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How it is actually done
The way we do error analysis here is OK, but it is not how it is usually done by a scientist. The scientist will, if necessary, linearize the data as we do. Then to find the error bound the scientist will resort to other methods than we have used. There are formulas for doing this. Look for example at
http://mathworld.wolfram.com/LeastSquaresFitting.html
In the Excel file Analysis5 V2, parts of the above are implemented. If one, for example, uses all the data from Data Analysis 1, and enters them as one long table with all the pairs of data we have, we get the slope 1.44 and the y-intercept –0.048, i.e. exactly the same values as the Excel linear line of best-fit tool gives you using the same set of points. That is of course because Excel uses the same algorithms.
The line we have found is actually the most likely line given the data. For the errors, the algorithms found 0.155 for the slope and 0.103 for the intercept.
This would give us the equation
R=(1.4±0.2)L –0.05±0.10
I.e. almost exactly what we found in Data analysis 1:
R=(1.4±0.2)L –0.03±0.08
The reason we don’t work with the methods mentioned above to find the errors is that it will be mainly a “plug-in values- out pops an answer” approach. Secondly, for the small number of data points we have, the method we use actually works very well, as we saw. Thirdly, the risk that the student will use the “plug in” formulas even when they are not well suited for the job is fairly high. When using the manual method one can basically see when it works and when not.
Anyhow, the columns marked “x” and “y” in the Excel file could be filled with other numbers, and then it will give you the line of best fit, including errors. You could possibly use this to see if your manual work gives reasonable values.
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