I have some data points of the form $(x_i,y_i,\delta y_i)$, where $y$ are counts and the error associated to each $y_i = N$ is $y_i = \sqrt{N}$. I want to create the cumulative distribution of these points and fit a function to it (for some reasons unrelated to my question, I was asked to try that). So if the function describing $y$ vs. $x$ if $f(x)$, now i want to fit $F(x) = \int_0^{x}{f(y)dy}$ to $(x_i,y_1+y_2+...y_i)$. How should I define the errors on each individual point in this case? Should I use the propagation of error for a sum and have $\delta y_i = \sqrt{\delta y_1^2 + \delta y_2^2 + ... + \delta y_i^2}$? Should I do something like this $\delta y_i = \sqrt{N_1+N_2+...N_i}$? Should I do something else? In both cases above, the error gets significantly bigger towards the end so I am not sure if that is the right thing to do. When I was fitting $f(x)$ (so before trying the cumulative approach), I was weighting in the fit, each point by the inverse of its error i.e. for the i-th point I would have $w_i = 1/\sqrt{N_i}$. If I do the same thing now, the points towards the end will have and extremely small weight, and contribute almost nothing to the fit. Yet, intuitively, I would expect them to count the most, as they contain more data than the points in the beginning (and hence they should reflect the real distribution better). What is the proper way to define the errors on the points in this case and use them in the fit? Thank you!

  • $\begingroup$ To get F you end up dividing by a count as well. Don't ignore the effect of that dividing. (Typically the total sample size is conditioned on.) $\endgroup$ – Glen_b Sep 26 at 6:31
  • $\begingroup$ Ordinarily one fits a density or probability function to the data. By fitting a CDF directly to the cumulative data you run into the (serious) problem of strong serial correlation among the values. $\endgroup$ – whuber Sep 26 at 14:49

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