how to calculate absolute uncertainty
Absolute uncertainty is a measure of the range of values within which the true value of a measurement likely falls. It quantifies the precision of a measurement by indicating the interval around the reported value. Here's a step-by-step guide to calculate absolute uncertainty:
1. Identify the source of uncertainty: Uncertainty in measurements can arise from various sources, including the precision of the measurement instruments, environmental factors, or systematic errors. For most basic measurements, the uncertainty is often taken as the smallest division on the measurement instrument scale.
For example, if you are using a ruler with millimeter markings to measure length, the absolute uncertainty could be ±0.5 mm if you estimate between the lines.
2. Combine uncertainties: If you have several measurements with their individual uncertainties, you may need to combine these uncertainties to find the total uncertainty for a derived measurement.
- For addition or subtraction, absolute uncertainties should be added.
E.g., if A = a ± u(a) and B = b ± u(b), then A + B = (a + b) ± (u(a) + u(b))
- For multiplication or division, relative uncertainties (percentage uncertainties) are typically added, and then this sum is multiplied by the final result to obtain the absolute uncertainty.
E.g., if C = A × B, then u(C)/C = u(A)/A + u(B)/B, hence u(C) = C × (u(A)/A + u(B)/B)
3. Use standard uncertainty formulas: For more complex calculations, you can use standard formulas of uncertainty propagation. For example, when dealing with functions of multiple variables where each variable has an uncertainty, you would apply partial differentiation and combine the uncertainties accordingly.
4. Take into account significant figures: After calculating uncertainty, it is important to report it (and your measurement) to the correct number of significant figures. The general rule is that the uncertainty should be reported to one significant figure (or at most two if the first digit is a 1) and the measurement should be reported to the same decimal place as the uncertainty.
Let's look at an example:
Imagine you're measuring the length of an object using a ruler that has markings at every 1 mm. If you've determined the object's length to be 20.0 cm (200 mm) but the measurement could be off by half the smallest division of the ruler, then the absolute uncertainty would be ±0.5 mm. Thus, you would report the length as 20.0 cm ± 0.05 cm or 200 mm ± 0.5 mm, depending on the unit.
When reporting results, always include the units of uncertainty, and remember that the absolute uncertainty provides a range rather than a single fixed error value.