Paul Berman

Significant Figures Part III

How Many Significant Figures or Decimal Places Are Correct?

The number of sig figs or decimal places that should be presented is often a decision to be made by the end user. In many instances, showing one or two sig figs is adequate; for example, in a simple comparison determining whether a result is less than or greater than a benchmark value. If the results will undergo substantial statistical evaluation or other arithmetic calculations, then more sig figs would be recommended. As a general rule, it is better to keep at least one additional sig fig through all calculations and round afterwards rather than rounding first.

One famous example for keeping extra sig figs and decimal places for calculations is the original “butterfly effect.” In 1961, a mathematician/meteorologist named Edward Lorenz was using a computer to simulate weather effects. He entered a value that had been rounded from six decimal places to only three, which represented about a 0.025% difference in the value. After he ran the program, he realized that the result represented a completely different weather pattern than if he had used all six decimal places. During later presentations, this effect took on its more popular meaning, which is still in use today.

At the other end of the spectrum, it is possible to provide far too many sig figs. Since computers and calculators became common, they have been used more and more frequently for data collection and analysis. If the data are being calculated using mathematical formulae (e.g., linear or quadratic regression), the computer could easily provide 32 or more sig figs in its evaluation. If this were presented as a single whole number, for example, it would be the equivalent of counting the number of grains of sand in a pile the size of Earth.

Some things to keep in mind when determining how many sig figs should be presented is how the data will be collected, limits in the ability to precisely measure a value, and if the data collected represents an exact measurement or is instead measuring a sample. For example, counting the number of children enrolled at each elementary, middle, and high school in a state is relatively straightforward. An exact count can be provided fairly easily, so a number that has six or seven sig figs would be appropriate.

On the other hand, counting the total number of people living in a state would be more challenging, since it would include births, deaths, and people moving into and out of the state. In addition, there may be a certain number of people who are temporary residents. Since all these changes can happen hundreds or even thousands of times per day, it would be more appropriate to provide only three or four sig figs for a statewide population count.

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