Average of an uarray in python uncertainties

Question

My problem:

I have an array of ufloats (e.g. an unarray) in pythons uncertainties package. All values of the array got their own errors, and I need a funktion, that gives me the average of the array in respect to both, the error I get when calculating the mean of the nominal values and the influence the values errors have.

I have an uarray:

2 +/- 1 3 +/- 2 4 +/- 3

and need a funktion, that gives me an average value of the array.

Thanks

Answer 1

Assuming Gaussian statistics, the uncertainties stem from Gaussian parent distributions. In such a case, it is standard to weight the measurements (nominal values) by the inverse variance. This application to the general weighted average gives,

$$ \frac{\sum_i w_i x_i}{\sum_i w_i} = \frac{\sum_i x_i/\sigma_i^2}{\sum_i 1/\sigma_i^2} $$.

One need only perform good 'ol error propagation on this to get an uncertainty of the weighted average as,

$$ \sqrt{\sum_i \frac{1}{1/\sum_i \sigma_i^2}} $$

I don't have an n-length formula to do this syntactically speaking on hand, but here's how one could get the weighted average and its uncertainty in a simple case:

    a = un.ufloat(5, 2)
    b = un.ufloat(8, 4)
    wavg = un.ufloat((a.n/a.s**2 + b.n/b.s**2)/(1/a.s**2 + 1/b.s**2), 
                     np.sqrt(2/(1/a.s**2 + 1/b.s**2)))
    print(wavg)
    >>> 5.6+/-2.5298221281347035

As one would expect, the result tends more-so towards the value with the smaller uncertainty. This is good since a smaller uncertainty in a measurement implies that its associated nominal value is closer to the true value in the parent distribution than those with larger uncertainties.

Answer 2

Unless I'm missing something, you could calculate the sum divided by the length of the array:

from uncertainties import unumpy, ufloat
import numpy as np
arr = np.array([ufloat(2, 1), ufloat(3, 2), ufloat(4,3)])
print(sum(arr)/len(arr))
# 3.0+/-1.2

You can also define it like this:

arr1 = unumpy.uarray([2, 3, 4], [1, 2, 3])
print(sum(arr1)/len(arr1))
# 3.0+/-1.2

uncertainties takes care of the rest.