============
Grubbs' Test
============
:doc:`/WorkProcessClassifiers/GlobalAlgorithm/index` - :doc:`/WorkProcessClassifiers/OneDimensionalAlgorithm/index`

*Grubbs' test* algorithm calculates the ratio of the deviation of each data point from the mean of the data set to the standard deviation of the data set. The basic formula is as follows:

.. math::

   G = \frac{\text{max}|(Y_i-\mu)|}{\sigma}

where :math:`G` is the Grubbs' test statistic, :math:`Y_i` is the :math:`i`\ th component of :math:`Y`\ , :math:`\mu` is the mean value, and :math:`\sigma` is the standard deviation of the data set. If

.. math::

   G > G^{\text{table}} \, \text{,}

then the data point can be considered outside the region of interest. :math:`G^{\text{table}}` includes literature values for a certain significance level.

.. rubric:: Input Parameters

+----------------------------------------+------------------------------------------------+----------------------------------------+--------------------------------------------------+---------------------------------------------------------------------------------+
| Parameter                              | Type                                           | Constraint                             | Description                                      | Remarks                                                                         |
+========================================+================================================+========================================+==================================================+=================================================================================+
| :math:`Y`                              | :math:`Y \in \mathbb R^N`                      | :math:`N \in \mathbb{N}`               | Input data sequence of length :math:`N`          | Not effective for a data vector with less than :math:`6`          constituents. |
+----------------------------------------+------------------------------------------------+----------------------------------------+--------------------------------------------------+---------------------------------------------------------------------------------+
| :math:`\mu`                            | :math:`\mu \in \mathbb{R}`                     |                                        | Mean distribution of :math:`Y`                   |                                                                                 |
+----------------------------------------+------------------------------------------------+----------------------------------------+--------------------------------------------------+---------------------------------------------------------------------------------+
| :math:`\sigma`                         | :math:`\sigma\in \mathbb{R}`                   |                                        | Standard deviation of :math:`Y`                  |                                                                                 |
+----------------------------------------+------------------------------------------------+----------------------------------------+--------------------------------------------------+---------------------------------------------------------------------------------+
| :math:`G^{\text{table}}`               |                                                |                                        | Grubbs' test values from literature              |                                                                                 |
+----------------------------------------+------------------------------------------------+----------------------------------------+--------------------------------------------------+---------------------------------------------------------------------------------+

.. rubric:: Output Parameters

+----------------------------+----------------------------------------------------+------------+-------------------------------------------------------------------------------------------+---------+
| Parameter                  | Type                                               | Constraint | Description                                                                               | Remarks |
+============================+====================================================+============+===========================================================================================+=========+
| :math:`\hat{Y}`            | :math:`\hat{Y} \in \mathbb R^N`                    |            | Values in the :math:`Y`          list which are outside the region of interest are marked |         |
+----------------------------+----------------------------------------------------+------------+-------------------------------------------------------------------------------------------+---------+

.. rubric:: Single Steps using the Algorithm

* :doc:`/DataPreprocessing/DataCleaning/OutlierDetection/OutlierDetectionWithGrubbsTest/index`

.. rubric:: References

- F.\  Grubbs, Procedures for Detecting Outlying Observations in Samples, Technometrics, vol. 11(1), pp. 1-21, 1969.