Asked about Probability by simulation
What does the central limit theorem say, and where does it break?
An answer that works
The average of many independent draws is approximately normal whatever the shape of the individual draws, centred on their mean, with a standard deviation equal to theirs divided by the square root of the sample size. It breaks when the draws are not independent, when the underlying distribution has no finite variance, and in practice long before that whenever the data is badly skewed: on exponential data, averages of 200 still have measurable skew and a lower tail 4 per cent short of the normal one.
Then they push further
Is thirty enough?
It depends entirely on the skew, which is why the rule is a rule of thumb and not a theorem. Simulate your own data: standardise the averages, check the 2.5th and 97.5th percentiles against minus and plus 1.96, and raise n until they agree. On revenue per user, where one customer in a thousand is worth a hundred times the rest, thirty is nowhere near enough.
Reading an answer is not being able to give one. The section that teaches this is Probability by simulation, 4 hours of Probability & statistics.
Learn this properlyAlso asked about Probability & statistics