Regardless of the measurement and the quality of the instrumentation, there is always uncertainty associated with a physical measurement. This uncertainty is a combination of that associated with the instrument and that related to the system being measured. An example of the former is the inability to exactly determine the position of a length measurement between the lines on a meterstick. An example of uncertainty related to the system being measured is the variation of temperature within a sample of water so that a single temperature for the sample is difficult to determine.

Uncertainties can be expressed in two ways: absolute uncertainty and percent uncertainty

Some laboratory experiments, such as one that measures free-fall acceleration, may involve finding a value that is already known. In this type of experiment, the accuracy of your measurements can be determined by comparing your results with the accepted value. The accuracy of a measurement refers to how close that measurement is to the accepted value for the quantity being measured. Precision depends on the instruments used to measure a quantity. A meterstick that includes millimeters, for example, will give a more precise result than a meterstick whose smallest unit of measure is a centimeter. Thus, a measurement of $9.61 \;\mathrm{m/s^2}$ for free-fall acceleration is more precise than a measurement of $9.8 \;\mathrm{m/s^2}$, but $9.8 \;\mathrm{m/s^2}$ is more accurate than $9.61 \;\mathrm{m/s^2}$.

Absolute uncertainty refers to an uncertainty expressed in the same units as the measurement. Therefore, the length of a computer disk label might be expressed as:

$(5.5 \pm 0.1) \;\mathrm{cm}$

The uncertainty of $\pm 0.1$ cm by itself is not descriptive enough for some purposes, however. This uncertainty is large if the measurement is $1.0$ cm, but it is small if the measurement is $100$ m. For this reason, the percentage uncertainty, or relative uncertainty, is often more meaningful than the absolute uncertainty because relative uncertainty takes the size of the quantity being measured into account.

Fractional uncertainty or percent uncertaintyis used to give a more descriptive account of the uncertainty. In this type of description, the uncertainty is divided by the actual measurement. Therefore, the length of the computer disk label could be expressed as:

$5.5 \;\mathrm{cm} \pm \dfrac{0.1}{5.5} = 5.5 \;\mathrm{cm} \pm 0.018$       (fractional uncertainty)

$5.5 \;\mathrm{cm} \pm 1.8\%$       (percent uncertainty)

When combining measurements in a calculation, the percent uncertainty in the final result is generally larger than the uncertainty in the individual measurements. This is called propagation of uncertainty and is one of the challenges of experimental physics.

Some simple rules can provide a reasonable estimate of the uncertainty in a calculated result:

 

Multiplication and division:

When measurements with uncertainties are multiplied or divided, add the percent uncertainties to obtain the percent uncertainty in the result.

 

Addition and subtraction:

When measurements with uncertainties are added or subtracted, add the absolute uncertainties to obtain the absolut uncertainty in the result.

 

Powers:

If a measurement is taken to a power, the percent uncertainty is multiplied by that power to obtain the percent uncertainty in the result.

 

For complicated calculations, many uncertainties are added together, which can cause the uncertainty in the final result to be undesirably large. Experiments should be designed such that calculations are as simple as possible.

Notice that uncertainties in a calculation always add. As a result, an experiment involving a subtraction should be avoided if possible, especially if the measurements being subtracted are close together. The result of such a calculation is a small difference in the measurements and uncertainties that add together. It is possible that the uncertainty in the result could be larger than the result itself!

Error Propagation Calculator (Absolute uncertainty)

 

Exercises

With help from the preceding uncertainty rules, try to solve the following exmples:

• Find the change in temperature, with associated uncertainty, when the temperature increases from $(27.6 \pm 1.5) \;\mathrm{^o C}$ to $(99.2 \pm 1.5) \;\mathrm{^oC}$    Answer

  Answer

  Because the result is a subtraction, add the uncertainties:

  $\Delta T=T_2 -T_1 = (99.2 \pm 1.5) \;\mathrm{^o C} - (27.6 \pm 1.5) \;\mathrm{^o C}$

  $\Delta T=T_2 -T_1 = (71.6 \pm 3.0) \;\mathrm{^o C} = 71.6 \;\mathrm{^o C} \pm 4.2 \%$.

• Find the area, with associated uncertainty, of a rectangular plate of dimensions $5.5 \;\mathrm{cm} \pm 1.8 \%$ by $6.4 \;\mathrm{cm} \pm 1.6 \%$    Answer

  Answer

  Because the result is a multiplication, add the percent uncertainties:
  $A=lw = (5.5 \;\mathrm{cm} \pm 1.8 \%)(6.4 \;\mathrm{cm} \pm 1.6 \%)$

  $A=lw = 35 \;\mathrm{cm} \pm 3.4 \% = (35 \pm 1) \;\mathrm{cm^2}$.
• Find the volume of a sphere of radius $6.20 \;\mathrm{cm} \pm 2.0 \%$    Answer

  Answer

  Because the result is determined by raising a quantity to a power, multiply the power by the percent uncertainty:
  $V =\dfrac{4}{3} \pi r^3 = \dfrac{4}{3} \pi (6.20 \;\mathrm{cm} \pm 2.0 \%)^3 \\ V= 998 \;\mathrm{cm^3} \pm 6.0 \% = (998 \pm 60) \;\mathrm{cm^3}$.