I have a set of values, S, that have a gaussian distribution.

I order the set in some way, assigning an integer "position" to each value in S.

I then sample the set S, by choosing values based on a position linearly at random (so the distribution of my position sampling is linear). My sampled values form a new set, Q.

Do the values in the new set, Q, also have a gaussian distribution? Do the relative sizes of the sets Q and S affect this? Does the order I choose to assign integer "positions" to values in S prior to sampling affect this?

I am not sure what concepts from mathematics this even deals with, beyond "stats". I am working with generating synthetic data, and I want to manipulate sets of values while preserving underlying statistical properties (in this case distributions). What should I search for to learn more about this?