The Central Limit Theorem

The one idea: No matter what shape a population's individual values follow — skewed, lumpy, uniform, anything — if you repeatedly take random samples of size n and compute each sample's mean, those sample means pile up into an approximately normal distribution as n grows, centered exactly on the population mean μ, with a standard deviation of σ/√n. This single fact is why the normal curve and z-tables can be used to make inferences about almost any average, for almost any kind of data.

⚠️ The trap: "The Central Limit Theorem means my data becomes normal with a big enough sample." No, it doesn't — your individual data points keep whatever shape the population has, forever, no matter how much data you collect. The CLT is a statement about the sampling distribution of the sample MEAN, not about individual observations — a histogram of 100,000 raw data points from a skewed population still looks skewed; a histogram of the means of 100,000 samples of size 30 drawn from that same population looks like a bell curve. These are two completely different objects, and mixing them up is the single most common CLT error in applied work.

The theorem, stated precisely

From the standard textbook formulation: if repeated random samples of size n are taken from a population with mean μ and standard deviation σ, the sampling distribution of the sample means will have a mean equal to μ and a standard error equal to σ/√n. Moreover, as n increases, the sampling distribution will approach a normal distribution — regardless of the shape of the original population.

Unpack the pieces:

• **Sampling distribution…