# Significant figures

It would be easy to report all numbers to the accuracy that your device or computer displays, but this is not necessarily the appropriate number to include in your report. It’s perfectly acceptable to do calculations using any number of digits. However, they should be reported to the number of *significant figures* that the data supports. The number of significant figures indicates the precision of the value.

The number of significant figures (sf) that a number is quoted to is determined by counting digits from the largest (leftmost) nonzero digit to the end of the number. For example, π written to 6 sf would be 3.14159. The number of sf indicates the accuracy of the value.

This table shows some numbers and their reporting to various significant figures. Note that you should use trailing zeroes (shown in bold) if needed to report to the appropriate number of significant figures.

Distance (m) | 1 sf | 2 sf | 3 sf | 4 sf |
---|---|---|---|---|

32.9798563 | 0.03 km | 33 m | 33.0 m |
32.98 m |

0.0001348546 | 0.1 mm | 0.13 mm | 0.135 mm or 135 µm |
0.1349 mm or 134.9 µm |

1384.9328 | 1 km | 1.4 km | 1.38 km | 1385 m or 1.385 km |

3.04579839×10^{6} |
3 Mm | 3.0 Mm |
3.05 Mm | 3.046 Mm or 3046 km |

_{Table 1: Reporting distances to a variety of significant figures.}

When reporting to significant figures, be sure to choose the size of your unit, using an appropriate prefix, so that the accuracy of the number is clear. For example, if you were to report 32.9798563m to 1 sf as 30m, your reader could think that the number is correct to 2 sf (between 29.5m and 30.5m) as indicated by the trailing zero.

Also be sure to use a unit that your audience will understand. For example, if your reader would not understand Mm you could use km instead (3.04579839×10^{6} m reported to 1sf is 3×10^{3} km).

### Let’s look at some examples…

You are measuring the length of a 25 mm-long rod with a ruler. You can read a ruler (if you’re careful) in fractions of a millimeter, so reporting the length as 25.0 mm is reasonable, but 25.00 mm implies that you are able to read the ruler to the nearest 0.01 mm and reporting the length as 25 mm would imply that it could be between 24.5 and 25.5 mm. In this case, 3 sf are appropriate.

If you now measure the rod’s 2.6 mm diameter then this should be reported as 2.6 mm (assuming you have used your ruler correctly). Now you are only able to report to 2 sf.

The circumference of the rod is π times the diameter. How should this be reported?

2.6π = 8.168141, but if you report the value using this number, it implies that the diameter was known to about 7 sf. The circumference should be reported as 8.2 mm; this uses the same number of significant figures as the diameter.

Note that the range of possible circumferences for a diameter reported as 2.6 mm is between 2.55π mm = 8.0 mm and 2.65π mm = 8.3 mm, so reporting a circumference of 8.2 mm implies more precision in your measurement than you started with; if this is important then you could write the answer as 8.2 ± 0.2 mm.

### Remember the rules for rounding numbers

In order to report to an appropriate amount of significant figures, you may need to round your results. The methods for rounding numbers should be familiar to all students studying engineering, but here’s a quick reminder.

The rule for rounding to the nearest positive whole number is: if the decimal part is > 0.5, round up; if it is < 0.5 round down; if it is exactly 0.5, then round toward the nearest even number.^{1} The round towards even rule is to avoid always rounding upward numbers ending in 5. This avoids producing a higher average from a data set than would be expected.

This rule is easily generalised for rounding to different numbers of decimal places.

Here are some examples:

- Rounding 2.135 to 2 dp produces 2.14 (i.e. an even last digit)
- Rounding 2.143 to 2 dp produces 2.14
- Rounding 2.145 to 2 dp produces 2.14 (i.e. an even last digit)
- Rounding 2.1451 to 2 dp produces 2.15

## Discussion

Should you do the error analysis before you know how many sf to use in your results?

_{[1] Recommended in BS 2846-1:1991 “Guide to statistical interpretation of data. Routine analysis of quantitative data”.}

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