Appendix B


Course Support
Lab Contents

Uncertainties in Graphs

Error Bars
To indicate the uncertainty of data on a graph on uses vertical and horizontal bars through the data point to indicate the range that the data might assume. For instance, \(5 \pm 1\) would be a bar drawn from 4 to 6. When the uncertainty is too small to be indicated on the graph then a single point is used.

Figure 5

Worst Fits
When determining parameters, such as slope, from a graph there is always an uncertainty associated with it - similar to making any sort of measurement. In determining the slope of a straight line graph one draws a best fit line. To estimate the uncertainty of the best fit one draws two worst fit lines. These indicate the largest and smallest slopes which might still reasonably fit the plot. The uncertainty of the best fit is then taken as half the difference between the worst fits. Note that the best fit may not lie exactly half way between the two worst fits.

Graph slope =\(m_{best} \pm \Delta m\)

where slope uncertainty is
\(\Delta m = \frac{1}{2}(m_{high} - m_{low})\)


Special Case, points dead-on
A special case may occur when the best fit line hits all the points dead-on, so that drawing worst fit lines is unfeasible. The uncertainty in the slope is now determined from the reading error of the graph by treating the graph paper as a measuring instrument. This is done by considering the x-values as fixed and measuring the uncertainty in reading of the y-values from the graph (as read from the graph, not from the data). This uncertainty is then propagated through the slope calculation. Note that the uncertainties in the data points themselves are not used here and the error bars are too small to be visible on the graph.

\[m \pm \Delta m = \frac{(y_{2} \pm \Delta y) - (y_{1} \pm \Delta y)}{x_{2} - x_{1}} \]