Chapter 5. Propagation of Uncertainty and Interval Arithmetic
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# Chapter 5. Propagation of Uncertainty and Interval Arithmetic

If interval arithmetic is not activated, propagation of uncertainty is estimated using the variance formula, using partial derivatives, as shown below. This applies to numbers, defined using the interval() function or “±” notation, or with “read precision” enabled, and variables, including physical constants, with specified standard uncertainty. The uncertainty will be calculated separately for non-differentiable functions.

Figure 5.1. Variance Formula

When interval arithmetic is activated, it replaces the variance formula for interval calculations and estaimation of uncertainty propagation. Interval arithmetic is also used implicitly for all approximate calculations to keep track of precision changes, and gracefully handle for example catastrophic cancellation (in subtraction two nearly equal numbers). Interval arithmetic can be (de)activated using Mode+Approximation → Interval Arithmetic or from the corresponding menu on the keypad.

Some non-invertible functions (including bessel and airy functions) are not properly supported and only the function values for the endpoints of the interval are calculated. Trigonometric functions should however give correct intervals for real and imaginary numbers, but will in some cases for complex numbers with both a real and imaginary part return a too wide interval.

Generally, the resulting interval will be guaranteed to include the true interval, but may for non-trivial expression (especially involving complex numbers) return an interval that is too wide.

The result is by default shown as an ordinary number with the number of significant digits determined by the size of the uncertainty (“2.11±0.03 = 2.1”), or the width of interval, unless the interval is too wide. If the interval() function has been used in the expression the result is by default displayed as an interval, and if “±” notation has been used, it will also be used in the result. The default behaviour can be changed fromMode+Interval Display. The midpoint alternative displays the value halfway between the lower and upper limit of the interval (“interval(2.075, 2.15) = 2.1125”). Note that, for the plus/minus notation, the same midpoint (note that when interval arithmetic is enabled this does not necessarily equal the result for the midpoint of intervals in the expression) is displayed in front of the plus/minus symbol. In plus/minus notation, the uncertainty is displayed with two significant digits, with the exception that all digits before the decimal separator are always shown.

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