I am working with several `TimeSeries`

containing `Around`

objects. I find that the propagated uncertainties for numerical values are larger than I expect them to be. In contrast, the propagated uncertainties for symbolic values are exactly what I expect them to be.

This is also true for ordinary `List`

objects, so I will use `List`

to demonstrate. Note that the original data values are all positive (`n(i) > 0`

) and their original uncertainties equal their square roots (`δn(i) = √n(i)`

)

First, generate a list of symbolic data:

`symbolicData = Table(Around(n(i), Sqrt(n(i))), {i, 10})`

Then, `Accumulate`

these data:

`symbolicTotals = Accumulate@symbolicData`

The uncertainty of the last value is the square root of that value:

`{symbolicTotals((-1))("Uncertainty"), Sqrt@symbolicTotals((-1))("Value")}`

Given my understanding of error propagation, this is correct.

Now, generate numerical data:

`numericalData = Table(Around(#, Sqrt(#)) &@RandomInteger(10), {i, 10})`

`Acculumate`

these data:

`numericalTotals = Accumulate@numericalData`

Compare the last value’s uncertainty and square root:

`{numericalTotals((-1))("Uncertainty"), Sqrt@numericalTotals((-1))("Value")}`

The uncertainty is larger than the square root. This happens for any long list of numbers.

Why? And how can I fix it? (Assuming there’s an actual problem here, and not simply a lack of knowledge on my end.)

My *guess* is that this discrepancy is a consequence of the fact that

When Around is used in computations, uncertainties are by default propagated using a first-order series approximation, assuming no correlations.

What exactly is being approximated here, and if `Around`

*always* does this, why doesn’t it affect symbolic values?

I understand that `AroundReplace`

allows for higher-order approximations, but it starts with a symbolic expression and substitutes numerical values. I’m *starting* with numerical values. Is my only option to generate `TimeSeries`

objects with symbolic values, do all my computations, and then `AroundReplace`

the symbols with the original numerical data?

Is there a better way?