The background asymmetry estimate has been 0.000 +/- 0.100 was used.  This was chosen as an very conservative number based on the studies in this blog and this blog last December, which most likely did not use all the available runs.  Due to the limited statistics, a fair amount of the observed fluctation seems due to statistics, and keeping with 0 +/- 0.1 would lead to the dominant systematic uncertainty.  Also, as some aspects of the analysis may have been updated since then, it is worth revisiting.  Since it was already decided a single value for all pT is sufficiently accurate, given the statistical fluctations, it is fair to consider the value of A_LL integrated over all pT.  The following show A_LL vs pT for each of three mass bins (0-0.1 GeV, 0.1-0.2 GeV, and 0.2-0.3 GeV) and the results for each mass bin are fit to a constant.

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Mass Bin 1: 0-0.1 GeV

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Mass Bin 2: 0.1-0.2 GeV

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Mass Bin 3: 0.2-0.3 GeV

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Mass Bin 4: 0.3-0.4 GeV

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Conclusions

The fits to a constant for A_L blue are (each row 1-3 correspond to mass bins 1-3)

Fit	result:	 0.015	+/-	0.010	chi^2/ndf	=	4.4884/6
Fit	result:	-0.003	+/-	0.005	chi^2/ndf	=	1.13235/6
Fit	result:	-0.014	+/-	0.009	chi^2/ndf	=	5.77578/6

The fits to a constant for A_L yellow are

Fit	result:	-0.002	+/-	0.009	chi^2/ndf	=	2.34014/6
Fit	result:	-0.001	+/-	0.005	chi^2/ndf	=	4.60174/6
Fit	result:	 0.003	+/-	0.009	chi^2/ndf	=	5.39672/6

The fits to a constant for A_LL are

Fit	result:	 0.007	+/-	0.016	chi^2/ndf	=	5.38901/6
Fit	result:	 0.001	+/-	0.009	chi^2/ndf	=	4.92826/6
Fit	result:	-0.020	+/-	0.016	chi^2/ndf	=	2.29198/6

The chi^2/ndf are very reasonable.  We taking the estimate of the background to be the average of the fit result for the 1st and 3rd mass bins (the bins immediately adjacent to the signal region mass bin) and set the uncertainty to the larger of either (1) the propagated uncertainties from the fits or (2) half the difference in the central values between the 1st and 3rd bin.

A_L blue background asymmetry:    0.001 +/- 0.014
A_L yellow background asymmetry:  0.000 +/- 0.006
A_LL background asymmetry:       -0.006 +/- 0.013

The significant decrease in uncertainty from last December's estimate is due to the increased number of runs used for this study and by using the average of the fits from the two mass regions rather than just judging a number by eye.