Calculating confidence in part of a sample (explain like I'm in the humanities please :) ) Working on a two-part project. First, we're estimating the total size of a very large population. Each member of the population has a standardized random 15-digit identifier, and we have guessed using a brute force method until discovering enough extant identifiers (20,000) to estimate the population size, with a confidence calculated using standard error of the mean. That part is ok.
The second part is a description of the 20k sample: demographics, etc. What I need to figure out is how to calculate the CI/error% of particular subsets of that sample. So if there's a group of 1,000 within the sample of 20,000 with unusual characteristics, I want to be able to describe those characteristics. Let's say it's a sample of people where one dimension of the data is income. There's a group of 1,000 CEOs in the group with a much higher income than everyone else. What I want to know how to figure out is how statistically significant the mean income of that 1,000 would be. It seems like it would require some compounding of the standard error of the sample with regard to the population and the sub-sample of CEOs in relation to the sample. Am I overthinking this?

I promised a serious answer (to supplement my gag answer) so here goes.  I think at present you are mixing several issues, including considerations relating to descriptive statistics for a sample and some (not clearly specified) population inferences.  The main thing I would recommend at this stage is to take some time to consider and describe your sampling method, and in particular, determine whether it can reasonable be regarded as a "non-informative" type of random sampling.  If you are using genuine random sampling then you will have available the entire suite of standard inference methods, including methods for construting confidence interval for population means, quantiles, etc.
With regard to your desire to describe aspects of your sample, this would typically be done using standard visualisations with accompanying descriptive statistics.  The usual rule here is to remember that "a picture tells a-thousand words", so you should primarily be thinking about how to illustrate as much useful information about your sample using appropriate plots.  When describing the distribution of continuous quantities across multiple discrete groups, a violin plot is a useful tool.  When dealing with heavily skewed variables like income/wealth, you might also consider showing these on a logarithmic scale.
At the moment your problem is not that you are "overthinking" things --- you just don't seem to have a clear description (or even consideration?) of your sampling method or a clear description of what population quantities are of interest in your inference problem, and what descriptive aspects of the sample you wish to illustrate.   Once you have a clear formulation of those things you will be in a better position to choose appropriate statistical methods for these tasks.
