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I'm unsure about the interpretation of $z_{\alpha}$.

I've seen some source claim that $z_{\alpha}$ is equal to the z value with $\alpha$ of the area to the right of it. Under this view, $z_{0.025}$ is equal to the 97.5th percentile, and therefore equals 1.96.

Alternatively, I've read that $z_{\alpha}$ is the $\alpha$th percentile. Under this view, $z_{0.025}$ is equal to -1.96.

Which is the preferred interpretation? Is $z_{0.025}$ equal to 1.96, or -1.96?

kjetil b halvorsen
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Guillaume F.
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    It has been traditional to use $z_{0.025}$ as the value $1.96$ that cuts probability $0.025$ from the _upper_ tail of a standard normal distribution. (This notation is often used in printed distribution tables. Sometimes this is called a 'percentage point'.) However, this is to be distinguished from quantile 0.025, which cuts probability 0.025 from the _lower_ tail of a distribution. In R, code `qnorm(.025)` (std normal quantile function, without additional parameters) returns $-1.959964$ and `pnorm(-1.96)` (CDF) returns $0.0249979.$ – BruceET Dec 06 '20 at 04:25

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Percentage points. It has been traditional to use $z_{0.025}$ as the value $1.96$ that cuts probability $0.025$ from the upper tail of a standard normal distribution. (This subscript notation is often used in printed tables for normal, t, chi-squared, and other distributions. So you may also see notations such as $t_{10; .025}, \chi^2_{15;.05}, F_{5,15;.05}.$ Sometimes these is called a percentage points.)

From R, $z_{0.025}, t_{10; .025)}, \chi^2_{15;.05},$ and $F_{5,15;.05}$ respectively, are as follows (to more decimal places than you would see in a printed table).

qnorm(.975)
[1] 1.959964
qt(.975, 10)
[1] 2.228139
qchisq(.95, 15)
[1] 24.99579
qf(.95, 5, 15)
[1] 2.901295

Quantiles. This is to be distinguished from quantile $0.025,$ which cuts probability $0.025$ from the lower tail of a distribution. In R, code qnorm(.025) (without additional parameters, the standard normal quantile function) returns $−1.959964$ and pnorm(-1.96) (CDF) returns $0.0249979.$

qnorm(.025)    # same as: qnorm(.025, 0, 1)
[1] -1.959964
pnorm(-1.96)
[1] 0.0249979
BruceET
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    Are you sure that $z_{0.025} = 1.96$ and not instead $z_{0.975} = 1.96$? – Sextus Empiricus Dec 07 '20 at 17:55
  • I haven't seen that. Maybe notations differ, especially now that R is more commonly used, but traditionally I believe the subscript notation has referred to _upper_ tails. In printed tables, there is often a small figure of the PDF with the relevant area shaded. – BruceET Dec 07 '20 at 18:00
  • I haven't seen that. Maybe notations differ, especially now that R is more commonly used, but traditionally I believe the subscript notation has referred to upper tails. In about half of printed t and chi-sq tables it's obviously upper 2.5% in the other half it says the same in words. Normal tables do not ordinarily use subscript notation. Many 95% z CIs are written $\bar X \pm z_{.025}\frac{\sigma}{\sqrt{n}},$ which may be ambiguous; but in parallel t CIs are $\bar X \pm t_{n-1;.025}\frac{S}{\sqrt{n}}.$ In table headers, there is often a small figure of the PDF with the relevant area shaded, – BruceET Dec 07 '20 at 21:31