I have a question regarding calculating z-scores.

suppose I sample 20 people and ask them to rate on a 7-point scale how satisfied they are with X. (imagine that this method is perfectly valid and reliably for the sake of this example...) sample mean = 5.77.

Previously I sampled 5000 people, and I know that the pop. mean is 4.89, and the pop. standard deviation is .64

I now want to convert my mean score to a normalized score on a 100-point scale (in reality, this is done for purpose of combining disparate metrics using different scales, but I'm simplifying here...) So, I now have a normalized score of something like 91.4.

(in excel, that looks like: normsdist((5.77 - 4.89) / .64)*100

Now, if instead I first convert each individual's score to a normalized score using the same process described above, and then compute the mean of the 20 normalized scores, I get a very different answer, approximately 83.6.

can anyone tell me why this is happening? why aren't they they the same?

Thank you!

Ryan