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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): [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!

Brais.

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# Statistical uncertainty of weighted mean

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