Central limit theorem

The central limit theorem is a statement about one object: the sampling distribution of $\bar{x}$ (x-bar), the distribution you would get by taking every random sample of size $n$ from a population and recording each sample's mean. For independent observations from a population with a finite mean $\mu$ (mu) and a finite standard deviation $\sigma$ (sigma), that distribution gets closer to normal as $n$ grows, centered at $\mu$ with standard deviation $\frac{\sigma}{\sqrt{n}}$.

Two sentences come up constantly and both are wrong: "with a large enough sample, the data become normal," and "a large sample makes the population normal." Nothing about the population or the sample changes shape. A right-skewed population stays right-skewed however much of it you collect, and a histogram of your 200 observations is a picture of that population, not of $\bar{x}$. The normal shape belongs to a distribution you never see in a single study.

The familiar $n \ge 30$ is a rule of thumb, not part of the theorem, and it guarantees nothing. Draw from a strongly right-skewed population (an exponential, $\mu = 1$, $\sigma = 1$) and build the usual 95% t-interval for the mean. Across two million simulated samples the interval captured $\mu$ 88.3% of the time at $n = 5$, 91.2% at $n = 15$, 92.7% at $n = 30$, 94.2% at $n = 100$, and 94.8% at $n = 500$.

The misses are lopsided too. At $n = 30$, 6.5 of those 7.3 missed percentage points were intervals sitting entirely below $\mu$. A sample that draws none of the long right tail has both a low mean and a small $s$, so it produces a short interval in the wrong place. Skew does not only widen the error, it aims it.

Two edges. The population needs a finite standard deviation. And if the population is already normal, the sampling distribution of $\bar{x}$ is exactly normal at every $n$, so the theorem is not needed. Topic 2.12 is Sampling Distributions and the Central Limit Theorem.