Sampling from non-normal distributions yields

samplingexample.jl

# julia code to show that the mean of 14 samples from a 50/50 mixture
# of normal and uniform has extremely close to "normal" distribution

using Distributions, StatsPlots

meanvals = [mean([rand(Normal(0,1),7); rand(Uniform(-1,1),7)]) for i in 1:100000]
histogram(meanvals)

pop = [rand(Normal(0,1),10000); rand(Uniform(-1,1),10000)];

histogram(pop)

expmeans = [mean(rand(Exponential(1.0),14)) for i in 1:100000]
histogram(expmeans)
  

Above is the histogram of the sample means.... Here is the population that you're drawing from... very NON normal:

Here's the histogram of mean values from 14 exponentially distributed random variables.