Minor changes

source/statistics/code/sem.R

@@ -0,0 +1,14 @@
+nooutput=sapply(1:3,function(i){
+    n=10
+    mu=10
+    sigma=4
+    sem=sigma/sqrt(n)
+
+    # Compute 1000 times the mean a random sample and get the standard deviation of the mean
+    means_sd=sd(sapply(1:1000,function(dummy){mean(rnorm(n,mean=mu,sd=sigma))}))
+
+    # See how SEM is a good approximation of the mean standard deviation
+    message(paste("----- Experiment",i,"-----"))
+    message(paste0("Means SD: ", round(means_sd,digits=2)))
+    message(paste0("SEM       ", round(sem,digits=2)))
+})

source/statistics/introduction.rst

@@ -32,3 +32,43 @@ Null
 .. 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