Significance arithmetic

Significance arithmetic is a set of rules (sometimes called significant figure rules) for approximating the propagation of uncertainty in scientific or statistical calculations. These rules can be used to find the appropriate number of significant figures to use to represent the result of a calculation. If a calculation is done without analysis of the uncertainty involved, a result that is written with too many significant figures can be taken to imply a higher precision than is known, and a result that is written with too few significant figures results in an avoidable loss of precision. Understanding these rules requires a good understanding of the concept of significant and insignificant figures.

The rules of significance arithmetic 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, however, have a limited number of significant digits, because these constants are known to us only by measurement. On the other hand, c (speed of light) is exactly 299,792,458 m/s by definition.

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—not the position of the significant figures. For instance, using significance arithmetic rules:

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 × 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 × 101 is the best one can give, as other possible answers give a false sense of accuracy. Further, the 6 × 101 is itself confusing (as it might be considered to imply 60 ±5, which is over-optimistic; more accurate would be 64 ±8).

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:

Transcendental functions

Transcendental functions have a complicated method for determining the significance of the result. These include the logarithm function, the exponential function and the trigonometric functions. The significance of the result depends on the condition number. In general, the number of significant figures for the result is equal to the number of significant figures for the input minus the order of magnitude of the condition number.

The condition number of a differentiable function f at a point x is see Condition number: One variable for details. Note that if a function has a zero at a point, its condition number at the point is infinite, as infinitesimal changes in the input can change the output from zero to non-zero, yielding a ratio with zero in the denominator, hence an infinite relative change. The condition number of the mostly used functions are as follows;[1] these can be used to compute significant figures for all elementary functions:

Rounding rules

Because significance arithmetic involves rounding, it is useful to understand a specific rounding rule that is often used when doing scientific calculations: the round-to-even rule (also called banker's rounding). It is especially useful when dealing with 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, bankers sometimes round 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.

Disagreements about importance

Significant figures are used extensively in high school and undergraduate courses as a shorthand for the precision with which a measurement is known. However, significant figures are not a perfect representation of uncertainty, and are not meant to be. Instead, they are a useful tool for avoiding expressing more information than the experimenter actually knows, and for avoiding rounding numbers in such a way as to lose precision.

For example, here are some important differences between significant figure rules and uncertainty:

In order to explicitly express the uncertainty in any uncertain result, the uncertainty should be given separately, with an uncertainty interval, and a confidence interval. The expression 1.23 U95 = 0.06 implies that the true (unknowable) value of the variable is expected to lie in the interval from 1.17 to 1.29 with at least 95% confidence. If the confidence interval is not specified it has traditionally been assumed to be 95% corresponding to two standard deviations from the mean. Confidence intervals at one standard deviation (68%) and three standard deviations (99%) are also commonly used.

See also


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