I have learned that z-scores are "Measures of Relative Standing" and that they can be used to compare two scores that are from different normal distributions.
For example if I wanted to compare how I performed
with a 85% in a math class with mean 70 and std dev 15;
and
with a 70% in a english class with mean 50 and std dev 15;
I would use the z-score formula to calculate a relative value for two situation and use it to compare them.
However why is it not enough just to compare each classes with the mean and use this as a basis for comparison? For example
85% -70% = 15%
70% - 50% = 20%
20 > 15 therefore I must have performed better in english
I know this is incorrect but I don't understand why. thx for the help
This works fine when they have the same standard deviation (they are both $15$ in this particular problem), but in general.
If the standard deviation were $5$ for the math class, you'd have performed $3$ standard deviations from the mean, which would be better than your $Z$-score for English.