product moment correlation). The precise details of how to move from a given effect size to determine power will depend on which family of effect size you have. That is, it matters whether your .3 and .5 are r (Pearson-type) or d.

As a general case consider that your $t = \frac{r}{\sqrt{1-r^2}}*\sqrt{df}$. That is to say that the magnitude of your test statistic is a multiple of your effect size and some scaling factor that reflects your sample size. So your sample size does not affect your effect size. Collectively your effect size and your sample size influence the magnitude of your test statistic. Larger test statistics are more likely to exceed the critical value for that test statistic. So, you are more likely to reject the null hypothesis as a consequence of the data you have (as opposed a decision you made, e.g. a change in $\alpha$). Therefore, you will have greater power for any increase in effect size or sample size. As a designer of the experiment you have only limited control over effect size, however you have some control over sample size.

Correct me if I'm wrong, but when you say that you "calculated the sample size given two different effect sizes", given the context of your last question, I take it to mean that you conducted a power analysis for those two effect sizes.

You can go ahead and use your full set of 45 students (so long as other experimental concerns, e.g. counterbalancing, don't preclude it). Given the information you've provided and the above assumption, it seems likely that you'll be able to obtain sufficient power (as specified by your earlier power analysis) so long as the true effect is not too much smaller than .5. To know exactly what effect size at which you will have a given power, you'll have to do another power analysis using your sample size of 45.

The details of how to write this into your proposal go beyond the scope of this website. I'd recommend talking to your colleagues and advisor to see if you can look through other sample proposals and use them as a guide.

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Thank you very much. Yes I conducted a power analysis for those effect sizes. –  Fran Jan 22 '13 at 17:25
If this answer covers your question completely you could mark this answer as accepted. –  rpierce Jan 31 '13 at 1:16
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