Sampling Distributions and the Central Limit Theorem
Take a sample of size $n$ from a population and compute its mean; repeat this many times. The distribution of these means is the sampling distribution of $\bar{x}$. Its mean is $\mu$, its standard deviation is $\dfrac{\sigma}{\sqrt{n}}$, and as $n$ grows it approaches a normal distribution whatever the population looks like. The dashed grey curve is the population, the red curve the normal distribution predicted by theory, and the bars are the means of the samples you drew.
What to notice
- Start with $n = 2$ on the right-skewed population: the distribution of the means is still skewed. Around $n = 30$ it settles onto the red curve.
- Quadrupling $n$ halves the spread, because the standard deviation is divided by $\sqrt{n}$.
- If the population is already normal, the sampling distribution is normal for every $n$, even $n = 1$.
- With the bimodal population no sample mean may look like a "typical" individual: the means pile up in the middle, where nobody is.