The formula for the t-test on significance of the correlation coefficient is given as follows (e.g. https://newonlinecourses.science.psu.edu/stat501/node/259/):
$t^* = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$
How to derive this formula? It is different from the standard t-test formula given e.g. here: https://en.wikipedia.org/wiki/Student%27s_t-test, which is likely due to the fact that correlation coefficients are analyzed.
A t-test is a test on a statistic that has a t-distribution under the null hypothesis. A variable $Z$ has a t-distribution if it is obtained by dividing a Normally-distributed variable $X$ by a $\chi^2$-distributed variable $Y$. For the familiar t-test, $X$ is the sample mean of some IID data, which by the central limit theorem is Normally distributed, while $Y$ is the standard error of the mean, which has a $\chi^2$-distribution, and thus $X/Y$ follows a t-distribution.
For correlation coefficients, under the null-hypothesis that the population correlation coefficient equals 0, the sample correlation is approximately Normally distributed with standard error $SE(r)=\sqrt{\frac{1-r^2}{n-2}}$, and the standard error is again $\chi^2$-distributed. Thus, the t-statistic is obtained by dividing the sample correlation coefficient $r$ by this standard error: $$ t=\frac{r}{SE(r)}=\frac{r}{\sqrt{\frac{1-r^2}{n-2}}} =\frac{r\sqrt{n-2}}{\sqrt{1-r^2}} $$ Note that in both cases we get the t-statistic by dividing a Normally-distributed variable by its $\chi^2$-distributed standard error, and so they're actually really not that different.
if both are same the calculated values in both cases should be same. But in many cases seen are not same. to find out the significance for correlation, the correlation value is converted to t and finding corresponding value. r value calculated between two groups for each corresponding pair value are having any pattern with other. where as t test formula from means and sd or covariance of two group values
both cases purposes and results differ
