Significance arithmetic

Significance arithmetic is a set of rules (sometimes called significant figure rules) for approximating the propagation of error in scientific or statistical calculations. These rules can be used to find the appropriate number of significant figures in the result of a calculation. This is important lest the result of a calculation be written with too many or too few significant figures and implicitly show an uncertainty that is either overconfident or underconfident. Understanding these rules requires a good understanding of the concept of significant and insignificant figures.

The rules are an approximation based on statistical rules for dealing with probability distributions. See the article on propagation of uncertainty for these more advanced and precise rules. Significance arithmetic rules rely on the assumption that the number of significant figures in the operands gives accurate information about the uncertainty of the operands and hence the uncertainty of the result. For an alternative see interval arithmetic.

An important caveat is that significant figures apply only to measured values. Values known to be exact should be ignored for determining the number of significant figures that belong in the result. Examples of such values include: Physical constants such as Avogadro's number have a limited number of significant digits, because these constants are only known to us by measurement.
 * integer counts (e.g., the number of oranges in a bag)
 * legally defined conversions, such as international currency exchange
 * constants that are defined arbitrarily (e.g., a millimeter is 0.001 meters)
 * scalar operations, such as "tripling" or "halving"
 * mathematical constants, such as π and e

Multiplication and division using significance arithmetic
When multiplying or dividing numbers, the result is rounded to the number of significant figures in the factor with the least significant figures. Here, the quantity of significant figures in each of the factors is important&mdash;not the position of the significant figures. For instance, using significance arithmetic rules:


 * 8 × 8 = 6 &times; 101
 * 8 × 8.0 = 6 &times; 101
 * 8.0 × 8.0 = 64
 * 8.02 × 8.02 = 64.3
 * 8 / 2.0 = 4
 * 8.6 /2.0012 = 4.3
 * 2 × 0.8 = 2

If, in the above, the numbers are assumed to be measurements (and therefore probably inexact) then "8" above represents an inexact measurement with only one significant digit. Therefore, the result of "8 × 8" is rounded to a result with only one significant digit, i.e., "6 &times; 101" instead of the unrounded "64" that one might expect. In many cases, the rounded result is less accurate than the non-rounded result; a measurement of "8" has an actual underlying quantity between 7.5 and 8.5. The true square would be in the range between 56.25 and 72.25. So 6 &times; 101 is the best one can give, as other possible answers give a false sense of accuracy. Further, the 6 &times; 101 is itself confusing (as it might be considered to imply 60 ±5, which is over-optimistic).

Addition and subtraction using significance arithmetic
When adding or subtracting using significant figures rules, results are rounded to the position of the least significant digit in the most uncertain of the numbers being summed (or subtracted). That is, the result is rounded to the last digit that is significant in each of the numbers being summed. Here the position of the significant figures is important, but the quantity of significant figures is irrelevant. Some examples using these rules:


 * 1 + 1.1 = 2
 * 1 is significant to the ones place, 1.1 is significant to the tenths place. Of the two, the least accurate is the ones place.  The answer cannot have any significant figures past the ones place.


 * 1.0 + 1.1 = 2.1
 * 1.0 and 1.1 are significant to the tenths place, so the answer will also have a number in the tenths place.


 * 100 + 110 = 200
 * 100 is significant to the hundreds place, while 110 is significant to the tens place. Therefore, the answer must be rounded to the nearest hundred.


 * 100. + 110. = 210.
 * 100. and 110. are both significant to the ones place (as indicated by the decimal), so the answer is also significant to the ones place.


 * 1×102 + 1.1×102 = 2×102
 * 100 is significant up to the hundreds place, while 110 is up to the tens place. Of the two, the least accurate is the hundreds place.  The answer should not have significant digits past the hundreds place.


 * 1.0×102 + 111 = 2.1×102
 * 1.0×102 is significant up to the tens place while 111 has numbers up until the ones place. The answer will have no significant figures past the tens place.


 * 123.25 + 46.0 + 86.26 = 255.5
 * 123.25 and 86.26 are significant until the hundredths place while 46.0 is only significant until the tenths place. The answer will be significant up until the tenths place.

Rounding rules
Because significance arithmetic involves rounding, it is useful to understand a specific rounding rule that it is wise to use when doing scientific calculations: the round-to-even rule (also called banker's rounding). It is especially useful when dealing with large data sets or doing calculations on large data sets.

This rule helps to eliminate the upwards skewing of data when using traditional rounding rules. Whereas traditional rounding always rounds up when the following digit is 5, Banker's sometimes rounds down to eliminate this upwards bias.

See the article on rounding for more information on rounding rules and a detailed explanation of the round-to-even rule.

Uncertainty and error

 * Uncertainty is not the same as error. If the outcome of a particular experiment is reported as 1.234±0.056 it does not mean the observer made an error;  it may be that the outcome is inherently statistical, and is best described by the expression 1.234±0.056.  To describe that outcome as 1.234±0.002 would be false and erroneous, even though it expresses less uncertainty.


 * Uncertainty is not the same as insignificance, and vice versa. An uncertain number may be highly significant (example: signal averaging).  Conversely, a completely certain number may be insignificant.


 * Significance is not the same as significant digits. Digit-counting is not as rigorous a way to represent significance as specifying the uncertainty separately and explicitly (such as 1.234±0.056).  Many examples of this can be seen in the NIST compendium of physical constants -- none of the values there conform to any "significant figures" rules.


 * Manual, algebraic propagation of uncertainty -- the nominal topic of this article -- is possible, but challenging. Alternative methods include the crank three times method and the Monte Carlo method.  Another option is interval arithmetic, which can provide a strict upper bound on the uncertainty, but generally it is not a tight upper bound (i.e. it does not provide a best estimate of the uncertainty).  For most purposes, Monte Carlo is more useful than interval arithmetic.

In order to express the uncertainty in any uncertain result, the uncertainty should be given separately and explicitly. There is generally not any rigorous way to express the uncertainty using significant figures.