My name is Karen. I am a junior in high school and I have a question:

If given a class of 30 people who take a test with a mean of 80. Can the standard deviation ever = 0? If so, why?

My supposition is that the only way the standard deviation can be 0 is if all the students in the class scored an 80 on the test.

You are precisely correct.

If you had the 30 scores you would calculate the standard deviation in 4 steps.

1. For each of the 30 student calculate (the student's score - 80)².
   
   
2. Add the 30 squares you calculated in step 1.
   
   
3. Divide the sum by 29 (= 30 - 1).
   
   
4. Take the square root of ths sum you found in step 3.

In step 2, each of the numbers you are adding is a square and hence can not be negative. Thus the only way that the sum can be zero is if each of the numbers you add is zero. That is for each student