# Combinations of measurements: obtain mean and errors

1. Jul 16, 2015

### ChrisVer

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 $\sigma_{tot} =\sqrt{(syst)^2+(stat)^2} =0.284 \approx 0.28$

My result reads:
$m_{h}^{(A+C,2\gamma)}(GeV) = 125.106 \pm 0.28 ( 0.25_{stat} \pm 0.13_{sys})$
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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Last edited: Jul 16, 2015
2. Jul 16, 2015

### Orodruin

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

3. Jul 16, 2015

### ChrisVer

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.

4. Jul 16, 2015

### Orodruin

Staff Emeritus
Of course you have, things are often very close to Gaussian. If this was all there was to combining ATLAS and CMS data, we would not need to wait for a combined analysis, anyone with a pocket calculator could do it.

5. Jul 16, 2015

### mathman

A guess: they had data to more significant figures (for the errors) than presented.

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