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I am a little bit confused about some tecniques concerning error-analysis.Consider this situatiuon:

During an experiment I collected a $x,y$ table of data that are expected to satisfy a linear relation $y=kx$. For any $x_i$ I have an associated error/uncertainty $\delta x_i$ and the same applies to the $y_i$'s. What is the best thing to do in order to have a good estimate of the parameter $k$ and the associated error $\delta k$?

  1. Maybe compute $k_i=y_i/x_i$ and compute the maximal error $\delta k_i$ with the error propagation furmula for division, then compute some kind of weighted average the $k_i$ according to the size of the $\delta k_i$? What would be a good estimate for $\delta k$ then?

  2. Or compute $k$ that minimizes $\sum_i (kx_i-y_i)^2$ with standard least square tecniques, i.e. $k=\frac{\sum y_ix_i}{\sum x_i^2}$ and compute $\delta k$ with the error propagation forumlas and add the additional error due to least square approximation?

Or something else?

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  • $\begingroup$ It depends on how your errors are distributed. If they follow a normal distribution, then the best predictor for the constant are given by a least-mean square fit, if the errors are not normally distributed, then they are not. $\endgroup$
    – CuriousOne
    Commented Jul 19, 2016 at 15:01
  • $\begingroup$ If you have measurement errors in both your $x$'s and $y$'s, then you want an "errors-in-variables" model $\endgroup$
    – Paul T.
    Commented Jul 19, 2016 at 15:15
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    $\begingroup$ This might get more attention/answers on the statistics SE. $\endgroup$
    – knzhou
    Commented Jul 19, 2016 at 16:50

1 Answer 1

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During an experiment I collected a $x,y$ table of data that are expected to satisfy a linear relation $y=kx$. For any $x_i$ I have an associated error/uncertainty $\delta x_i$ and the same applies to the $y_i$'s. What is the best thing to do in order to have a good estimate of the parameter $k$ and the associated error $\delta k$?

The typical solution to this problem is total least squares regression(TLS). Like ordinary least squares regression, this method assumes a linear relationship between the dependent variable and the independent variable. Unlike ordinary least squares, it accounts for normally distributed errors in the independent variable.

TLS is an example of an errors-in-variables method, as mentioned in the comments.

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  • $\begingroup$ It is strange: my problem is one of the most elementary kind of experiment one can see in a physics introductory course but this solution seems to be really advanced $\endgroup$ Commented Jul 19, 2016 at 17:17
  • $\begingroup$ @MarcoDisce, Often the errors in the independent variable are small enough (or can be improved by better experimental technique) so we can ignore them and just use ordinary least squares. If the $\delta{}x$ are small you won't see much different between the OLS and TLS results. $\endgroup$
    – The Photon
    Commented Jul 19, 2016 at 17:27
  • $\begingroup$ Ok but one would need some kind of estimate about how much the result can be wrong with OLS when you have a small but finite error on the dependent valuable $\endgroup$ Commented Jul 19, 2016 at 17:48

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