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I was performing an optics experiment for measuring the focal length of a concave mirror by "removing the parallax" method.

So we are using not very accurate and high precision instruments. My instructor asked to take three readings and then reverse the object and image distance ($u$ and $v$ ) by principle of reversibility of light and make a total of six readings.

Now I doubt how good a practice this is because a single error in any one reading will affect two readings and then will let a lot of error propagate into amy further operations like drawing a graph etc. Should I discard this practise? If yes, how can I make people understand it is an imprecise method?

What does statistics say?

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  • $\begingroup$ ...how can I make people understand it is an imprecise method? is a question about psychology, not physics. $\endgroup$ – sammy gerbil May 7 '18 at 18:03
  • $\begingroup$ @sammygerbil: Providing a reason why it's a bad method (or at least not a good one) is a way to do it, and that is a question about physics. $\endgroup$ – Michael Seifert May 7 '18 at 20:33
  • $\begingroup$ @sammygerbil I couldn't agree with you... $\endgroup$ – tatan May 8 '18 at 2:23
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There is a general rule in experimental physics that "more data means less uncertainty." This is because the estimate of the uncertainty is usually assumed to be $$ \sigma = \sqrt{ \frac{1}{n-1} \sum_{i = 1}^n (x_i - \bar{x})^2 }, $$ and a higher $n$ leads to a smaller $\sigma$.

However, the uncertainty in your overall measurement will not decrease if you simply count the data points twice. This is because the standard deviation of the data is only a good estimate of the uncertainty in $\bar{x}$ if the individual measurements $x_i$ are uncorrelated with each other. This basically means that the value of $x_i$ is independent of the value of $x_j$ for all $i$ and $j$. In your experiment, the values of the "duplicated" data are highly (in fact, perfectly) correlated with the original data points, so the equation doesn't allow you to reliably estimate the uncertainty in your measurement of $\bar{x}$.

In fact, it's not too hard to show that "duplicating" the measurements like this doesn't change your uncertainty at all. To get into this, you have to generalize the ideas of error propagation to the cases where errors are correlated with each other. This isn't usually done in intro physics courses, but it's not that much of a stretch. To get a flavor of it, take a look at the Wikipedia page on error propagation; the $\sigma_{AB}$ terms are the ones you usually neglect in intro physics labs, but if the measurements are correlated with each other, you need to consider them.

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  • $\begingroup$ I didn't understand what you mean by "uncertainty will not decrease..." $\endgroup$ – tatan May 7 '18 at 17:14
  • $\begingroup$ @tatan: Rewritten to (hopefully) be clearer. $\endgroup$ – Michael Seifert May 7 '18 at 20:31
  • $\begingroup$ So basically that means that uncertainty would not decrease and its not the best way to do an experiment, right? $\endgroup$ – tatan May 8 '18 at 2:10
  • $\begingroup$ u vs v graph should ideally be a hyperbola. With this type of experimental approach and instrumentations, would you as a teacher, expect something highly accurate (a more or less perfect hyperbola) from students? $\endgroup$ – tatan May 8 '18 at 2:28
  • $\begingroup$ Your answer really helped. Could you elaborate on the generalisation of ideas of error propagation to the cases where the errors are correlated. I would better like to see in terms of numbers how much error can really propagate by taking a sample data. $\endgroup$ – tatan May 8 '18 at 4:24

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