Standard error of the mean of a non-normal distribution It seems like a basic question, but I haven't find an answer yet.
How do you calculate the standard error of the mean when you have a discrete non-normal distribution?
Actually, I have a discrete distribution that I represent as a box plot because it is non-normal and has outliers. Therefore, I display the median, IQR, etc. In this case, what is the equivalent of the standard error of the mean that we employ with normal distributions?
 A: Per the central limit theorem, the sample mean is asymptotically normally distributed even for non-normal sampled distributions under some quite weak assumptions, which can in practice usually be assumed to hold. And the asymptotics kick in even with quite small $n$. So you can estimate the SEM in the standard way, no need for the underlying distribution to be normal (or even continuous).
For additional peace of mind, consider bootstrapping the mean and calculating the standard deviation of the bootstrapped means.
As an illustration, assume we have a sample containing $0$ fourteen times and $1$ six times. It doesn't get much more non-normal than that. We have $\bar\mu=0.3$ with SEM $0.105$. Bootstrapping the mean 1,000 times yields a standard deviation of the bootstrapped means of $0.103$. The difference is minuscule. Also, here is a histogram of the bootstrapped means:

R code:
> foo <- c(rep(1,6),rep(0,14))
> sd(foo)/sqrt(length(foo))
[1] 0.1051315
> 
> set.seed(1)# for reproducibility
> bb <- boot::boot(foo,statistic=function(xx,index)mean(xx[index]),R=1000)
> sd(bb$t[,1])
[1] 0.1034599
> 
> hist(bb$t[,1],main="Bootstrapped means",xlab="")

