# Error propagation and data linearization

I am fairly new to teaching error propagation to students but came across a question I couldn’t answer myself. Say we have a set of data that produces a parabola. In order to linearize the data and get a straight line we plot the $$x$$-data as $$x^2$$. My question is how do I deal with the uncertaintities in the x-data?

• Do you have error in both the x and y data? – David White Feb 14 at 3:37
• In some cases yes. – greenplasticdave Feb 14 at 5:27
• Related. If you're serious about uncertainty analysis, this kind of "linearization" may introduce more problems than it solves. – rob Feb 14 at 6:11
• Wold Cross Validated be a better home for this question? – Qmechanic Feb 14 at 7:19
• @rob Usually in intro-level classes, it's a hard-and-fast rule that everything has to be linearized before you can do a fit. Probably a solution that results in the best possible uncertainty analysis given those confines is most interesting in this case... – Bunji Feb 14 at 16:51

The uncertainty in the slope can be found from the $$R^2$$ value using $$\sigma_m = m\sqrt{\frac{(1/R^2)-1}{n-2}},$$ where $$m$$ and $$\sigma_m$$ are the slope and uncertainty in the slope respectively, and $$n$$ is the number of data points used to make the fit. Students should be reminded (several times!) to be careful - $$R^2$$ is already squared! Don't square it again.
The uncertainty in y-intercept can be found using the above result: $$\sigma_b = \sigma_m \sqrt{\frac{\sum x^2}{n}}.$$
While the $$R^2$$ value should account for the precision of data (i.e. more precise data will tend to have a higher $$R^2$$ value -- especially when $$n$$ is big), this approach completely disregards errors that affect the accuracy of the data. For example, if you used a very low-quality ruler that had the potential to be 20% too short or too long, but got really precise results, the $$R^2$$ value (and hence $$\sigma_m$$ would be high even though the results might be quite biased!