The T score and Z score are both standardized test scores that we use when we test for equality of means for one or two samples.
The difference between the t score and z score lies in the fact that when we do hypothesis testing using normal distribution we calculate the z score whereas when we do hypothesis testing using the student’s T distribution we calculate the t score.
How do we know whether to calculate a Z score or a T score?
Generally speaking, whenever we draw a large random sample when testing the hypothesis we calculate the Z score.
This is because the sample mean follows a normal distribution for large samples (this is called the Central Limit Theorem).
But if the sample size is small then we calculate t scores because the sample mean tends to follow the students’ T distribution for small samples.
How do we know whether the sample size is large or small?
As an empirical rule, we consider the sample size to be large if it is greater than 30 and small if the sample size is less than 30.
As the sample size increases the T distribution tends to approach the normal distribution more and more closely. Beyond the sample size of 30, it has been observed the T distribution is almost identical to the normal distribution.
This is called the limiting property of the T distribution. Hence it is much simpler to use normal distribution and calculate Z scores if sample sizes are greater than 30.