.. _propagation_of_uncertainties:

Propagation of uncertainties
============================

If present, scipp propagates uncertainties (errors) as described in `Wikipedia: Propagation of uncertainty <https://en.wikipedia.org/wiki/Propagation_of_uncertainty>`_.
The implemented mechanism assumes *uncorrelated data*.

An overview for key operations is:

.. table::
    :widths: 50 50

    +---------------------+-------------------------------------------------------------------------+
    |Operation :math:`f`  |Variance :math:`\sigma^{2}_{f}`                                          |
    +=====================+=========================================================================+
    |:math:`-a`           |:math:`\sigma^{2}_{a}`                                                   |
    +---------------------+-------------------------------------------------------------------------+
    |:math:`|a|`          |:math:`\sigma^{2}_{a}`                                                   |
    +---------------------+-------------------------------------------------------------------------+
    |:math:`\sqrt{a}`     |:math:`\frac{1}{4} \frac{\sigma^{2}_{a}}{a}`                             |
    +---------------------+-------------------------------------------------------------------------+
    |:math:`a + b`        |:math:`\sigma^{2}_{a} + \sigma^{2}_{b}`                                  |
    +---------------------+-------------------------------------------------------------------------+
    |:math:`a - b`        |:math:`\sigma^{2}_{a} + \sigma^{2}_{b}`                                  |
    +---------------------+-------------------------------------------------------------------------+
    |:math:`a * b`        |:math:`\sigma^{2}_{a}b^{2} + \sigma^{2}_{b}a^{2}`                        |
    +---------------------+-------------------------------------------------------------------------+
    |:math:`a / b`        |:math:`\frac{\sigma^{2}_{a} + \sigma^{2}_{b} \frac{a^{2}}{b^{2}}}{b^{2}}`|
    +---------------------+-------------------------------------------------------------------------+
    |:math:`e^{a}`        |:math:`e^{2a} \sigma^{2}_{a}`                                            |
    +---------------------+-------------------------------------------------------------------------+
    |:math:`\log(a)`      |:math:`\sigma^{2}_{a} / a^{2}`                                           |
    +---------------------+-------------------------------------------------------------------------+
    |:math:`\log_{10}(a)` |:math:`\sigma^{2}_{a} \log^{2}(10) / a^{2}`                              |
    +---------------------+-------------------------------------------------------------------------+

The expression for division is derived from :math:`(\frac{\sigma_{a/b}}{a/b})^{2} = (\frac{\sigma_{a}}{a})^{2} + (\frac{\sigma_{b}}{b})^{2}`.