Difference between Z-score and Z-test

What is the difference between Z-score and Z-test?

Often this question is asked in SAS interviews, so what should be the perfect answer ... Remember ... Always answer to impress.







1). Z-score is also known as standard score. it is calculated for population (overall)

Where x is raw score, μ is population mean and σ is standard deviation of population. 

Usage :

Z Score is free of any scale, hence it is used as a transformation technique while we need to make any variable unit free in various statistical techniques.

Also, it is used to identifying outliers in a univarite way. If abs(z score) > 2.5, the observation is considered as an outlier.


2). Z-test is a statistical technique to test the Null Hypothesis against the Alternate Hypothesis.


Where M is sample mean, μ is hypothesized mean and SE is standard error of mean.

Key Difference

Z-score requires population parameters (mean, standard deviation) and Z-test works on sample parameters (mean, standard deviation, standard error).


Example:

Father enters the room and finds,  his first son is happy and second one is sad.



When asked , happy boy says I scored 80 in English, so is the reason of my happiness.
Sad boy says, I scored 80 in Maths, so is the reason I am sad.

Confused ???


Boy is happy as 80 in the English test is excellent as compared to others' marks (others secured less)
Boy 2 is sad as 80 in the Maths test is bad as compared to others' marks (others secured more).


English score of 80 (above average) in comparison to average score of 70 of the entire class.
Happy boy did better than most ( average  score = 70) in the class. You can see that most people in the distribution are below him.


                          
Maths score of 80 (below average) in comparison to average score of 90 of the entire class.
Sad boy did worse than most (score of 80) in the class. You can see that most people in the distribution are above him.



After calculating Z - score we can get either positive or negative value. Positive is above the mean. Negative is below the mean. So basically it tells you positive relative to mean.