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Statistical uncertainty of weighted mean

by Brais
Tags: statistical, uncertainty, weighted
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Nov9-12, 11:56 AM
P: 7

I am using physical data to do an analysis (~30k measurements). These measurements include energies, momenta, angles... of particles.
I am calculating a value (call it v) at the end after a lengthy process, and if I introduce all the data into my program I did, the result is vσ.
If, however, I "bin" my events in energy and angle (say I made four bins in total), when I calculate "v", I get v1σ1, v2σ2, v3σ3, v4σ4. Then I combine these values into one using a weighted average (c stands for combined): [itex] v_c = \sum(v_i/\sigma^2_i)/\sum(1/\sigma^2_i)[/itex], and [itex]\sigma_c = 1/\sqrt{\sum(1/\sigma^2_i)}[/itex] (as can be seen here).
When I do this, it turns out that [itex]\sigma_c \simeq \sigma/2[/itex]. How can this be? I am using the same amount of statistics!

Any reply or idea will be very welcome!!

Thank you!

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Nov9-12, 05:52 PM
P: 4,579
Hey Brais.

The wiki article looks straight-forward, but perhaps you could just outline your calculations in a little more detail step by step to show the simplifications and assumptions you used.
Nov10-12, 04:49 AM
P: 7
Hi, thanks for your reply!
I am calculating a fit. If I put all my data together I get an error that is higher than that of fitting different sets of points separately and then combining them with a weighted mean.
I didn't do any simplification, just applied the expression seen in wikipedia.


Stephen Tashi
Nov10-12, 09:28 AM
Sci Advisor
P: 3,319
Statistical uncertainty of weighted mean

Quote Quote by Brais View Post
I didn't do any simplification, just applied the expression seen in wikipedia.
You are using a formula from that article that applies when you know the standard deviations of the distributions that are involved, but I'd guess that you don't. Your [itex] \sigma_i [/itex] are probably estimators of standard deviations that you computed from the sample. (The term "standard deviation" is ambiguous. It has at least 5 different meanings in statistics, depending on the context where it appears.)
Nov10-12, 06:49 PM
P: 4,579
Following Stephen Tashi's post, you should probably just clarify exactly what attribute you are using.
Feb1-13, 09:32 AM
P: 7
Hi again!

A long time ago I had to stop this analysis and so my doubt wasn't importantr for some time :)
I use the errors that my minimization algorithm "MINUIT" gives. Unfortunatelly I cannot find anything except that it (obviously) calculates a covariance matrix and error matrix...



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