Percent Uncertainty

One method of expressing uncertainty is as a percent of the measured value. If a measurement \(A\) is expressed with uncertainty, \(\delta A\), the percent uncertainty (%unc) is defined to be

There is an uncertainty in anything calculated from measured quantities. For example, the area of a floor calculated from measurements of its length and width has an uncertainty because the length and width have uncertainties. How big is the uncertainty in something you calculate by multiplication or division? If the measurements going into the calculation have small uncertainties (a few percent or less), then the method of adding percents can be used for multiplication or division. This method says that the percent uncertainty in a quantity calculated by multiplication or division is the sum of the percent uncertainties in the items used to make the calculation. For example, if a floor has a length of \(4.00\;m\) and a width of \(3.00\;m\), with uncertainties of \(2\%\) and \(1\%\), respectively, then the area of the floor is \(12.0\; m^2\) and has an uncertainty of \(3\%\). (Expressed as an area this is \(0.36\;m\), which we round to \(0.4\;m^2\) since the area of the floor is given to a tenth of a square meter.)

An important factor in the accuracy and precision of measurements involves the precision of the measuring tool. In general, a precise measuring tool is one that can measure values in very small increments. For example, a standard ruler can measure length to the nearest millimeter, while a caliper can measure length to the nearest 0.01 millimeter. The caliper is a more precise measuring tool because it can measure extremely small differences in length. The more precise the measuring tool, the more precise and accurate the measurements can be.

When we express measured values, we can only list as many digits as we initially measured with our measuring tool. For example, if you use a standard ruler to measure the length of a stick, you may measure it to be 36.7 \(\mathrm{cm}\). You could not express this value as 36.71 \(\mathrm{cm}\) because your measuring tool was not precise enough to measure a hundredth of a centimeter. It should be noted that the last digit in a measured value has been estimated in some way by the person performing the measurement. For example, the person measuring the length of a stick with a ruler notices that the stick length seems to be somewhere in between 36.6 \(\mathrm{cm}\) and 36.7 \(\mathrm{cm}\), and he or she must estimate the value of the last digit.

Using the method of significant figures, the rule is that the last digit written down in a measurement is the first digit with some uncertainty. In order to determine the number of significant digits in a value, start with the first measured value at the left and count the number of digits through the last digit written on the right. For example, the measured value 36.7 \(\mathrm{cm}\) has three digits, or significant figures. Significant figures indicate the precision of a measuring tool that was used to measure a value.

Special consideration is given to zeros when counting significant figures. The zeros in 0.053 are not significant, because they are only placekeepers that locate the decimal point. There are two significant figures in 0.053. The zeros in 10.053 are not placekeepers but are significant—this number has five significant figures. The zeros in 1300 may or may not be significant depending on the style of writing numbers. They could mean the number is known to the last digit, or they could be placekeepers. So 1300 could have two, three, or four significant figures. (To avoid this ambiguity, write 1300 in scientific notation.) Zeros are significant except when they serve only as placekeepers.