Metrics
==================

* **Expected value/Espérance**: Noted :math:`\mathbb{E}[X]`, is a **theorical value**. For example, when playing coin
  flipping, the expected value for getting heads or tails is 0.5.

Variance
------------------

Variance can be seen as the expected squared deviation from the expected value of a random variable :math:`X`.

.. math::
   \mathbb{V}[X]=\mathbb{E}[X-\mathbb{E}[X]]^2=\frac{\sum_{i=1}^n (x_i - \mathbb{E}[X])^2}{n}=\mathrm{Cov}(X,X)

Covariance
------------------

Covariance is a way to quantify the relationship between two random variables :math:`X` and
:math:`Y` (`source <https://www.youtube.com/watch?v=qtaqvPAeEJY>`_). Covariance **DOES NOT**
quantify how strong this correlation is! If covariance is:

Positive
    For large values of :math:`X`, :math:`Y` is also taking large values
Negative
    For large values of :math:`X`, :math:`Y` is also taking low values
Null
    No correlation

.. math::
   \mathrm{Cov}(X,Y)=\mathbb{E}[(X-\mathbb{E}[X])(Y-\mathbb{E}[Y])]=\frac{\sum_{i=1}^n (x_i - \mathbb{E}[X])(y_i - \mathbb{E}[Y])}{n}
    
Standard deviation
-----------------------

Standard deviation provides a way to interprete the variance using the unit of :math:`X`.

.. math::
   \sigma=\sqrt{\mathbb{V}[X]}


Standard Error of the Mean
-----------------------------

Standard Error of the Mean (SEM) quantifies the error that is potentially made when computing the mean.

.. math::
   \mathrm{SEM}=\sigma_X^{-}=\sqrt{\frac{\mathbb{V}[X]}{n}}=\frac{\sigma}{\sqrt{n}}


Here is how to interpret it.
If :math:`n=1`, the error is at most :math:`\sqrt{\mathbb{V}[X]}=\sigma_X` which is the standard deviation or :math:`X`.
The more :math:`n` increases, the lower the error becomes.
More infos in `this video <https://www.youtube.com/watch?v=BwYj69LAQOI>`_.
If it is still unclear, see the following R code:

.. literalinclude:: code/sem.R
   :language: R

Output example:

.. code-block:: console
   
   ----- Experiment 1 -----
   Means SD: 1.22
   SEM       1.26
   ----- Experiment 2 -----
   Means SD: 1.26
   SEM       1.26
   ----- Experiment 3 -----
   Means SD: 1.27
   SEM       1.26