**Instructions:** Enter the sample size \(n\) and the significance level \(\alpha\) and the solver will compute the critical correlation coefficient \(r_c\).

#### How to Use this Critical Correlation Calculator

The significance of a sample correlation coefficient \(r\) is tested using the following t-statistic:

\[t = r \sqrt{\frac{n-2}{1-r^2}}\]For a given sample size \(n\), the number of degrees of freedom is \(df = n-2\), and then, a critical t-value for the given significance level \(\alpha\) and \(df\) can be found. Let us call this critical t-value \(t_c\). Using the expression of the t-statistic:

\[t_c = r \sqrt{\frac{n-2}{1-r^2}} = r \sqrt{\frac{df}{1-r^2}}\]and now if we solve for \(r\) we find that

\[r_c = \sqrt{\frac{\frac{t_c^2}{df}}{\frac{t_c^2}{df}+1}}\]and this value of \(r_c\) is the so called *critical correlation value* used to assess the significance of the sample correlation coefficient \(r\). These critical correlation values are usually found in specific tables.

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