Suppose we measure the sides of a rectangle and find them to be
45.0 ± 0.1 cm and 544 ± 1 cm.
The area of the rectangle will be 45 X 544 = 24480 sq. cm. But, how accurate is our knowledge of the area? The question causes us to find a way to determine the uncertainty of a result that is calculated from uncertain inputs.
To estimate the uncertainty in a quantity computed from measurements each with a known uncertainty, you may use the following simplified rules. These rules yield an uncertainty estimate slightly larger than the probable uncertainty. See next section for a more rigorous procedure for uncertainty propagation (often called error propagation by other authors).
1. If two quantities are added or subtracted, we add the individual uncertainties to get the uncertainty in the results.
a. (324 ± 1 ) cm + (670 ± 1 ) cm = 994 ± 2 cm
b. (764 ± 1) cm – (670 ± 1 ) cm = 94 ± 2 cm
2. When multiplying or dividing, we add the percentage uncertainties to get the percentage uncertainty in the result. Note: you can use the relative uncertainty in place of the percentage uncertainty in these calculations and it will save you many factors of one hundred. Using the dimensions of the rectangle from above…
c. (544 ± 1) cm X (45.0 ± 0.1) cm
= (544 ± 0.2%) cm X (45.0 ± 0.2%) cm = 24480 ± 0.4% cm2 (=24480 ± 97.92 cm2)
= 24480 ± 100 cm2 (one sig. fig. in uncertainty!)
= (2.45 ± 0.01) x 102 cm2
Notice that the number of significant figures is automatically correct when we report uncertainties to one significant figure and make the uncertainty and the result agree in their least significant figure.
3. The procedure is the same for division.
d. (544 ± 1) cm/(45.0 ± 0.1) cm
= (544 ± 0.2%) / (45.0 ± 0.2%) = 12.089 ± 0.4%
= 12.089 ± 0.048
= 12.09 ± 0.05 (one sig. fig. in uncertainty!)
4. When finding the square root of a quantity, we divide the percentage uncertainty by 2 . When squaring we multiply the percentage uncertainty by two . Similar rules apply to other powers.
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Last Updated November 19, 2014
