NIST TN 1297: Appendix D5. tp and the quantile t(1-alpha)

D.5.1 As pointed out in the Guide, the t-distribution is often tabulated in quantiles. That is, values of the quantile t_1-α are given, where 1 - α denotes the cumulative probability and the relation $$1 - \alpha = \int_{-\infty}^{t_{1-\alpha}} f(t,\nu) {\rm d}t $$ defines the quantile, where f is the probability density function of t. Thus t_p of this Technical Note and of the Guide and t_1-α are related by p = 1 - 2α. For example, the value of the quantile t_0.975, for which 1 - α = 0.975 and α = 0.025, is the same as t_p(ν) for p = 0.95, It should be noted, however, that in reference [D.2] the symbol p is used for the cumulative probability 1 - α, and the resulting t_p(ν) is called the "quantile of order p of the t variable with ν degrees of freedom." Clearly, the values of t_p(ν) defined in this way differ from the values of t_p(ν) defined as in this Technical Note and in the Guide, and given in Table B.1 (which is of the same form as that given in reference [10]). Thus, one must use tables of tabulated values of t_p(ν) with some care.