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I'm trying to figure out the difference between an A/B split test and an ANOVA test and I came across this article. It suggests that the mean of $n$ independent Bernoulli random variables is normally distributed. Is this true?

If I have a random variable $y$ that follows a Bernoulli distribution:

$F(\theta) = \theta^{y}(1-\theta)^{1-y}$

And it takes values of 1 or 0, will the mean of $n$ of these variables be normally distributed?

$\frac{1}{n}\sum y_{i}$

JoeDanger
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    You can find out a lot about this subject by searching our site. – whuber Nov 19 '13 at 20:13
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    The central limit theorem applies to the Bernoulli (indeed, that's how the Gaussian distribution was first discovered ... by De Moivre in the mid-1700s, as a large sample approximation to the binomial); but take care - the CLT describes what happens in the limit, to a standardized mean (or a standardized sum). As whuber suggests, there's a lot of material on it here. – Glen_b Nov 20 '13 at 00:54
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    To add to @Glen's comment, note that the mean can't possibly be exactly normally distributed for any finite $n$ since it can only take on $n+1$ possible (rational!) values. – cardinal Nov 20 '13 at 03:45

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