Editing Significant figures (section)

== Rounding to significant figures ==
[[Rounding]] to significant figures is a more general-purpose technique than rounding to ''n'' digits, since it handles numbers of different scales in a uniform way. For example, the population of a city might only be known to the nearest thousand and be stated as 52,000, while the population of a country might only be known to the nearest million and be stated as 52,000,000. The former might be in error by hundreds, and the latter might be in error by hundreds of thousands, but both have two significant figures (5 and&nbsp;2). This reflects the fact that the significance of the error is the same in both cases, relative to the size of the quantity being measured.

To round a number to ''n'' significant figures:<ref>{{cite web
|title = Rounding Decimal Numbers to a Designated Precision
|author = Engelbrecht, Nancy|display-authors=etal
|year = 1990
|publisher = U.S. Department of Education
|location = Washington, D.C.
|url = https://archive.org/download/ERIC_ED327701/ERIC_ED327701.pdf
}}</ref><ref name="Numerical Mathematics and Computing, by Cheney and Kincaid">[https://books.google.com/books?id=ZUfVZELlrMEC&dq=Condition+Number+Rule+of+Thumb&pg=PA321 Numerical Mathematics and Computing, by Cheney and Kincaid].</ref>

# If the ''n'' + 1 digit is greater than 5 or is 5 followed by other non-zero digits, add 1 to the ''n'' digit. For example, if we want to round 1.2459 to 3 significant figures, then this step results in 1.25.
# If the ''n'' + 1 digit is 5 not followed by other digits or followed by only zeros, then rounding requires a [[Rounding#Tie-breaking|tie-breaking]] rule. For example, to round 1.25 to 2&nbsp;significant figures:
#* [[Rounding#Round half away from zero|Round half away from zero]] rounds up to 1.3. This is the default rounding method implied in many disciplines{{Citation needed|date=August 2018|reason=How many, and which, disciplines?}} if the required rounding method is not specified.
#* [[Rounding#Round half to even|Round half to even]], which rounds to the nearest even number. With this method, 1.25 is rounded down to 1.2. If this method applies to 1.35, then it is rounded up to 1.4. This is the method preferred by many scientific disciplines, because, for example, it avoids skewing the average value of a long list of values upwards.
# For an integer in rounding, replace the digits after the ''n'' digit with zeros. For example, if 1254 is rounded to 2&nbsp;significant figures, then 5 and 4 are replaced to 0 so that it will be 1300. For a number with the decimal point in rounding, remove the digits after the ''n'' digit. For example, if 14.895 is rounded to 3&nbsp;significant figures, then the digits after 8 are removed so that it will be 14.9.

In financial calculations, a number is often rounded to a given number of places. For example, to two places after the [[decimal separator]] for many world currencies. This is done because greater precision is immaterial, and usually it is not possible to settle a debt of less than the smallest currency unit.

In UK personal tax returns, income is rounded down to the nearest pound, whilst tax paid is calculated to the nearest penny.

As an illustration, the [[decimal]] quantity '''12.345''' can be expressed with various numbers of significant figures or decimal places. If insufficient precision is available then the number is [[rounding|rounded]] in some manner to fit the available precision. The following table shows the results for various total precision at two rounding ways (N/A stands for Not Applicable).
{| class="wikitable"
|-
! Precision !! Rounded to<br />significant figures !! Rounded to<br />decimal places
|-
| style=text-align:center |6
| 12.3450
| 12.345000
|-
| style=text-align:center |5
| 12.345
| 12.34500
|-
| style=text-align:center |4
| 12.34 or 12.35
| 12.3450
|-
| style=text-align:center |3
| 12.3
| 12.345
|-
| style=text-align:center |2
| 12
| 12.34 or 12.35
|-
| style=text-align:center |1 || 10 || 12.3
|-
| style=text-align:center |0 || {{n/a}} || 12
|}

Another example for '''0.012345'''. (Remember that the leading zeros are not significant.)

{| class="wikitable"
|-
! Precision !! Rounded to<br />significant figures !! Rounded to<br />decimal places
|-
| style=text-align:center |7
| 0.01234500
| 0.0123450
|-
| style=text-align:center |6
| 0.0123450
| 0.012345
|-
| style=text-align:center |5
| 0.012345
| 0.01234 or 0.01235
|-
| style=text-align:center |4
| 0.01234 or 0.01235
| 0.0123
|-
| style=text-align:center |3
| 0.0123
| 0.012
|-
| style=text-align:center |2
| 0.012
| 0.01
|-
| style=text-align:center |1
| 0.01
| 0.0
|-
| style=text-align:center |0 || {{n/a}} || 0
|}

The representation of a non-zero number ''x'' to a precision of ''p'' significant digits has a numerical value that is given by the formula:{{Citation needed|date=July 2017}}

<math display="block">10^n \cdot \operatorname{round}\left(\frac{x}{10^n}\right)</math>

where

<math display="block">n=\lfloor \log_{10} (|x|) \rfloor + 1 - p</math>

which may need to be written with a specific marking as detailed [[#Significant figures rules explained|above]] to specify the number of significant trailing zeros.