# Is it correct to say that a z-value follows a standard normal distribution?

As the title reveals; is it correct to say that a z-value follows a standard normal distribution? The sentence that I use it in, is as follows:

Moreover, all p-values are reported with the corresponding test statistic: the z-value that follows a standard normal distribution.

Not necessarily. Sometimes the term "z value" is used to describe a standardized variable $Z$ (or a single value $z$), which is a set of observed values (realizations of some distribution) with a population mean of exactly 0, and a population standard deviation of exactly 1 derived from another variable $X$ through the transformation:
$$Z = \frac{X-\mu_{X}}{\sigma_{X}}$$
If the distribution of $X$ is normal, then $Z$ will have a standard normal distribution. However, if $X$ is not normal, then $Z$ will not be either, even though it still has a population mean of exactly 0, and population standard deviation of exactly 1 (i.e. $\mu_{Z} = 0, \sigma_{Z}=1$).
• You are welcome! (Also: Welcome to CV!) Ask a new question! :) Although, given "the assumption is being made that each observed / measured variable is normally distributed," do you actually have $\mu_{X}$ and $\sigma_{X}$ (population statistics)? Or do you just have $\bar{X}$ and $s_{X}$ (sample statistics)? If the latter you do not have a $Z$ variable, but a $T$ variable: $T=\frac{X-\bar{X}}{s_{X}}$. Aug 23, 2018 at 18:07
• "The z-value is simply the parameter estimate divided by the standard error" says nothing about whether that is a sample or population standard error. I am unfamiliar with the package. The authors may have made a mistake, or may be assuming things that make their statement true given the assumptions. Under the null hypothesis, the population mean and variance of Bernouli data are known, so $Z$ can be calculated. You should post another question, rather than trying to get new questions answered in the comments. :) Aug 23, 2018 at 18:25