z-scores

#1
Dear Forum,
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
 

Dason

Ambassador to the humans
#2
In general you can't switch the order in which you apply functions to a set of data. There really isn't much more to explain. Another example: 2^2 + 3^2 = 13 but (2+3)^2 = 25. If I square my two numbers then add them together it doesn't give the same as adding them together and then squaring.
 
#3
That was surprisingly more simple than I thought, and perhaps I should have first opened to page 5 in an Algebra 1 textbook...
 

rogojel

TS Contributor
#4
hi,
strictly seen you are comparing two different things and this will introduce an extra error. You have a sample mean of a sample of 20 and you normalize it with the population mean and the population std deviation - that is you are treating the sample mean as if it was an individual measurement, which it is not.

I think that it would be more consistent to correct for the sample by dividing the population standard deviation by the square root of the sample size, SQRT(20) in your case.

regards
rogojel