Why are we interested in asymptotics if the real-world data is almost always finite?

Question

Why are we interested in asymptotics if the real-world data is almost always finite?

Answer 1

Asymptotic theory tells us about the statistical properties of a sample as it grows to an arbitrarily large size $n$. Often datasets are sufficiently large that theorems like the law of large numbers and the central limit theorem apply in practice. Think of doing a census of tree heights in a forest or the number of time the house wins at a casino at craps over a day.

On important thing to note is that asymptotic theory is largely concerned with limiting behaviour of random variables, so there are no infinite datasets involved. For instance, if a sequence of random variables (like sample averages of a dataset) $\hat{X}_1, \hat{X}_2, \ldots$ converges in probability to the true mean $\mu$ (an asymptotic result), then that merely states that for any arbitrarily small error tolerance $\epsilon$ I select, there is some large sample size $n$ so that there is no chance that $\hat{X}_n$ is more than $\epsilon$ away from $\mu$.