Significant or Otherwise

Consider the following statements:

The number of registered voters in Hennepin County in the last election was 729,527.

Presuming that this statistic is based on an official database, it is precise by definition for a particular time and place. This number is accurate to 6 significant figures.

The city of St. Paul has a population of 285,068.

This statement is not likely to be correct. Even if a city's population could be defined in a precise way (permanent residents? homeless? warm bodies?), how can we account for the minute by minute changes that occur as people move in and out, or are born and die?

It is entirely possible that the last census yielded precisely 285,068 records, and this may be the population for legal purposes, but it is highly unlikely that this is the true population. To better reflect this fact, one might list the population as 284,000 or even 290,000. These two quantities have been rounded off to three and two significant figures respectively and have the following meanings:

• 284,000 - implies that the population is believed to be within the range of 283,500 to 284,500 (or 284,000±500). This number is accurate to 3 significant figures.
• 290,000 - implies that the population is believed to be within the range of 285,000 to 295,000 (or 290,000±5,000). This number is accurate to 2 significant figures.

Which of these numbers is chosen depends on the degree of confidence we have in the original census figure, how long ago the census was conducted, migration trends, and a host of other considerations.

The point this exercise is that the concept of significant digits has less to do with mathematics than with our confidence in a particular measurement.