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I have a set of "degree of polarization (DOP)" values for a star. Assume there are 10 DOP values in the set.

DOP is defined as square root of sum of squares of fractional Stokes parameters, namely q and u, i.e.

$$DOP=P = \sqrt{q^2+u^2}$$

where $q$ and $u$ are directly measured quantities; and not $P$.

I have to find error/uncertainty in this set. I am assuming that DOP values are spread Gaussially.

My query is this: From the following two methods, which is correct method to find error/uncertainty in DOP?

(a) Just find RMS (standard deviation) of the 10 DOP values.
This is error in DOP set.

OR

(b) Using error-propagation technique, find individual error
in each of 10 DOP values. Then find average of these 10 individual
errors. This average is final error/uncertainty in DOP set.

Next, Which is correct way to find average of DOP set:

(i) Just find average of 10 DOP values. 

OR

(ii) First find average of q's and average of u's and then put them in DOP definition.

In other words, does Standard Deviation/RMS has any significance when a physical quantity is not directly measured but it is some function of another directly measured physical quantities ?

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My query is this: From the following two methods, which is correct method to find error/uncertainty in DOP?

If you know the individual errors for each p and q, then I would use (b) and propagate the errors since the relationship is not linear. Be careful with averages though. It might be better to use the median, for example, if you have one or two significant outliers.

Next, Which is correct way to find average of DOP set:

Since the relationship between (p,q) and DOP is not linear, use method (i). If it were linear, the two methods would be identical (at least they should be).

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