Population distribution
Sampling distribution of x̅
Watch the Central Limit Theorem in action, then dig into the t-distribution
sampling distribution CLT t-distributionThe CLT says sample means pile up in a bell curve, regardless of the population shape.
Watch what happens to the distribution of sample means as you draw more samples. Does the shape of the resulting sampling distribution depend on the population shape? Peruse the Explorer to find out!
Ultimately, we want to make a claim about how weird it would be to observe a sample mean as extreme as ours...IF our sample really came from some reference population . For example, if we observe a 2cm difference in mean height between two groups, how unusual would that be in a population with a true mean height difference of 0cm? That's our null hypothesis.
To answer that, we need to know where our x̅? falls relative to the null distribution—the sampling distribution of x̅? assuming the null is true (e.g., assuming we live in a world where there's no height difference between groups). And to locate x̅ in that distribution, we need know its spread. But do we know that?? YES we do! The CLT section above showed us that the spread of the sampling distribution is σ/√n.
Awesome. Things are looking good. But there's a little wrinkle...we don't know σ ☹️. Luckily, we can estimate it from the sample as s. But s varies from sample to sample, which means our characterization of the null distribution's spread is itself uncertain. Some samples give us an s that's too small, making our x̅? look more extreme than it really is; others give an s that's too large, making x̅? look less extreme. That extra wobble pushes probability into the tails, giving rise to the t-distribution. Scroll down to see it in action!