Sampling and the Central Limit Theorem

From Sample to Population

Statistics uses a sample (a subset of collected data) to infer properties of a population (the whole group of interest).

For valid inference, use a random sample where every member is equally likely to be chosen, avoiding bias (systematic error from non-random selection).

Because samples vary, $\bar{x}$ is a random variable. Its sampling distribution has a centre equal to $\mu$ and a spread called the standard error:

$SE = \frac{\sigma}{\sqrt{n}}$

Here, $\sigma$ is population standard deviation and $n$ is sample size. Larger samples yield more precision, but only by a factor of $\sqrt{n}$.

The Central Limit Theorem

The Central Limit Theorem (CLT) states that for a sufficiently large sample ($n \ge 30$ typically), the sampling distribution of the mean is approximately normal, regardless of the population's underlying shape.

Common pitfall: Thinking the CLT makes the raw data normal. It only applies to the sampling distribution of the mean. The population shape remains unchanged; only the distribution of the statistic turns normal as $n$ grows.