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# Critical Correlation Calculator

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

Type the sample size ($$n$$):

Type the significance level ($$\alpha)$$:

Type of tail:

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 $$\Box$$

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