## Significant Figures and Rounding

Engineers are numbers people, and they sure do like to include lots of numbers in their reports. They eat calculus for breakfast. They pepper their writing with equations just to make it look good. Some include lots of decimal places just to make their numbers look more accurate. This is a no-no.

When reporting numbers or when converting from one unit of measure to another, the SPE Metric Standard says “… the number of significant digits retained should be such that accuracy is neither sacrificed nor exaggerated.”

Significant digits, also called significant figures (or Sig Figs, as Mr. Gerlach often referred to them in high school physics class), are defined as those digits that carry meaning about a number’s precision or repeatability. Leading and trailing zeros used as placeholders to establish the order of magnitude do not count as Sig Figs.

Example:
The population of a small town might be 1,200.
The population of a rural county might be 12,000.
The population of a small city might be 120,000.
The population of a big city might be 1,200,000.
In each of these cases, there are two significant figures, the 1 and the 2.

The true precision only allows for two Sig Figs because of births, deaths, people moving in, people moving out, etc. From one day to the next, the third digit would likely vary, so reporting the big city population as 1,204,687 would probably be true for about 30 seconds. And the small town would have to get a new sign made every time a baby was born if it said “Welcome to Krancie, population 1,206.”

Here’s another example:
Say you want to calculate the circumference of a 4-in. pipe with an outer diameter of 4.5 in. You know you multiply the diameter times π, which is 3.141592653…. Would you report the circumference as 14.137166941…? No, that would be too many Sig Figs.

You would have to round the number to the proper number of Sig Figs. How do we do this rounding business?

If the first non-significant digit is a 5 followed by other non-zero digits, round the last significant digit up a number.
Example: 2.8456 rounded to three Sig Figs would be 2.85.

If the first non-significant digit is less than 5, merely truncate at the last Sig Fig.
Example: 2.8446 rounded to three Sig Figs would be 2.84.

But what if it’s exactly halfway? What if the number is 2.8450 and you need to round it to three Sig Figs? Here is the tie-breaker
rule: Round half to even. This is called “banker’s rounding, and is the default rounding method used in IEEE 754.
Examples:
2.8450 would be rounded to 2.84 to make the last Sig Fig even.
2.8550 would be rounded to 2.86 to make the last Sig Fig even.

Now back to our pipe circumference calculation.
Our 4.5-in. diameter pipe has two Sig Figs, and this is the smallest number of Sig Figs in our calculation product. Thus, our answer can only have two Sig Figs. The correct precision for the circumference would be 14 in.

When performing calculations, use as many Sig Figs as are available in the various steps, not rounding in between, but rounding only the final results to the correct number of Sig Figs.