Suppose a football stadium with 60,000 seats looked to be mostly filled on a Saturday afternoon, and, though it lacked working turnstiles, an accurate tally was needed of how many people were there. The Census Bureau of old would to try to count everyone.
In the first quarter, census takers could walk up and down the aisles, counting the number of fans in each row. They could also ask fans if anyone had been in the empty seats. Adding up those counts would yield a total.
Critics might say this count is flawed because some people were not in their seats when the census takers came by. They had gone to buy food, to visit the restroom or perhaps had wandered elsewhere in the stadium to see friends. The Census Bureau of today, heeding the critics, would try for a better tally with a mix of counting and sampling.
In the first quarter, the census takers could quickly count every other row and multiply by two. This would yield a preliminary total. Then, to check the accuracy, the counters could choose one row in the middle of each section and carefully interview everyone there. By taking more time, they could find those who had left their seats for a short while. Then, the numbers for these recounted rows could be used to adjust the numbers for the entire section.
Suppose the count turned up a total of 43,000. If, in the sample rows, 15 of 20 seats were occupied, this would suggest the stadium was three-quarters full and the count would be adjusted to 45,000.