Z = the raw score minus the mean divided by the standard deviation:
(In this formula s = standard deviation)
Let us say that on her first exam in a QM course, Jane got a mark of 70, when the class mean was 80 and the standard deviation was 5.
This would mean that Jane would have had a z-score of −2 (i.e., 70 − 80 / 5 = −2).
If we look at the handout entitled, Rule of 68-95-99, we can see that 2.5% of the class got a lower mark than did Jane and correspondingly 97.5% of the class got a better mark. Compared to the rest of the class Jane did not do very well.
For the second test, Jane worked very hard going carefully over all the material. She got a mark of 85, when the class mean was 75 and the standard deviation was 10. As a result Jane had a z-score of +1 (i.e., 85 − 75 / 10 = 1), which means that she did better than 84% of the class.