F. Percentage Uncertainty

The uncertainty of a measured value can also be presented as a percent or as a simple ratio.(the relative uncertainty). The percent uncertainty is familiar. It is computed as:


The percent uncertainty can be interpreted as describing the uncertainty that would result if the measured value had been100 units . A similar quantity is the relative uncertainty (or fractional uncertainty). It is simpler to compute and is given by:


The relative uncertainty can be interpreted as describing the uncertainty that would result if the measured value had been just one unit. With these two new representations for uncertainty, we must be careful in speech and writing so that our audience is clear about which one is being used. The following list describes accepted usage.

  • Absolute uncertainty: This is the simple uncertainty in the value itself as we have discussed it up to now. It is the term used when we need to distinguish this uncertainty from relative or percent uncertainties. If there is no chance of confusion we may still simply say “uncertainty” when referring to the absolute uncertainty. Absolute uncertainty has the same units as the value. Thus it is:3.8 cm ± 0.1 cm.
  • Relative Uncertainty: This is the simple ratio of uncertainty to the value reported. As a ratio of similar quantities, the relative uncertainty has no units. In fact there is no special symbol or notation for the relative uncertainty, so you must make it quite clear when you are reporting relative uncertainty.2.95 kg ± 0.043 (relative uncertainty)
  • Percent Uncertainty: This is the just the relative uncertainty multiplied by 100. Since the percent uncertainty is also a ratio of similar quantities, it also has no units. Fortunately there is a special notation for the percent uncertainty (%), so it will be easily recognized in writing.2.95 kg ± 4.3%
    Note that it is acceptable to report relative and percent uncertainties to two figures. This is to prevent rounding errors when we convert back to absolute uncertainty.

The percentage uncertainty is of great importance in comparing the relative accuracy of different measurements. For example, if we limit ourselves to 0.1 percent accuracy we know the length of a meter stick to 1 mm, of a bridge 1000 meters long to 1 meter, and the distance to the sun (93 million miles) to no better than 93,000 miles. Thus in giving the result of a measurement, one should carry enough figures to show the accuracy of the measurement, no more and no less , and should in addition state the A.D. or the percentage uncertainty. For the three examples given above one should write:

1.000 ± 0.001 meters (or 1.000 meters ± 0.l%)
1000 ± 1 meter (or 1000 meters ± 0.1%)
(93.00 ± 0.09) x 106 miles (or 93.00 x 106 miles ± 0.1%)

Last Updated November 19, 2014