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I realize that this question is most likely not too interesting and challenging to discuss here, it's rather a matter of different notations, but I would like to clarify that if possible.

When conducting Z/t tests, I got used to the convention that $z_{\frac{\alpha}{2}}$ is such a value that $P(Z>z_\frac{\alpha}{2}) = \frac{\alpha}{2}$. It also follows that $-z_{\frac{\alpha}{2}} = z_{1-\frac{\alpha}{2}}$. The same is true for the t-test and is just a matter of swapping the letters.

Visually speaking, this is just such a value that the area to the right equals $\frac{\alpha}{2}$.

I noticed, however, that sometimes these cutoff values are introduced as the inverse CDFs and then the order of the values on the X-axis is reversed, of course.

What is interesting, some authors mix the two and then their notation gets a bit confusing. See, for example, below:

https://online.stat.psu.edu/stat500/lesson/6b/6b.1

$t_{0.005}$ is positive here in the solution, but the illustrations place $t_{1-\alpha}$ on the right of the X-axis, which doesn't make a lot of sense to me unless there is this notation failure.

What is the more widely accepted notation? Well, I understand that this is not a big deal at all and it's always clear from context which notation the author prefers, but I just stumbled upon the online book above and this inconsistency made me wonder.

Thanks!

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    $\begingroup$ I get the feeling that half of statisticians think each way is right and think the other half is fools for believing otherwise. $\endgroup$ – Dave Mar 22 at 14:12
  • $\begingroup$ It just surprises me that there can be a second opinion on a thing like this one. In the book above, this duality only resulted in somewhat confusing illustrations. Honestly speaking, currently they are just wrong $\endgroup$ – Don Draper Mar 22 at 14:14
  • $\begingroup$ This is indeed a big problem in statistics, especially for students. I am a strong proponent of a consistent notation for quantiles of the (continuous) distribution of $X$: $q_{\alpha}$ is always the value so that $\alpha = \text{Prob}[X\leqslant q_{\alpha}]$. Or for other distributions: $q_{\alpha}=F^{-1}(\alpha)$ for some suitably defined inverse of the cumulative distribution function $F(x)$ of $X$. $\endgroup$ – StijnDeVuyst Mar 22 at 15:45
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Traditionally, printed tables for distributions t, chi-squared, and F have used the subscript notation you show to indicate right-tail probability. However, software such as R often uses a quantile function (inverse CDF) to find the value with a certain percentage below.

For example, in R, code qt(.975, 24) returns $2.063899,$ called 'quantile 0.975'. You can probably find something like $2.064$ on line 24 of your t table under header $0.025$ (sometimes called a 'percentage point').

As far as I know, textbook authors have not yet developed a standard notation for this value that is a good match for both printed tables and software output.

I am not seriously proposing this, but maybe it should be something like $t_{0.025} = t_{24; 0.025} \approx 2.064$ and ${}_{0.975}t = {}_{0.975}t_{24} = 2.063899.$

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