Sample Size and the Width of the Margin of Error

A margin of error measures how far a sample's estimate might realistically sit from the true value in the full population. Asking more people doesn't usually change the estimate itself much, because a well-chosen sample already reflects the population fairly well. What it changes is the confidence in that estimate: more data means less random wobble, so the plus-or-minus shrinks and the interval tightens around the same central number.

The mechanism is a square root relationship, not a straight-line one. Margin of error scales with 1 over the square root of the sample size n. That means to cut the margin of error in half, you need to multiply n by 4, not by 2. Doubling the sample only divides the margin of error by the square root of 2, about 1.41, so it shrinks to roughly 71 percent of its old width. This is why pollsters chasing a tighter margin need samples that grow much faster than the precision they're buying: going from a margin of 4 points to 2 points takes four times the respondents, not twice.

This sits alongside other ideas about sampling and estimation: a larger sample narrows the margin of error without guaranteeing the estimate moves closer to the truth, since bias from a flawed sampling method isn't fixed by adding more people. Margin of error also pairs with causal claims and scatterplot models in the sense that all three ask the same underlying question: how much should a number drawn from limited data be trusted as a stand-in for the real pattern.