Significant figures

The significant figures (also known as the significant digits, precision or resolution) of a number written in positional notation are digits that carry meaningful contributions to its measurement resolution. This includes all digits except:[1]

All leading zeros. For example, "013" has two significant figures: 1 and 3
Trailing zeros when they are merely placeholders to indicate the scale of the number (exact rules are explained at identifying significant figures)
Spurious digits introduced, for example, by calculations carried out to greater precision than that of the original data, or measurements reported to a greater precision than the equipment supports.

Of the significant figures in a number, the most significant is the position with the highest exponent value (the left-most in normal decimal notation), and the least significant is the position with the lowest exponent value (the right-most in normal decimal notation). For example, in the number "123", the "1" is the most significant figure as it counts hundreds (10²), and "3" is the least significant figure as it counts ones (10⁰).

Significance arithmetic is a set of approximate rules for roughly maintaining significance throughout a computation. The more sophisticated scientific rules are known as propagation of uncertainty.

Numbers are often rounded to avoid reporting insignificant figures. For example, it would create false precision to express a measurement as 12.34525 kg (which has seven significant figures) if the scales only measured to the nearest gram and gave a reading of 12.345 kg (which has five significant figures). Numbers can also be rounded merely for simplicity rather than to indicate a given precision of measurement, for example, to make them faster to pronounce in news broadcasts.

Radix 10 is assumed in the following.

Identifying significant figures

Significant figures rules explained

Digits in light blue are significant figures; those in black are not

The rules for identifying significant figures when writing or interpreting numbers are as follows:[2]

All non-zero digits are considered significant. For example, 91 has two significant figures (9 and 1), while 123.45 has five significant figures (1, 2, 3, 4 and 5).
Zeros appearing anywhere between two significant figures are significant: 101.1203 has seven significant figures: 1, 0, 1, 1, 2, 0 and 3.
Zeros to the left of the significant figures (leading zeros) are not significant. For example, 0.00052 has two significant figures: 5 and 2.
Zeros to the right of the non-zero digits (trailing zeros) are significant if they are to the right of the decimal point as these are only necessary to indicate precision. However trailing zeros in the ones place or higher may or may not be significant, depending on the precision of the measurement. Thus 1.20 and 0.0980 have three significant figures whereas 45,600 may have 3, 4 or 5 significant figures. Note that 120.00 would have five significant figures - the zero to the left of the decimal is significant because it is between two significant figures (the 2 and the zeros to the right of the decimal point).

The significance of trailing zeros in a number not containing a decimal point can be ambiguous. For example, it may not always be clear if the number 1300 is precise to the nearest unit (and just happens coincidentally to be an exact multiple of a hundred) or if it is only shown to the nearest hundreds due to rounding or uncertainty. Many conventions exist to address this issue. However, these are not universally used and would only be effective if the reader is familiar with the convention:

An overline, sometimes also called an overbar, or less accurately, a vinculum, may be placed over the last significant figure; any trailing zeros following this are insignificant. For example, 1300 has three significant figures (and hence indicates that the number is precise to the nearest ten).

Less often, using a closely related convention, the last significant figure of a number may be underlined; for example, "1300" has two significant figures.

A decimal point may be placed after the number; for example "1300." indicates specifically that trailing zeros are meant to be significant.[3]

As the conventions above are not in general use, the following more widely recognized options are available for indicating the significance of number with trailing zeros:

Eliminate ambiguous or non-significant zeros by changing the unit prefix in a number with a unit of measurement. For example, the precision of measurement specified as 1300 g is ambiguous, while if stated as 1.30 kg it is not. Likewise 0.0123 L can be rewritten as 12.3 mL

Eliminate ambiguous or non-significant zeros by using Scientific Notation: For example, 1300 with three significant figures becomes 1.30×10³. Likewise 0.0123 can be rewritten as 1.23×10⁻². The part of the representation that contains the significant figures (1.30 or 1.23) is known as the significand or mantissa. The digits in the base and exponent (10³ or 10⁻²) are considered exact numbers so for these digits, significant figures are irrelevant.

Explicitly state the number of significant figures (the abbreviation s.f. is sometimes used): For example "20 000 to 2 s.f." or "20 000 (2 sf)".

State the expected variability (precision) explicitly with a plus-minus sign, as in 20 000 ± 1%. This also allows specifying a range of precision in-between powers of ten.

Rounding and decimal places

The basic concept of significant figures is often used in connection with rounding. Rounding to 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. 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 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 to n significant figures:[4][5]

Identify the significant figures before rounding. These are the n consecutive digits beginning with the first non-zero digit.
If the digit immediately to the right of the last significant figure is greater than 5 or is a 5 followed by other non-zero digits, add 1 to the last significant figure. For example, 1.2459 as the result of a calculation or measurement that only allows for 3 significant figures should be written 1.25.
If the digit immediately to the right of the last significant figure is a 5 not followed by any other digits or followed only by zeros, rounding requires a tie-breaking rule. For example, to round 1.25 to 2 significant figures:
- Round half away from zero (also known as "5/4") rounds up to 1.3. This is the default rounding method implied in many disciplines if not specified.
- Round half to even, which rounds to the nearest even number, rounds down to 1.2 in this case. The same strategy applied to 1.35 would instead round 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.
Replace non-significant figures in front of the decimal point by zeros.
Drop all the digits after the decimal point to the right of the significant figures (do not replace them with zeros).

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 digits or decimal places. If insufficient precision is available then the number is rounded in some manner to fit the available precision. The following table shows the results for various total precisions and decimal places.

Precision	Rounded to significant figures	Rounded to decimal places
6	12.3450	12.345000
5	12.345	12.34500
4	12.34 or 12.35	12.3450
3	12.3	12.345
2	12	12.34 or 12.35
1	10	12.3
0	N/A	12

Another example for 0.012345:

Precision	Rounded to significant figures	Rounded to decimal places
7	0.01234500	0.0123450
6	0.0123450	0.012345
5	0.012345	0.01234 or 0.01235
4	0.01234 or 0.01235	0.0123
3	0.0123	0.012
2	0.012	0.01
1	0.01	0.0
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:

10^{n}\cdot \operatorname {round} \left({\frac {x}{10^{n}}}\right)

where

n=\lfloor \log _{10}(|x|)\rfloor +1-p

which may need to be written with a specific marking as detailed above to specify the number of significant trailing zeros.

Arithmetic

As there are rules for determining the number of significant figures in directly measured quantities, there are rules for determining the number of significant figures in quantities calculated from these measured quantities.

Only measured quantities figure into the determination of the number of significant figures in calculated quantities. Exact mathematical quantities like the $π$ in the formula for the area of a circle with radius $r$ , $π r 2$ has no effect on the number of significant figures in the final calculated area. Similarly the $½$ in the formula for the kinetic energy of a mass $m$ with velocity $v$ , $½ mv 2$ , has no bearing on the number of significant figures in the final calculated kinetic energy. The constants $π$ and $½$ are considered for this purpose to have an infinite number of significant figures.

For quantities created from measured quantities by multiplication and division, the calculated result should have as many significant figures as the measured number with the least number of significant figures.[6] For example,

1.234 × 2.0 = 2.468... ≈ 2.5,

with only two significant figures. The first factor has four significant figures and the second has two significant figures. The factor with the least number of significant figures is the second one with only two, so the final calculated result should also have a total of two significant figures. However see below regarding intermediate results.

For quantities created from measured quantities by addition and subtraction, the last significant decimal place (hundreds, tens, ones, tenths, and so forth) in the calculated result should be the same as the leftmost or largest decimal place of the last significant figure out of all the measured quantities in the terms of the sum. For example,

100.0 + 1.234 = 101.234... ≈ 101.2

with the last significant figure in the tenths place. The first term has its last significant figure in the tenths place and the second term has its last significant figure in the thousandths place. The leftmost of the decimal places of the last significant figure out of all the terms of the sum is the tenths place from the first term, so the calculated result should also have its last significant figure in the tenths place.

The rules for calculating significant figures for multiplication and division are opposite to the rules for addition and subtraction. For multiplication and division, only the total number of significant figures in each of the factors matters; the decimal place of the last significant figure in each factor is irrelevant. For addition and subtraction, only the decimal place of the last significant figure in each of the terms matters; the total number of significant figures in each term is irrelevant. However, greater accuracy will often be obtained if some non-significant digits are maintained in intermediate results which are used in subsequent calculations.

In a base 10 logarithm of a normalized number, the result should be rounded to the number of significant figures in the normalized number. For example, log₁₀(3.000×10⁴) = log₁₀(10⁴) + log₁₀(3.000) ≈ 4 + 0.47712125472, should be rounded to 4.4771.

When taking antilogarithms, the resulting number should have as many significant figures as the mantissa in the logarithm.

When performing a calculation, do not follow these guidelines for intermediate results; keep as many digits as is practical (at least 1 more than implied by the precision of the final result) until the end of calculation to avoid cumulative rounding errors.[7]

Estimating tenths

When using a ruler, initially use the smallest mark as the first estimated digit. For example, if a ruler's smallest mark is 0.1 cm, and 4.5 cm is read, it is 4.5 (±0.1 cm) or 4.4 – 4.6 cm. However, in practice a measurement can usually be estimated by eye to closer than the interval between the ruler's smallest mark, e.g. in the above case it might be estimated as between 4.51 cm and 4.53 cm (see below).

It is also possible that the overall length of a ruler may not be accurate to the degree of the smallest mark, and the marks may be imperfectly spaced within each unit. However assuming a normal good quality ruler, it should be possible to estimate tenths between the nearest two marks to achieve an extra decimal place of accuracy.[8] Failing to do this adds the error in reading the ruler to any error in the calibration of the ruler.[9]

Estimation

When estimating the proportion of individuals carrying some particular characteristic in a population, from a random sample of that population, the number of significant figures should not exceed the maximum precision allowed by that sample size.

Relationship to accuracy and precision in measurement

Traditionally, in various technical fields, "accuracy" refers to the closeness of a given measurement to its true value; "precision" refers to the stability of that measurement when repeated many times. Hoping to reflect the way the term "accuracy" is actually used in the scientific community, there is a more recent standard, ISO 5725, which keeps the same definition of precision but defines the term "trueness" as the closeness of a given measurement to its true value and uses the term "accuracy" as the combination of trueness and precision. (See the Accuracy and precision article for a fuller discussion.) In either case, the number of significant figures roughly corresponds to precision, not to either use of the word accuracy or to the newer concept of trueness.

In computing

Computer representations of floating-point numbers use a form of rounding to significant figures, in general with binary numbers. The number of correct significant figures is closely related to the notion of relative error (which has the advantage of being a more accurate measure of precision, and is independent of the radix, also known as the base, of the number system used).

References

Chemistry in the Community; Kendall-Hunt:Dubuque, IA 1988
Giving a precise definition for the number of correct significant digits is surprisingly subtle, see Higham, Nicholas (2002). Accuracy and Stability of Numerical Algorithms (PDF) (2nd ed.). SIAM. pp. 3–5.
Myers, R. Thomas; Oldham, Keith B.; Tocci, Salvatore (2000). Chemistry. Austin, Texas: Holt Rinehart Winston. p. 59. ISBN 0-03-052002-9.
Engelbrecht, Nancy; et al. (1990). "Rounding Decimal Numbers to a Designated Precision" (PDF). Washington, D.C.: U.S. Department of Education.
Numerical Mathematics and Computing, by Cheney and Kincaid.
"Significant Figure Rules". Penn State University.
de Oliveira Sannibale, Virgínio (2001). "Measurements and Significant Figures (Draft)" (PDF). Freshman Physics Laboratory. California Institute of Technology, Physics Mathematics And Astronomy Division. Archived from the original (PDF) on 2013-06-18.
Experimental Electrical Testing. Newark, NJ: Weston Electrical Instruments Co. 1914. p. 9. Retrieved 2019-01-14. Experimental Electrical Testing..
"Measurements". slc.umd.umich.edu. University of Michigan. Retrieved 2017-07-03.

External links

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[1] Chemistry in the Community; Kendall-Hunt:Dubuque, IA 1988

[2] Giving a precise definition for the number of correct significant digits is surprisingly subtle, see Higham, Nicholas (2002). Accuracy and Stability of Numerical Algorithms (PDF) (2nd ed.). SIAM. pp. 3–5.

[Chemistry_Significant_Figures-3] Myers, R. Thomas; Oldham, Keith B.; Tocci, Salvatore (2000). Chemistry. Austin, Texas: Holt Rinehart Winston. p. 59. ISBN 0-03-052002-9.

[4] Engelbrecht, Nancy; et al. (1990). "Rounding Decimal Numbers to a Designated Precision" (PDF). Washington, D.C.: U.S. Department of Education.

[Numerical_Mathematics_and_Computing,_by_Cheney_and_Kincaid-5] Numerical Mathematics and Computing, by Cheney and Kincaid.

[6] "Significant Figure Rules". Penn State University.

[7] Oliveira Sannibale, Virgínio (2001). "Measurements and Significant Figures (Draft)" (PDF). Freshman Physics Laboratory. California Institute of Technology, Physics Mathematics And Astronomy Division. Archived from the original (PDF) on 2013-06-18.

[Weston-8] Experimental Electrical Testing. Newark, NJ: Weston Electrical Instruments Co. 1914. p. 9. Retrieved 2019-01-14. Experimental Electrical Testing..

[UMmeasurements-9] "Measurements". slc.umd.umich.edu. University of Michigan. Retrieved 2017-07-03.