For example, you they try to calculate the likelihood of lines of different slopes and look at the width of the likelihood function as an estimate for the error in slope. Students could use other methods to find the uncertainty in the slope as well. It appears the error bars are too big we should question our procedure for generating them, and justify why we only have y-error and not x-error as well. If these are 1-sigma error bars, that's very unlikely with eight data points. In your plot, the best-fit line easily fits through every single error bar. This is an heuristic based on the basic ideas of error proagation you can also look up more detailed formulas for the uncertainty in the slope in least squares linear regression.Īdditionally, it's worth pointing out that I would want students with that data to be skeptical of the error bars. Like all averages, the error in the slope should decrease as $\dfrac.$$ Instead, students should recognize that finding a line of best fit is a form of averaging the data points. That completely ignores $n$, the number of data points! If the size of a typical error bar is $\delta y$ and the gap from the first to last $x$-value is $\Delta x$, then all your proposed estimates give numbers of order $\delta y/\Delta x$. I would give credit for them in an introductory class, but I would hope that at least some students would recognize why they aren't good error estimates for a line's slope, and in an advanced class I would definitely encourage students to go further. On the other hand, all these suggestions are a bit off the mark in terms of a nuanced understanding of error. There's no reason to favor one over the other very much. You'll get approximately the same answer no matter what you do, and since the error is only approximate anyway, that's fine. There are a range of such lines all these procedures are estimates of that range. This encourages them to do the same sort of critical thinking that you showed in formulating the question.Īll the things you suggested are pretty similar they all involve the idea that error bars tell you a range of plausible values, so roughly speaking a line that goes through the error bars is plausible. From a teaching perspective, I think what's most important is that students come up with their own procedure, that's it's reasonable, and that they justify it. I like that this question is asking for some intuitive justification and calling for more understanding in error analysis. If you do so, you would likely end up with an unsymmetric error, which you could report like that, but what to do if you want to keep it simple and your students in the beginner lab report just one error? Take the largest of both errors? Or a mean value between them? What would be the most reasonable approach for this and why?ĭo you have any further references where this method is discussed in physics or in papers about physics education? Wouldn't it be better to determine the lowest and the highest possible slopes and calculate each time the difference to the "best slope"? Why just stick to one case? To estimate the error for example Dana Roberts suggests to draw a line which barely fits within the error bars, calculate the slope and take the difference as error like this: Then you may draw the line which visually fits best through all error bars and just calculate the slope. Suppose you are interested in the slope of this linear function. Suppose you analyze experimantal data with pencil and piece of paper, one simple method is to plot the data in a linearized way with error bars.
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