Percent Uncertainty

In other science courses, you may have seen the concept of percent error:

    \[ \frac{\mathrm{measured} - \mathrm{true}}{\mathrm{true}} \times 100 = \mathrm{percent \, error} \]

However, as described in the Introduction to Statistical and Systematic Uncertainty, the true value is actually generally unknowable.  Thus, this concept of percent error really only makes sense if there is some accepted value without an uncertainty to which you can compare.

A much more useful idea is the :

    \[ \frac{\sigma}{\bar{x}} \times 100 = \mathrm{percent \, uncertainty} \]

where x is the mean as already discussed in Mean and σ is the uncertainty (the reason for this choice of symbol will be made clear later).


Consider two distance measurements each with an uncertainty of

    \[ \sigma = 10 \, \mathrm{cm} = 0.1 \, \mathrm{m.} \]

The first is Dr. Toggerson’s height:

    \[ \bar{x}_{\mathrm{Toggerson} = 1.67 \, \mathrm{m.} \]

The second measurement is the height of the library

    \[ \bar{x}_{\mathrm{library} = 93 \, \mathrm{m.} \]

The percent uncertainty on Dr. Toggerson’s height is

    \[ \frac{0.1 \, \mathrm{m}}{1.67 \, \mathrm{m}} \times 100 = 6.0 \% \]

Meanwhile, the same uncertainty on the library is

    \[ \frac{0.1 \, \mathrm{m}}{93 \, \mathrm{m}} \times 100 = 0.1 \% \]

The same uncertainty leads to different percent uncertainty depending on the mean value.




Icon for the Creative Commons Attribution-ShareAlike 4.0 International License

Physics 132 Lab Manual by Brokk Toggerson and Aidan Philbin is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License, except where otherwise noted.

Share This Book