Why are these standard deviations nearly identical?

[Sam Lawrence]

set seed 80085

set obs 200
gen score = rnormal(28,11)
replace score = 50 if score > 50

sum score, detail

local mean = r(mean)
local sd   = r(sd)

gen z = (score-`mean')/`sd' //doing this because the randomly generated mean and sd are not EXACTLY 28 and 11

sum z, detail

gen score_adj = score+(50-score)/2

sum score_adj, detail

local mean_adj = r(mean)
local sd_adj   = r(sd)

gen z_adj = (score_adj-`mean_adj')/`sd_adj' //again, doing this for the same reason; mean isn't exactly 39 with sd of 11/2) or 5.5

//-----Interpretation of Z-Scores-----//

*both are standardized distributions of score and score_adj. They are almost identical, though.

gen zdif = z - z_adj

sum zdif

please run this code and look at the differences in z-scores. I cannot understand why they are NOT different, beyond saying that "both transformations are linear". Is that correct?

Thank you so much for your help,

sam

[Clyde Schechter]

I don't understand your question. As you have noted, z-scores are invariant under linear transformations. You have two linear transformations of the original variable (score) and have found that the corresponding z-scores agree to within very tiny rounding errors. So it's just what you said you would expect. So what is the question here?