If the mean, actual observed values, and standard deviation are multiplied by #2#, does the z-score stay the same?

Answer 1

Yes, it does.

A z-score is a measure of how many standard deviations from the mean an observation is. Its formula is:

#z=(x-mu)/sigma#

where

If we double the observation (to #2x#), double the mean (to #2mu#), and double the standard deviation (to #2sigma#), the new z-score will be:
#z_"new"=(2x-2mu)/(2sigma)#
#color(white)(z_"new")=(2(x-mu))/(2sigma)=(cancel2(x-mu))/(cancel2sigma)#
#color(white)(z_"new")=(x-mu)/(sigma)#
which is exactly what #z# would be before we doubled everything. Thus, if the observations, mean, and standard deviation all double, the z-score does not change.
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