=============
Weighted Mean
=============
:doc:`/WorkProcessClassifiers/GlobalAlgorithm/index` - :doc:`/WorkProcessClassifiers/OneDimensionalAlgorithm/index`

*Weighted Mean* of a data set is similar to the mean, but the values in the data set are weighted differently. The basic formula is stated as follows:

.. math::

    \mu_{\text{weighted}} = \frac{\sum_{i=1}^{N} w_i \cdot Y_i}{\sum_{i=1}^{N} w_i} \, \text{,}

where :math:`Y_i` is the :math:`i`\ th value in the data set, :math:`N` is the length of the data vector and :math:`\mu_{\text{weighted}}` is the weighted mean value.

.. rubric:: Input Parameters

+--------------------+--------------------------------------------+----------------------------------------+-----------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| Parameter          | Type                                       | Constraint                             | Description                                         | Remarks                                                                                                                                                            |
+====================+============================================+========================================+=====================================================+====================================================================================================================================================================+
| :math:`Y`          | :math:`Y \in \mathbb R^N`                  | :math:`N \in \mathbb{N}`               | Input data vector of length :math:`N`               |                                                                                                                                                                    |
+--------------------+--------------------------------------------+----------------------------------------+-----------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| :math:`W`          | :math:`W \in \mathbb R^N`                  | :math:`N \in \mathbb{N}`               | Input weighting vector of length :math:`N`          | If :math:`\sum w_i = 1`              \         , the algorithm simplifies to :math:`\mu_{\text{weighted}} = \sum w_i \cdot Y_i`                        \         . |
+--------------------+--------------------------------------------+----------------------------------------+-----------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------+

.. rubric:: Output Parameters

+------------------------------------------------+--------------------------------------------------------------------+------------+------------------------------------------------+--------------------------------------------------------------------------------+
| Parameter                                      | Type                                                               | Constraint | Description                                    | Remarks                                                                        |
+================================================+====================================================================+============+================================================+================================================================================+
| :math:`\mu_{\text{weighted}}`                  | :math:`\mu_{\text{weighted}} \in \mathbb R`                        |            | Weighted mean value of data :math:`Y`          | The result is sensitive to improper values such as 'nan', 'inf', 'null', etc.  |
+------------------------------------------------+--------------------------------------------------------------------+------------+------------------------------------------------+--------------------------------------------------------------------------------+

.. rubric:: Single Steps using the Algorithm

* :doc:`/DataPreprocessing/MathematicalComputation/ComputingWeightedMeanValueWithWeightedMean/index`

* :doc:`/DataPreprocessing/DataCleaning/HandlingImproperValues/ReconstructingImproperValues/ReconstructingImproperValuesWithLocalCenteredMovingWeightedMean/index`