Uncertainties and graphing

Whenever you measure the length of a stick, or the period of a pendulum, or the mass of a whale, or ANYTHING, you should end up with two results:

1. the value of the measurement itself: for example, this stick has a length L = 45 cm .
2. the UNCERTAINTY in that measurement: for example, my measurement of the stick might be wrong by +/- 1 cm

How do you determine the uncertainty in a measurement? The long answer is given in the guides to uncertainties (see them listed in the For more information section below). The short answer is that you may either

• use the SMALLEST DIVISION in your measuring tool
or
• make several measurements, then look at the RANGE or STANDARD DEVIATION

So, you have a measurement, and the uncertainty. Be careful to write the values down using the proper number of significant figures.

Okay. Suppose that you have made some measurements, and written down the value and uncertainty of each one. If you have to combine the measurements to derive some result, how do you combine the uncertainties to find the uncertainty in the result? There are several approaches one can take. Again, read the guides below for detailed descriptions. We will follow two of these methods in class today.

Mathematical propagation of errors

There are mathematical rules for combining the uncertainties in several measurements, depending on whether the measurements are added, or multiplied, or whatever.

• a simple example: area of a rectangle
• Experiment: identify a metal using its density

Graphical propagation of errors

• Using graphs to describe data
  • What makes a graph Good? What makes a graph Bad?
  • How do you calculate the uncertainty in the slope of a line?

• Experiment: predict the mass of a yardstick

For more information

• My guide to determining and using uncertainties
• My ultra-brief, condensed guide to propagation of errors
• Professor Lindberg has written very detailed guides to uncertainties and graphing. You might want to read over his very nice material to see things explained in a slightly different manner.
  • Professor Lindberg's guide to uncertainties (PDF)
  • Professor Lindberg's guide to graphing (PDF)

Copyright © Michael Richmond. This work is licensed under a Creative Commons License.