71 lines
2.1 KiB
ReStructuredText
71 lines
2.1 KiB
ReStructuredText
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
|