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

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D.1 Terminology
D.2 Identification of uncertainty components
D.3 Equation (A-2)
D.4 Measurand defined by the measurement method; characterization of test methods; simple calibration
D.5 tp and the quantile t1-α
D.6 Uncertainty and units of the SI; proper use of the SI and quantity and unit symbols
D.7 References

D.5.1 As pointed out in the Guide, the t-distribution is often tabulated in quantiles. That is, values of the quantile t1-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 tp of this Technical Note and of the Guide and t1-α are related by p = 1 - 2α. For example, the value of the quantile t0.975, for which 1 - α = 0.975 and α = 0.025, is the same as tp(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 tp(ν) is called the "quantile of order p of the t variable with ν degrees of freedom." Clearly, the values of tp(ν) defined in this way differ from the values of tp(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 tp(ν) with some care.

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Created November 6, 2015, Updated November 25, 2019