# Sample size calculation for correlation study

I am planning a study to assess the correlation of perceived stress in medical students and their academic performance. After a literature review, it appears that one previously done recent study has reported a correlation coefficient value of 0.47. But when I use this value to calculate my required sample size through some online calculators, my N value comes out to be only 45.

Not sure if I'm doing something wrong, can someone kindly point me out to a standard formula for sample size calculation in correlational studies?

• What alpha and power did you specify? And you testing 0.47 against 0? Or some other value? – Penguin_Knight May 28 '15 at 18:09

I would be wary of using the published value of r, 0.47, as the basis for your sample size calculations.

What if the true correlation is say 0.25? If that were the true population correlation, would you want your study to find a "significant" result? If so, compute the sample size for r = 0.25 (or even smaller). More generally, try to find the sample size that can detect (with reasonable power) the smallest effect (correlation coefficient for this example) you would care about.

To run the power analysis, you need to know three of the four to calculate the last one:

1. Number of observations
2. Effect size (correlation coefficient)
3. Significance level
4. Power

You have stated the effect size (correlation coefficient) in your question to be 0.47. Next, let's decide to use the conventional significance level $\alpha = 0.05$. A typical choice for the power is $p = 0.8$. Using the library pwr in R, we get

> pwr.r.test(n=NULL, r=0.47, sig.level=0.05, power=0.80, alternative="two.sided")

approximate correlation power calculation (arctangh transformation)

n = 32.38727
r = 0.47
sig.level = 0.05
power = 0.8
alternative = two.sided


Alternatively, we could set the threshold upper:

> pwr.r.test(n=NULL, r=0.47, sig.level=0.05, power=0.95, alternative="two.sided")

approximate correlation power calculation (arctangh transformation)

n = 52.12905
r = 0.47
sig.level = 0.05
power = 0.95
alternative = two.sided


You don't need very large sample size because $r=0.47$ is already quite strong relationship.

Here is another example using GPower expressed in graph, with sample size versus power:

A sample of 45 seems to be reasonable, with power > 0.9.