[Rivet] Rivet "collaboration meeting"?

[Andy Buckley]

On 21/05/13 21:01, Leif Lönnblad wrote:
> On 2013-05-21 16:50, Frank Siegert wrote:
>> Why is this not using the normal variance a la
>> s^2 = 1/(n-1) <(w-<w>)^2>
>>     = 1/(n-1) (<w^2> - <w>^2)
>>     = 1/(n-1) (sum(w^2)/n - sum(w)^2/n^2)
> 
> If we use this, the error will be zero for uniform weights...

Yes, this would be a measure of the width of the distribution of weights.

> For uniform weights, the number of entries in a bin is typically given
> by a Poissonian distribution , which means that the variance is the same
> as the mean, so the error on the number of entries is given by
> sqrt(number of entries).
> 
> For weighted events, one way of thinking is the following: Imagine there
> is a discrete set of weights w_i. For each of these there are n_i
> entries giving the error w_i*sqrt(n_i) for the sum of weight, n_i*w_i.
> The total height of the histogram is sum_i(n_i*w_i), and the error in
> that number is the square root of the sum of the squared errors for each
> w_i: sqrt(sum_i(n_i*w_i^2)). This generalizes to the error
> sqrt(sum(w^2)) for the bin height sum(w).

Exactly. But with "height" replaced by "area" when the bin widths need
to be accounted for to converge to the physical distribution.

> Of course, this "derivation" only holds for a large number of entries in
> a large number of bins, but it gives a reasonable error estimate also
> for a few entries per bin. But maybe there is a better estimate out
> there...

If there is, I'd like to know. Hardly a proof of correctness, but ROOT
uses sqrt(sum(w^2)) too... well, it does if you explicitly tell it to
before starting to fill:

"If Sumw2 has been called, the error per bin is computed as the sqrt(sum
of squares of weights), otherwise the error is set equal to the sqrt(bin
content)."
from http://root.cern.ch/root/html/TH1.html

Where was this causing a problem, Frank/David?

Andy

-- 
Dr Andy Buckley, Royal Society University Research Fellow
Particle Physics Expt Group, University of Edinburgh / PH Dept, CERN