# Definition of Z sub alpha

In my experience, $$Z_{\alpha}$$ indicates the critical value where the right-tailed area under a standard normal distribution is $$\alpha$$, i.e.

$$P(Z > Z_\alpha) = \alpha$$

With this rule, if $$\alpha = 0.05$$, then

$$Z_{\frac{\alpha}{2}} = Z_{0.025} = 1.96 ~~~\text{and}~~~~ Z_{1-\frac{\alpha}{2}} = Z_{0.975} = -1.96.$$

However, in some internet sources, $$Z_{\alpha}$$ is defined as the inverse function of the standard normal CDF, i.e.

$$P(Z < Z_\alpha) = \alpha$$

which makes the example above have opposite signs:

$$Z_{\frac{\alpha}{2}} = -1.96 ~~~\text{and}~~~~ Z_{1-\frac{\alpha}{2}} = 1.96.$$

They make me confused about how to use correct symbols in the confidence intervals or test statistics. Is there a precise mathematical definition of $$Z_{\alpha}$$?

• Clearly both definitions are precise and mathematical. It's a matter of convention. Since the most popular convention in statistics is to define the distribution function in terms of $\Pr(Z \le z),$ the latter meaning is usually what is intended.
– whuber
Commented Nov 2, 2020 at 20:53

No it is not correct to say that $$P(Z It is $$P(Z which is equivalent to $$P(Z>z_{\alpha})=\alpha.$$
• Both are correct, depending on how $Z_\alpha$ is defined. It would be a mistake to assume the whole world has adopted your convention.