Central Limit Theorem: Sample Means Become Normal

The Central Limit Theorem (CLT) states: the distribution of sample means approaches a normal distribution as sample size $n$ increases, regardless of the original population's shape. The mean of the sampling distribution equals $\mu$ and its standard deviation is $\sigma/\sqrt{n}$.

This is why the normal distribution appears everywhere — any measurement that is the average of many independent factors will be approximately normal. Height (average of many gene effects), test scores (average of many question performances), manufacturing tolerances (average of many process variations).

Try it in the visualization

Select a source distribution (uniform, skewed, bimodal). Take samples of increasing size and watch the histogram of sample means morph from the original shape into a bell curve. The larger the sample size, the narrower and more normal the sampling distribution becomes.