In statistics, if you are given the correlation coefficient $r = 0.5$ and sample size $n = 10$, and if you're looking for $R^2$, this represents the coefficient of determination. The coefficient of determination is found by squaring the correlation coefficient $r$.

$R^2 = r^2$

Given: $r = 0.5$

Calculating $R^2$: $R^2 = (0.5)^2 = 0.25$

Thus: $R^2 = 0.25$

This means that 25% of the variability in the dependent variable is explained by the independent variable in this study.

Would you like further details, or do you have any additional questions?

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Here are some related questions for further exploration:

1. How is the coefficient of determination interpreted in regression analysis?
2. What does a higher or lower $R^2$ value imply about the strength of a relationship?
3. How do you test the significance of the correlation coefficient $r$?
4. How does the sample size $n$ impact the reliability of $r$ and $R^2$?
5. What are the assumptions underlying the use of the correlation coefficient?

Tip: Remember, the $R^2$ value does not imply causation—only association strength between variables.