There are uncertainties associated with any measurement, and correct reporting of the results of any experiment will include these uncertainties. This document describes the type of uncertainties, the way to calculate them, and how they should be reported.
When the uncertainty is reported as a value in the same units as the measured quantity, it is referred to as an absolute uncertainty. The following are examples of values written with absolute uncertainty:
20 ± 0.2 meters 140 ± 12 kilograms 400 ± 13 days
Absolute uncertainties are often easy to determine, especially if you make the measurement and there is some uncertainty associated with the device used for measuring. For instance, if you use a ruler with millimeter increments to measure a piece of wood, you can only be certain of your results to within the millimeter. In this case, you would report a measured value of 10 cm as: 10 ± 0.1 cm
If the absolute uncertainty is converted to a percentage of the measured value, then the uncertainty can be given as a percent uncertainty. The same three examples above expressed as percent uncertainties would be written as:
20 m ± 1% 140 kg ± 8.6% 400 d ± 3.25%
Both absolute and percent uncertainties are important, since addition and multiplication of uncertain values will require using each.
Very often you may want to compare a result to a known value. One method by which you can report accuracy is percent error, which is a measure of how different your result is from a known value. The following formula is used to calculate percent error:
% Error = Measured −Known Known
When you calculate percent error, you use the absolute measured value, ignoring the uncertainty. Note that while the percent equation result is always positive, you may wish to report the error as negative if it is below the known value.
Example of Percent Error Calculation
A student weighs a block using a scale with 1 g increments, and records the mass as 343 ± 1 grams. If the known value of the mass is 351 grams, then the percent error is:
% Error = 343 − 351 351
×100% = 2.3%
The percent error could be reported as “2.3% below the known value”.
Computation of Uncertainties When Adding or Multiplying
Suppose you wish to add or multiply many different values, each of which has an associated uncertainty. What is the uncertainty of the final result? In general, the following two rules apply when doing such calculations:
1) If you are adding or subtracting measurements, the absolute uncertainty of the result is the sum of the absolute uncertainties of each measurement.
2) If you are multiplying or dividing measurements, the percent uncertainty of the result is the sum of the percent uncertainties of each measurement.
Note that one calculation gives absolute uncertainty and the other percent uncertainty. You may have to convert between the two depending on how you wish to present your results.
Example of Uncertainty When Adding
A student measures the voltage across three resistors and gets the following results:
Resistor 1: 6.5 ± 0.3 V Resistor 2: 8.3 ± 0.5 V Resistor 3: 1.5 ± 0.1 V The total voltage drop across all three resistors is the sum of all three voltages, and in this case would be equal to:
Total Voltage = Resistor 1 + Resistor 2 + Resistor 3 = 16.3 ± 0.9 V Expressed as a percent uncertainty, this result would be 16.3 V ± 5.5%
Example of Uncertainty When Multiplying or Dividing
You wish to determine the density of a block of wood by measuring the mass and volume.
The volume is calculated by measuring the three sides of the block. Including uncertainties, you measure the following three side lengths:
A = 6.3 ± 0.1 cm B = 8.4 ± 0.1 cm C = 2.7 ± 0.1 cm
The volume of the block is simply the product of the three sides: A * B * C. To correctly report the uncertainty of the volume, you need to add the percent uncertainty of each of the three sides. Written as percent uncertainties, the side lengths are:
A = 6.3 cm ± 1.6% B = 8.4 cm ± 1.2% C = 2.7 cm ± 3.7%
The volume of the block should therefore be reported as:
Volume = (6.3 * 8.4 * 2.7) cm ± (1.6 + 1.2 + 3.7)% = 142.9 cm3 ± 6.5%
Expressed using absolute uncertainty, the volume would be:
Volume = 142.9 ± 9.3 cm3
Using a scale with 1 g increments, you measure the mass as:
Mass = 82 ± 1 g
Which, expressed as a percent uncertainty, is:
Mass = 82 g ± 1.2%
The final calculation of density equals the mass divided by the volume. The percent uncertainty of the density is the sum of the percent uncertainty of volume and mass:
Density = (82/142.9) g/cm3 ± (6.5 + 1.2)% = 0.57 g/cm3 ± 7.7%
Expressed using absolute uncertainty:
Density = 0.57 ± 0.044 g/cm3