Graphical Representations of Uncertainties

Uncertainty is far easier to judge on a graph than in a table, and a graph lets you find the uncertainty in a quantity you never measured directly.

Error bars — a line through each point extending one absolute uncertainty either side. Vertical bars show uncertainty in the y-value, horizontal bars in the x-value. If one uncertainty is too small to draw, say so rather than silently omitting it.

What error bars tell you — a point whose error bar the line of best fit misses is inconsistent with the trend. If the line passes through every bar, the data are consistent with that relationship. Error bars turn "the point looks a bit off" into a statement you can defend.

Line of best fit and worst acceptable line — draw the best-fit line first. Then draw the steepest and the shallowest lines that still pass through all the error bars. The steeper or shallower of these two is the worst acceptable line, and the difference between its gradient and the best-fit gradient is the uncertainty in your gradient.

uncertainty in gradient = |gradient of best line − gradient of worst line|

The same method gives the uncertainty in the intercept, using the intercepts of the two lines.

Example: a best-fit gradient of 2.40 N m⁻¹ with a worst acceptable gradient of 2.52 N m⁻¹ gives an uncertainty of 0.12 N m⁻¹, so the result is 2.40 ± 0.12 N m⁻¹, or 2.4 ± 0.1 N m⁻¹ to a sensible precision. As a percentage that is about 5%.

Why the triangle should be large — reading a gradient from a small triangle means small differences in Δy and Δx, and the uncertainty in reading each one is then a large fraction of the value. Using the full length of the line minimises this.

Interpreting the spread — if the error bars are large but the points sit tightly along the line, you have probably overestimated the uncertainties. If the bars are small but the points scatter widely, you have underestimated them, or something you assumed was controlled is not.

Quoting the final answer — a result is given as value ± uncertainty, with the uncertainty to one significant figure and the value rounded to the same decimal place. Writing 2.4013 ± 0.1 makes no sense: the uncertainty says the second decimal place is already unknown.