# Combinations of measurements: obtain mean and errors

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## Main Question or Discussion Point

I was looking at the attached picture, where it gives the Higgs Mass obtained from the two different channels from ATLAS and CMS.

Let's only talk about the diphoton channel : $H\rightarrow \gamma \gamma$
From the ATLAS the mass value is:
$m_{Ah} =126.02 \pm 0.51$
and the CMS:
$m_{Ch} =124.70 \pm 0.34$

Now it gives the combined result from ATLAS+CMS:
$m_{ACh}=125.07 \pm 0.29$

How can obtain the mean and error for the ATLAS+CMS?

I tried getting the weights $w_i = \sum_j (C^{-1})_{ij} \Big/ \sum_{kl} (C^{-1})_{kl}$ with $C$ the covariance matrix. Since they are different detectors they are not correlated and so the covariance matrix only has the variances on the diagonal. I obtain:
$w_A \approx 3.84468/12.4952$
$w_C \approx 8.65052/12.4952$
And I calculate the combined mass:
$\bar{m}_{ACh} = \sum_i w_i m_{ih} = 125.106$
I also tried to combine the errors. For the errors I used the statistical and systematic, given by:
$syst= \sqrt{\sum_{ij} w_i w_j C_{ij}^{sys}}=\sqrt{w_A^2 0.27^2 + w_C^2 0.15^2}=0.133 \approx 0.13$
$stat= \sqrt{\sum_{i} w_i^2 C_{ii}^{stat}}=\sqrt{w_A^2 0.43^2 + w_C^2 0.31^2}=0.252 \approx 0.25$
and $\sigma_{tot} =\sqrt{(syst)^2+(stat)^2} =0.284 \approx 0.28$

$m_{h}^{(A+C,2\gamma)}(GeV) = 125.106 \pm 0.28 ( 0.25_{stat} \pm 0.13_{sys})$
in comparison to
$m_{h}^{(A+C,2\gamma)}(GeV) = 125.07 \pm 0.29 ( 0.25_{stat} \pm 0.14_{sys})$
given ..The problem appears in the systematic error...
Any idea? Mine is that the measurements are considered somehow correlated in the systematics?

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Orodruin
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I strongly suspect they are using a more sophisticated statistical analysis than simply assuming everything is Gaussian ...

Gold Member
I don't know, I have seen several times giving the results from experiment A, experiment B etc... and then giving the A+B+... total result in similar figures.
So I'm trying to understand how they get the total result.

Orodruin
Staff Emeritus