Significance Arithmetic - Disagreements About Importance

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 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.