Need help simplifying standard error formula for redshift

[Rageuke]

TL;DR

Filled in all the equations for standard error in redshift (standard error formula, standard deviation formula, mean formula), would like to simplify it, too complicated otherwise thanks in advance :)

SE = standard error, expressed as sigma(x)
n = number of observations we take into account (from a total population) when calculating the standard error
sigma-index-x = standard deviation
N = total population
x-index-i = element of that population
mu = mean of the population
-> filled in all the equations and replaced x by z (redshift) to determine the standard error in z (sigma(z))
-> as you can see, last equation is way too complicated, can anyone help me simplify it?
PS. In the finale expression, x-index-i should be substituted with z-index-i

 

[Ibix]

Where you substituted your third expression into your second, why did you introduce a square root?

 

[Rageuke]

Small mistake of mine, I corrected it, thank you for your sharp eye!

 

[Ibix]

Right. And what's left is a textbook formula for standard deviation, although you are using the biased formula. Whether that matters or not depends on the size of your $N$.