Significant figures

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Rounding to n significant figures is a form of rounding. Significant figures (also called significant digits) can also refer to a crude form of error representation based around significant figure rounding. For this use, see Significance arithmetic.

Rounding to n significant figures is a more general-purpose technique than rounding to n decimal places, since it handles numbers of different scales in a uniform way. Computer representations of floating point numbers typically use a form of rounding to significant figures, but with binary numbers.

Contents 1 Identifying significant digits 2 Rounding conventions 3 Examples 4 Importance 4.1 Superfluous precision 4.2 Comprehension 5 Unimportance 6 Method 7 See also 8 References

[edit] Identifying significant digits

1. All non-zero digits are significant. Example: '123.45' has five significant figures: 1,2,3,4 and 5.

2. Zeros appearing in between two non-zero digits are significant. Example: '101.12' has five significant figures: 1,0,1,1,2.

3. All zeros appearing to the right of an understood decimal point or zeroes appearing to the right of non-zero digits after the decimal point are significant. Example: '12.2300' has six significant figures: 1,2,2,3,0 and 0. The number '0.00122300' still only has six significant figures (the zeros before the '1' are not significant). In addition, '12.00' has 4 significant figures.

4. All zeros appearing in a number without a decimal point and to the right of the last non-zero digit are not significant unless indicated by a bar. Example: '1300' has two significant figures: 1 and 3. The zeros are not considered significant because they don't have a bar. However, 1300.0 has five significant figures.

However, this last convention is not universally used; it is often necessary to determine from context whether trailing zeros in a number without a decimal point are intended to be significant.

Digits may be important without being 'significant' in this usage. For instance, the zeros in '1300' or '0.005' are not considered significant digits, but are still important as placeholders that establish the number's magnitude. A number with all zero digits (e.g. '0.000') has no significant digits, because the uncertainty is larger than the actual measurement.

[edit] Rounding conventions

When rounding to n significant digits, there are a few general rules that are followed:^[1]

• If the digit immediately to the right of the n^th significant digit is greater than 5, the number is rounded up.
• If the digit immediately to the right of the n^th significant digit is less than 5, the number is rounded down.
• If the digit immediately to the right of the n^th significant digit is 5 and there are non-zero digits after the 5, the number is rounded up.
• If the digit immediately to the right of the n^th significant digit is 5 and there are no subsequent non-zero digits, there are two commonly-used conventions (see rounding for longer discussion). In 'common rounding', such a digit is always rounded up; in 'unbiased rounding' (also known as 'round-to-even'), it is rounded in whichever direction leaves the n^th digit even. For instance, under unbiased rounding, 51.5 would be rounded up to 52, but 54.5 would be rounded down to 54.

[edit] Examples

Rounding to 2 significant figures:

• 12,300 becomes 12,000
• 0.00123 becomes 0.0012
• 0.1 becomes 0.10 (the trailing zero indicates that we are rounding to 2 significant figures).
• 0.02084 becomes 0.021
• 19,800 becomes 20,000
• 0.0125 becomes 0.012 in unbiased rounding, while it is 0.013 in biased.

One issue with rounding to n significant figures is that the value of n is not always clear. This occurs when the last significant figure is a zero to the left of the decimal point. For example, in the final example above, when 19,800 is rounded to 20,000, it is not clear from the rounded value what n was used - n could be anything from 1 to 5. The level of rounding can be specified explicitly. The abbreviation s.f. is sometimes used, for example "20,000 to 2 s.f.".^{[citation needed]} Scientific notation could be used to reduce the ambiguity, as in (2.0 × 10⁴). As always, the best approach is to state the uncertainty separately and explicitly, as in 20,000±1%, so that significant-figures rules do not apply. A less common method of presenting ambiguous significant figures is underlining the last significant figure of a number, for example "20000"

[edit] Importance

[edit] Superfluous precision

If a sprinter is measured to have completed a 100 m race in 11.71 seconds, what is the sprinter's average speed? By dividing the distance by the time using a calculator, we get a speed of 8.53970965 m/s.

The most straightforward way to indicate the precision of this result (or any result) is to state the uncertainty separately and explicitly, for example as 8.540±0.085 m/s or equivalently 8.540(85) m/s. This is particularly appropriate when the uncertainty itself is important and precisely known. In this case, it is safe and indeed advantageous to provide more digits than would be called for by the significant-figures rules.

If the degree of precision in the answer is not important, it is again safe to express trailing digits that are not known exactly, for example 8.5397 m/s.

If, however, we are forced to apply significant-figures rules, expressing the result as 8.53970965 m/s would seem to imply that the speed is known to the nearest 10 nm/s or thereabouts, which would improperly overstate the precision of the measurement. Reporting the result using three significant figures (8.54 m/s) might be interpreted as implying that the speed is somewhere between 8.535 and 8.544 m/s. This again overstates the accuracy, but not nearly so badly. Reporting the result using two significant figures (8.5 m/s) would introduce considerable roundoff error and degrade the precision of the result.

[edit] Comprehension

Numbers are often rounded off to make them easier to read. It's easier for someone to compare (say) 18% to 36% than to compare 18.148% to 35.922%. Similarly, when reviewing a budget, a series of figures like:

 Division A: $185,000 Division B: $ 45,000 Division C: $ 67,000 

is easier to understand and compare than a series like:

 Division A: $184,982 Division B: $ 44,689 Division C: $ 67,422 

To reduce ambiguity, such data are sometimes represented to the nearest order of magnitude, like:

 Revenue (in thousands of dollars):  Division A: 185  Division B:  45  Division C:  67 

[edit] Unimportance

People who are not experts in metrology or statistics can overestimate the usefulness of significant figures. The topic receives much more emphasis in high-school and undergraduate chemistry texts^{[2] [3]} than it does in real-world research laboratories.^{[4] [5]}

Practicing scientists commonly express uncertain quantities in the form 1.234±0.055 or equivalently 1.234(55). The point is that the uncertainty is expressed by one numeral (0.055), while the nominal value of the quantity is expressed by a separate numeral (1.234). Expressing these two things explicitly and separately is more sensible than trying to encode both the nominal value and the uncertainty into a single numeral.

As mentioned at the top of this article, "significant figures" primarily refers to a type of rounding, and is arguably appropriate when roundoff of the final answer is the dominant contribution to the uncertainty. However, there are innumerable important situations where roundoff of the final answer is not the dominant contribution to the uncertainty. Indeed, in experimental research (especially metrology), only in a very badly designed experiment would such roundoff error be dominant, because roundoff errors are so easily reduced. Furthermore, even when roundoff error is dominant, it is preferable to indicate this explicitly, as in 1.24(½) or equivalently 1.24(⁄).

Secondarily, "significant figures" may refer to a crude scheme for significance arithmetic, but as discussed in the significance arithmetic article and elswhere,^[6] there is generally not any rigorous way to express the uncertainty using significant figures.

In computer science and numerical analysis, good practice demands the use of guard digits. This is incompatible with any notion of significant figures. For a discussion, see Acton.^[7]

Good examples of how real scientists express uncertain quantities can be found in the NIST compendium of physical constants.^[4] None of the values there conform to any "significant figures" rules.

Procedures for how to properly represent uncertainty, and the rationale for these procedures, can be found in the references.^{[6] [5]}

[edit] Method

• Start with the left-most non-zero digit. E.g. the '1' in 1,000, or the '2' in 0.02.
• Keep n digits. Replace the rest with zeros.
• Round up by one if appropriate. For example, if rounding 0.039 to 1 significant figure, the result would be 0.04. There are several different rules for handling borderline cases - see rounding for more details.

Zeros appearing between nonzero digits are significant, for example:

• 60.8 has three significant figures
• 39008 has five significant figures

Zeros appearing in front of nonzero digits are not significant, for example:

• 0.093827 has five significant figures
• 0.0008 has one significant figure
• 0.012 has two significant figures

Zeros at the end of a number and to the right of a decimal are significant, for example:

• 35.00 has four significant figures
• 8,000.000000 has ten significant figures

Zeros at the end of a number without a decimal point may or may not be significant, and are therefore ambiguous, for example:

• 1,000 could have between one and four significant figures.

This ambiguity could be resolved by placing a decimal after the number, e.g. writing "1,000." to indicate specifically that four significant figures are meant.^[3]

To specify unambiguously how many significant figures are implied, scientific notation can be employed:

• 1×10³ or 1e3 has one significant figure, while
• 1.000×10³ has four.