Significance arithmetic
Encyclopedia
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 operand
s 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
, on the other hand, have a limited number of significant digits, because these constants are known to us only by measurement.
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).
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.
For example, many see these as 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.
Significant figures
The significant figures of a number are those digits that carry meaning contributing to its precision. This includes all digits except:...
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
Significant figures
The significant figures of a number are those digits that carry meaning contributing to its precision. This includes all digits except:...
.
The rules of significance arithmetic are an approximation based on statistical rules for dealing with probability distributions. See the article on propagation of uncertainty
Propagation of uncertainty
In statistics, propagation of error is the effect of variables' uncertainties on the uncertainty of a function based on them...
for these more advanced and precise rules. Significance arithmetic rules rely on the assumption that the number of significant figures in the operand
Operand
In mathematics, an operand is the object of a mathematical operation, a quantity on which an operation is performed.-Example :The following arithmetic expression shows an example of operators and operands:3 + 6 = 9\;...
s gives accurate information about the uncertainty of the operands and hence the uncertainty of the result. For an alternative see interval arithmetic
Interval arithmetic
Interval arithmetic, interval mathematics, interval analysis, or interval computation, is a method developed by mathematicians since the 1950s and 1960s as an approach to putting bounds on rounding errors and measurement errors in mathematical computation and thus developing numerical methods that...
.
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:
- integerIntegerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
counts (e.g., the number of oranges in a bag) - definitions of one unit in terms of another (e.g. a minute is 60 seconds)
- actual prices asked or offered, and quantities given in requirement specifications
- legally defined conversions, such as international currency exchange
- scalar operations, such as "tripling" or "halving"
- mathematical constants, such as π and eE (mathematical constant)The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...
Physical constants such as Avogadro's number
Avogadro's number
In chemistry and physics, the Avogadro constant is defined as the ratio of the number of constituent particles N in a sample to the amount of substance n through the relationship NA = N/n. Thus, it is the proportionality factor that relates the molar mass of an entity, i.e...
, on the other hand, have a limited number of significant digits, because these constants are known to us only by measurement.
Multiplication and division using significance arithmetic
When multiplying or dividing numbers, the result is roundedRounding
Rounding a numerical value means replacing it by another value that is approximately equal but has a shorter, simpler, or more explicit representation; for example, replacing $23.4476 with $23.45, or the fraction 312/937 with 1/3, or the expression √2 with 1.414.Rounding is often done on purpose to...
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:
- 8 × 8 = 6 × 101
- 8 × 8.0 = 6 × 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 × 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:- 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 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
Rounding
Rounding a numerical value means replacing it by another value that is approximately equal but has a shorter, simpler, or more explicit representation; for example, replacing $23.4476 with $23.45, or the fraction 312/937 with 1/3, or the expression √2 with 1.414.Rounding is often done on purpose to...
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, many see these as important differences between significant figure rules and uncertainty:
- Uncertainty is not the same as a mistake. If the outcome of a particular experiment is reported as 1.234±0.056 it does not mean the observer made a mistake; 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 incorrect, 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).
- Manual, algebraic propagation of uncertaintyPropagation of uncertaintyIn statistics, propagation of error is the effect of variables' uncertainties on the uncertainty of a function based on them...
—the nominal topic of this article—is possible, but challenging. Alternative methods include the crank three times method and the Monte Carlo methodMonte Carlo methodMonte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
. Another option is interval arithmeticInterval arithmeticInterval arithmetic, interval mathematics, interval analysis, or interval computation, is a method developed by mathematicians since the 1950s and 1960s as an approach to putting bounds on rounding errors and measurement errors in mathematical computation and thus developing numerical methods that...
, 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 explicitly express the uncertainty in any uncertain result, the uncertainty should be given separately.
See also
- RoundingRoundingRounding a numerical value means replacing it by another value that is approximately equal but has a shorter, simpler, or more explicit representation; for example, replacing $23.4476 with $23.45, or the fraction 312/937 with 1/3, or the expression √2 with 1.414.Rounding is often done on purpose to...
- Propagation of uncertaintyPropagation of uncertaintyIn statistics, propagation of error is the effect of variables' uncertainties on the uncertainty of a function based on them...
- Significant figuresSignificant figuresThe significant figures of a number are those digits that carry meaning contributing to its precision. This includes all digits except:...
- Accuracy and precisionAccuracy and precisionIn the fields of science, engineering, industry and statistics, the accuracy of a measurement system is the degree of closeness of measurements of a quantity to that quantity's actual value. The precision of a measurement system, also called reproducibility or repeatability, is the degree to which...
- MANIAC IIIMANIAC IIIThe Maniac III was a second-generation electronic computer , built in 1961 for use at the Institute for Computer Research at the University of Chicago.It was designed by Nicholas Metropolis and constructed by the staff of the Institute for Computer...
External links
- The Decimal Arithmetic FAQ — Is the decimal arithmetic ‘significance’ arithmetic?
- Advanced methods for handling uncertainty and some explanations of the shortcomings of significance arithmetic and significant figures.
- Significant Figures Calculator – Displays a number with the desired number of significant digits.
- Measurements and Uncertainties versus Significant Digits or Significant Figures – Proper methods for expressing uncertainty, including a detailed discussion of the problems with any notion of significant digits.