Statistical uncertainty of weighted mean

[Brais]

Hello!

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 v±σ.
If, however, I "bin" my events in energy and angle (say I made four bins in total), when I calculate "v", I get v1±σ1, v2±σ2, v3±σ3, v4±σ4. Then I combine these values into one using a weighted average (c stands for combined): $v_c = \sum(v_i/\sigma^2_i)/\sum(1/\sigma^2_i)$, and $\sigma_c = 1/\sqrt{\sum(1/\sigma^2_i)}$ (as can be seen here).
When I do this, it turns out that $\sigma_c \simeq \sigma/2$. How can this be? I am using the same amount of statistics!

Any reply or idea will be very welcome!

Thank you!

Brais.

 

[chiro]

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.

 

[Brais]

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.

Brais

 

[Stephen Tashi]

I didn't do any simplification, just applied the expression seen in wikipedia.
Brais

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 $\sigma_i$ 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.)

 

[chiro]

Following Stephen Tashi's post, you should probably just clarify exactly what attribute you are using.

 

[Brais]

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...

Thanks,

Brais