Calculate the percentile from a T-score [duplicate]

Question

Is it possible to calculate the percentile from a T-score?

If yes, what formula can be used for it, and what other variables (e.g. standard deviation, degrees of freedom, etc.) are needed for it?

Answer 1

It is possible to calculate the percentile given a t-statistic and the degrees of freedom. This is the CDF at the given t-value.

The formula for this is as follows, where $\nu$ is the degrees of freedom, $t^{2} < \nu $, and $_{2}F_{1}$ is a hypergeometric function.

$\int^{t}_{-\inf} f(u) du = \frac{1}{2} + t [ \frac{\Gamma(\frac{\nu + 1}{2})}{\sqrt{\pi\nu}\Gamma(\frac{\nu}{2})}] [ _{2}F_{1}(\frac{1}{2}, \frac{\nu + 1}{2}; \frac{3}{2}; -\frac{t^2}{\nu})]$

This can be easily done in R using pt().

Answer 2

Yes - to do so would require a complicated calculus calculation, however.

Textbooks usually include a table of t-values and their corresponding approximate percentiles based on the degrees of freedom.

There are also many free web-based technologies that can do this quickly, such as

https://www.t-to-percentile.uk