$-z_\frac{\alpha}{2}$ and $z_\frac{\alpha}{2}$ in Confidence Intervals I was watching a Youtube video about Confidence Intervals and came across a slide that threw me:

I get that the blue region is $1-\alpha$ (often 0.95 in practice) while the bits to the edge are each $\frac{\alpha}{2}$ (often 0.025 in practice).
However, I don't understand the tick marks down the bottom. Why can't they instead be respectively $z_{\frac{\alpha}{2}}$ and $z_{1-\frac{\alpha}{2}}$, so that in the case of a 95% Confidence Interval they'd come out as $z_{0.025}$ and $z_{0.975}$. Those values come out to -1.96 and 1.96, as per the following R code
qnorm(0.025)
qnorm(0.975)

I must be missing something obvious because this video has many comments but nobody has previously tried to 'correct' it. It'd be much appreciated if someone could explain how the tick-marks in the video are in fact correct, and why my interpretation is wrong or not as accurate.
There is a related question, in which the poster says

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 $-z_{\frac{\alpha}{2}} = z_{1-\frac{\alpha}{2}}$

However, I still don't understand the logic of the convention, or why it should be preferred to just writing the tick marks as $z_{\frac{\alpha}{2}}$ and $z_{1-\frac{\alpha}{2}}$.
 A: I suspect the convention arose when people had to look up z-scores by hand in a printed table. But it's also a convention that makes the arithmetic & concepts slightly easier to work with. Nothing deeper than that.

*

*Why write in terms of $-z_{\alpha/2}$ instead of $z_{1-(\alpha/2)}$?
Since the standard Normal distribution is symmetric, it's faster to look up one value e.g. $z_{\alpha/2}$ and negate it to get the other value. As soon as you've found one value, it's a waste of time to also look up $z_{1-(\alpha/2)}$ in a table.
Also, conceptually, I find it more useful to think about a CI as "estimate plus/minus margin of error" rather than "estimate plus two arbitrary other numbers (of which one might be negative)."


*Why treat $z_{\alpha/2}$ as the upper $\alpha/2$ percentile, not the lower percentile?
In my experience, Intro Stats students seem to find it sliiiightly easier to think about "There's 2.5% in each tail, so let's look up the tail percentile that gives us a positive number" compared to thinking "...so let's look up the lower percentile, and then we'll have to negate a negative number."
A: What you asked demonstrates that you had a great understanding of the general concept "$\alpha$-quantile of a distribution $F$/a random variable $X$".  Tacitly, the notation $\xi_\alpha$ is used to denote the lower $\alpha$-quantile of $X$ such that
\begin{align}
\alpha = P(X \leq \xi_{\alpha}) = F(\xi_\alpha), \tag{1}
\end{align}
assuming $X$ is continuous$^\dagger$.
However, it is not uncommon at all to see $\xi_\alpha$ as upper $\alpha$-quantiles (typically of standard normal distribution, $t$-distributions and $\chi^2$ distributions) in the multiplier of a confidence interval or in the rejection region of a hypothesis testing problem, as in your post's example.  If $\xi_\alpha$ is an upper $\alpha$-quantile of $F$ or $X$, then
\begin{align}
\alpha = P(X > \xi_\alpha) = 1 - F(\xi_\alpha). \tag{2}
\end{align}
In my opinion, there are probably two (not-so-convincing) reasons of this "abuse of notation":

*

*As the screenshot in your post shows, if interpreting the notation "$z_{\alpha/2}$" as the upper $\alpha/2$-quantile, the subscript $\alpha/2$ has a 1-to-1 correspondence to the right-tail area under the density curve.  In contrast, if recognizing the same quantity as the lower $(1 - \alpha/2)$-quantile (note that for any distribution, its lower $(1 - p)$-quantile is always equal to its upper $p$-quantile by comparing $(1)$ and $(2)$) and using the (conventional notation) $z_{1 - \alpha/2}$ to denote it, then there seems no such direct correspondence between its subscript $1 - \alpha/2$ and a single area-under-curve with intuitive interpretation related to the confidence level (you need to add the left-tail region and the mid-range region together to get $1 - \alpha/2$).

*Compared with $\bar{X} \pm z_{1 - \alpha/2}\frac{\sigma}{\sqrt{n}}$, the version $\bar{X} \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$ has slightly less typing (which, hopefully, makes memorizing this form easier).

Again, I want to re-emphasize that these two reasons are quite superficial and it does not make much sense to consider one convention is more superior or more preferred than the other.
To summarize, denoting the $1 - \alpha$ confidence interval of $\mu$ by $\bar{X} \pm z_{1 - \alpha/2}\frac{\sigma}{\sqrt{n}}$ (where $z_{1 - \alpha/2}$ stands for the lower $(1 - \alpha/2)$-quantile of $N(0, 1)$) or $\bar{X} \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$ (where $z_{\alpha/2}$ stands for the upper $\alpha/2$-quantile of $N(0, 1)$) is merely a notational difference (that is, with the same data, both conventions give exactly the same numerical confidence interval). Each convention is widely seen in the statistical literature. Below I listed two prestigious references for each convention (it is easy to find more examples for each category):

*

*Books that use upper $\alpha/2$-quantiles in confidence intervals

*

*Statistical Models by A. C. Davison

*Design and Analysis of Experiments by D. Montgomery



*Books that use lower $(1 - \alpha/2)$-quantiles in confidence intervals

*

*Applied Linear Statistical Models by H. Kutner, et al.

*The Elements of Statistical Learning by T. Hastie, et al.



It is indeed very interesting though, from statistics history perspective, to find out which publication first introduced upper quantiles into the confidence interval/hypothesis testing and its motivation of doing so.
By the way, by setting the the lower.tail argument in qnorm to F, it then returns upper quantiles:
qnorm(0.025, lower.tail = F)  # This gives 1.96.


For general random variable, $\xi_\alpha$ is defined such that
\begin{align}
P(X \leq \xi_\alpha) \geq \alpha \text{ and }
P(X \geq \xi_\alpha) \geq 1 - \alpha. \tag{2}
\end{align}
Note that both $(1)$ and $(2)$ may give non-unique $\alpha$-quantiles.
A: People sometimes find it easier to work with positive numbers and use explicit +/- signs
Firstly, as you may already be aware, the use of $-z_{\alpha/2}$ and $z_{\alpha/2}$ is allowed here because of the symmetry of the standard normal distribution.  The symmetry of this distribution around zero allows us to conclude that the respective critical points with upper and lower tail areas of $\alpha/2$ are the negation of one another.
As to why this convention has evolved (as opposed to just using the standard normal quantile function as you propose), it has probably come about because people find it easier to work with fundamental quantities that are positive numbers and then impose a negative sign on them when they want to turn them negative.  By framing the analysis in terms of the critical point $z_{\alpha/2}$ (defines as having upper tail area $\alpha/2$) you always get a positive critical point value, so when you put a minus-sign in front of it you get a negative value.  The aesthetic advantage of this framing is that the sign of the critical points in the analysis is obvious from the notation, rather than having to be inferred from knowledge of when the quantiles are positive or negative.  For experienced mathematicians and statisticians this is unlikely to matter, but people with less familiarity with the problem might find it helpful.
