circumstances, 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 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. Kahan considers significance arithmetic to be unreliable as a form of automated error analysis. 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 Rounding Propagation of uncertainty Significant figures Accuracy and precision MANIAC III Loss of significance References Further reading ASTM E29-06b, Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications 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. Numerical analysis Elementary arithmetic