What is the difference between the critical z-value, z-score, and z-test statistic?

Because an individual needs to have a firm foundation in statistics in order to become a good data scientist. Because I never studied statistics in secondary school, college, or university, I have been studying statistics on my own for the past few years.

When studying inferential statistics, it is important to be familiar with the normal distribution, which is also called a z distribution by some. The z-score and z-value are derived from the normal distribution.

One question that I have in this regard is what is the difference between the z-score and the critical z-value.

The critical z-value and z-score are closely related but serve different purposes in the field of statistics. Both the z-score and the critical z-value are defined in the normal distribution, but the critical z-value sets the threshold and the z-score is a standardised value of a datapoint.

The critical differences between the critical z-value and the z-score are as follows:-

The z-score and z-test statistic are closely related, but are not exactly the same.

The difference between the z-score and the z-test statistic can be cited in the table below:-

I have been working on a normal distribution where I have created a critical z-value and a z-test statistic. The example below is to test whether the mean test score of students differs from the national average when the sample size is large (n > 30).

The population mean is 75 while the sample mean is 72. Although it is pretty much common sense that the class average is not the same as the population average, but it is a good exercise to calculate the z-score, z-test statistic, and critical z-value.

The critical z-value, 1.96, defines the value at which the null hypothesis will be rejected. The z-test statistic, -2.12, is beyond the critical z-value, which means that the null hypothesis was rejected in this instance.

The diagram below, prepared in Excel, provides a visualisation of the statistical values calculated above:-

In summary, the z-value, z-score, and z-test statistic are very useful in statistics, especially in the field of hypothesis testing. It is useful to be able to calculate statistical values in order to provide a scientific basis for a decision.