Say I took twenty measurements of the wind velocity at a point. This is a sample of the infinite number of measurements I would need to get the true velocity at that point. I then calculate the mean and standard deviation. I now want to know how close the mean and standard deviation from the sample are, to the true mean and true standard deviation of the population.
I can calculate a standard error and a confidence interval. This will give me the uncertainty, due to sampling, of my mean. But I've read that the sample data must typically be normal distributed and the population standard deviation must be known, see link below: http://www.stat.yale.edu/Courses/1997-98/101/confint.htm
If the sample data is not normally distributed and/or the population standard deviation not known, a standardized "t distribution" must be used. Where can I find a simple link which explains what are the confidence intervals using a t distribution? (e.g. for sample data which is normally distributed, 95% confidence interval uses 1.96 from the standardized normal distribution).
Next, how do I calculate the uncertainty of the measured standard deviation of the sample? Another user asked this question here: Calculating uncertainty in standard deviation
It was replied to with a formula that resembles the standard error. My question is, where does this formula come from? I can't find it in the statistics books I am using. I don't want a derivation or proof of it, just a source so I can read a little about it and reference it. What assumptions does it use?
I have also found how I can estimate intervals for the measured standard deviation of the sample data: https://faculty.elgin.edu/dkernler/statistics/ch09/9-3.html
But it seems to only be applicable if the sample data is normally distributed. What do I do if it isn't normally distributed or I don't have enough samples?
EDIT: I have found some advice on the t-distribution that has helped me: https://stattrek.com/probability-distributions/t-distribution.aspx