[Hps-analysis] combined errors

[Nelson, Timothy Knight]

Stepan, *as long as the errors on the two measurements are uncorrelated*, the *relative* error on the combination cannot be worse than either measurement. Just work out the math on paper: it’s simply impossible. Holly was quoting relative (%) errors. My point was that the difference between the two means is irrelevant in the combined *relative* error.

I rather think the puzzle here will be solved by realizing that some of the events in the tail of one distribution are not in the tail for the other: that the combination pulls some tail events into the fit region and that the combined gaussian fits have different yields.

Tim

> On Mar 25, 2016, at 10:31 AM, Stepan Stepanyan <stepanya at jlab.org> wrote:
> 
> Tim,
> 
> I disagree, your example has nothing to do with what we are discussion.
> The problem that we saw is not that we measured 1 m to be 1.05 after
> combining them. The problem is that in one case you device measured
> 1 m +/- 0.01 , another 1.1 +/- 0.05. When you combine, average may
> get to 1.05 but measurement accuracy will be worse than 0.01.
> 
> Stepan
> 
> On 3/25/16 12:02 PM, Timothy Nelson wrote:
>> Stepan,
>> 
>> When combining multiple independent measurements of a quantity, the error on the combined value depends only on the errors of the individual measurements and the number of measurements, and not on any systematic shift between the two independent techniques. I understand that it is natural to think otherwise, but this is basic statistics.
>> 
>> To make this more intuitive, consider the following thought experiment:
>> 
>> - I have two perfectly precise methods for measuring a length, but they are systematically different by 10%
>> - I use those methods to measure something that is exactly one meter long.
>> - Therefore, one technique always gives exactly 1.0 meters and the other technique always gives exactly 1.1 meters
>> - When I combine those two measurements, the combination is always exactly 1.05 meters, no more, no less.
>> 
>> Obviously, from this extreme example, one can see that a systematic shift, even one that is infinitely larger than the error of either measurement alone, does not result in a wider distribution for the combined result.
>> 
>> Tim
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