Sampling and Representativeness

The Sampling Distribution

If you draw many random samples from a population and compute each sample's mean, those means form their own distribution: the sampling distribution of the mean.

Even if the original population is heavily skewed, the Central Limit Theorem guarantees this sampling distribution approaches a normal distribution as sample size grows large enough, conventionally at $n \geq 30$.

This lets researchers use normal-distribution statistics even when the underlying population is non-normal.

Standard Error and Pitfalls

The spread of the sampling distribution is captured by the standard error (SE), measuring how much sample means vary from the true population mean due to random chance.

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

Where $\sigma$ is the population standard deviation and $n$ is the sample size. Standard error shrinks as $n$ grows, which is why larger samples yield more precise estimates.

Common pitfall: Confusing standard deviation (SD) with standard error (SE). SD describes the spread of individual data points around the mean. SE describes the spread of sample means across many hypothetical samples, and SE is always smaller than SD for the same data.