Calculating Weighing Uncertainty, Significant Figures
Raj had a question for Paul concerning significant figures. When his group measured the weight in a four decimal place balance, i.e., 0.0001 grams, he wondered how they should go about rounding the value.
First, determine the linearity for the balance from the calibration certificate. For this example, we will assume ± 0.15 mg for a 4-place balance. Balance manufacturers recommend the use of a rectangular distribution to convert the linearity contribution to a standard uncertainty. Therefore, the balance linearity contribution is 0.15mg/(3)^1/2 = 0.09 mg. This contribution may have to be counted twice if the sample is weighed by difference -- once for the tare and once for the gross weight, giving a standard uncertainty that equals (2 x (0.09)2)^1/2 = 0.13 mg.
Thusly, a weight of 0.0413 g would have a standard uncertainty of 0.13 mg. Using a coverage factor of 2, an uncertainty of 0.26 mg (that would be rounded to 0.3 mg) would be reported. Using this example, the weight would be reported as 0.0413 grams ± 0.0003 grams and the number of significant figures is 3.
I hope this helps. Please see this our corresponding article, Significant Figures and Uncertainty, for additional help with significant figures and uncertainty calculations.
Serving you in chemistry,
Paul R. Gaines, Ph.D.
CEO of Inorganic Ventures & Fellow Chemist
DISCLAIMER: Advice offered by the chemists at Inorganic Ventures is intended for the individual posing the question. Feel free to contact us to verify whether these suggestions apply to your unique circumstances.