Significant figures

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.

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 and non-zero digits 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).

4. All zeros appearing in a number without a decimal point and to the right of a 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 without a decimal point are intended to be significant.

Rounding conventions

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

• If the digit immediately to the right of the n^th digit is greater than 5, the number is rounded up.
• If the digit immediately to the right of the n^th digit is less than 5, the number is rounded down.
• If the digit immediately to the right of the n^th 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 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.

Examples

Rounding to 2 significant figures:

• 12,300 becomes 12,000
• 0.00123 become 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

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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. 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.". More commonly, scientific notation is used to handle the ambiguity.

Importance

The number of significant digits in a stated number is important for several reasons:

• Spurious accuracy: If a sprinter is measured to have completed a 100.0 m race in 11.71 seconds, what is his average speed? By dividing the distance by the time using a calculator, we get a speed of 8.53970965 m/s Obviously, the speed of the sprinter is not known to the nearest 10 nm/s, so it is more sensible to report it to four significant figures (8.540 m/s), because the time is only measured to four significant figures.
• Comprehension - 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

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

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

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.

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.

See also

• Arithmetic precision