Sample Statistics and Margin of Error
A sample can only ever estimate the truth about a population — margin of error describes how much that estimate might be off. Watch the confidence interval tighten as the sample size grows.
A sample statistic (like a sample mean or sample proportion) is a real number you can calculate from data you actually collected. A population parameter is the true value for the entire population — almost always unknown, since surveying everyone is rarely possible. The sample statistic is your best estimate of it.
Margin of error describes how far that estimate might reasonably be from the truth. A larger sample size shrinks the margin of error, but not in a straight line — it shrinks with the square root of the sample size, so cutting the margin of error in half takes four times as much data, not just twice as much.
Margin of error only measures random sampling variability — it says nothing about whether the sample was collected fairly in the first place. A poll of gym visitors will never accurately represent an entire city’s exercise habits, no matter how large the sample or how small the margin of error gets. That’s a question about the sampling method, not the math.