Central Limit Theorem for proportion - np >=10 or 5? While trying to understand deeper about Central Limit Theorem for proportions, I learned CLT for proportions is based on the fact that binomial distributions can be approximated by normal distributions when some requirements are met.
For the approximation to work this requirement has to be met -

*

*$np \ge 5$ and $n(1-p) \ge 5$
(for example, statisticshowto.com mentioned this requirement)
However, when I looked at the Central Limit Theorem for proportions, it's said that -

*

*$np \ge 10$ and $n(1-p) \ge 10$
Is the requirement.
(for example, onlinestatbook.com mentioned this requirement)
Which one of these two is true or used more? Why the difference? Or can we say the CLT is more strict than just using approximating binomial with normal distribution?
 A: The two situations are the same except for a constant factor. The proportion $X/n$ is the number of counts $X$ divided by the total number of draws $n$.
The reason for the discrepancy in the rules $np<N$ and $nq<N$ with $N$ either $5$ or $10$ is because it is a rule of thumb. It is not an exact boundary.

For the approximation to work this requirement has to be met

This is a very strong statement.

*

*The expression 'to work' is not very clear. The approximation is never exactly correct. The expression 'to work' means that the approximation is not bad. But when does an expression work in that sense? When the error is 5%, when it is 10%? There is no clear definition when an approximation works.

*The requirement is a rough measure. The product $np$ doesn't tell everything. It is gonna be different when we have $n=20$ and $p=0.5$ in comparison to $n=20000$ and $p=0.0005$. Also it is gonna be different depending on the $z$ value that we try to approximate.

So the $np>10$ is not a strict requirement. It is a rule of thumb. It will depend on the situation whether it works or not.
