The Relative Uncertainty Formula and How to Calculate It

The relative uncertainty or relative error  formula is used to calculate the uncertainty of a measurement compared to the size of the measurement. It is calculated as: relative uncertainty absolute error / measured value If a measurement is taken with respect to a standard or known value, calculate relative uncertainty as follows: relative uncertainty absolute error / known value Absolute error is the range of measurements in which the true value of a measurement likely lies. While absolute error carries the same units as the measurement, relative error has no units or else is expressed as a percent. Relative uncertainty is often represented using the lowercase Greek letter delta (ÃŽ ´). The importance of relative uncertainty is that it puts error in measurements into perspective. For example, an error of /- 0.5 centimeters may be relatively large when measuring the length of your hand, but very small when measuring the size of a room. Examples of Relative Uncertainty Calculations Example 1 Three 1.0 gram weights are measured at 1.05 grams, 1.00 grams, and 0.95 grams. The absolute error is  ± 0.05 grams.The relative error (ÃŽ ´) of your measurement is 0.05 g/1.00 g 0.05, or 5%. Example 2 A chemist measured the time required for a chemical reaction and found the value to be 155 /- 0.21 hours. The first step is to find the absolute uncertainty: absolute uncertainty 0.21 hoursrelative uncertainty Δt / t 0.21 hours / 1.55 hours 0.135 Example 3 The value 0.135 has too many significant digits, so it is shortened (rounded) to 0.14, which can be written as 14% (by multiplying the value times 100). The relative uncertainty (ÃŽ ´) in the measurement for the reaction time is: 1.55 hours /- 14% Sources   Golub, Gene, and Charles F. Van Loan. Matrix Computations – Third Edition. Baltimore: The Johns Hopkins University Press, 1996.Helfrick, Albert D., and William David Cooper. Modern Electronic Instrumentation and Measurement Techniques. Prentice Hall, 1989.