Minor changes

source/statistics/bessel_correction.rst

@@ -1,4 +1,7 @@
+.. _bessel_correction:
+
 Bessel's Correction
 -----------------------
 
-TODO
+Bessel's correction is the use of :math:`n-1` instead of :math:`n` in the formula for the sample
+variance and sample standard deviation.

source/statistics/metrics.rst

@@ -12,6 +12,11 @@ occurring we have:
 .. math::
    \mathbb{E}[X]=x_1p_1+x_2p_2+\cdots+x_np_n
 
+When working with a sample, the following is an unbiased estimator of the expected value (`source <https://stats.stackexchange.com/questions/518084/whats-the-difference-between-the-mean-and-expected-value-of-a-normal-distributi>`_):
+
+.. math::
+   \overline{x}=\frac{\sum_{i=1}^n x_i}{n}
+   
 Variance
 ------------------
 
@@ -20,6 +25,13 @@ Variance can be seen as the expected squared deviation from the expected value o
 .. 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)
 
+When working with a sample, the following is an unbiased estimator of the variance:
+
+.. math::
+   s=\frac{\sum_{i=1}^n(x_i-\overline{x})^2}{n-1}
+
+To understand why denominator is :math:`n-1` see :ref:`Bessel's correction <bessel_correction>`.
+   
 Covariance
 ------------------
 
@@ -45,11 +57,17 @@ Standard deviation provides a way to interprete the variance using the unit of :
 .. math::
    \sigma=\sqrt{\mathbb{V}[X]}
 
+When working with a sample, the following is an unbiased estimator of the standard deviation:
+
+.. math::
+   s=\sqrt{\frac{\sum_{i=1}^n(x_i-\overline{x})^2}{n-1}}
+
+To understand why denominator is :math:`n-1` see :ref:`Bessel's correction <bessel_correction>`.
 
 Standard Error of the Mean
 -----------------------------
 
-Standard Error of the Mean (SEM) quantifies the error that is potentially made when computing the mean.
+Standard Error of the Mean (SEM) quantifies the error that is potentially made when computing the mean of a population.
 
 .. math::
    \mathrm{SEM}=\sigma_X^{-}=\sqrt{\frac{\mathbb{V}[X]}{n}}=\frac{\sigma}{\sqrt{n}}
@@ -79,6 +97,11 @@ Output example:
    Means SD: 1.27
    SEM       1.26
 
+When working with a sample of :math:`n` individuals, an estimator of the SEM is:
+
+.. math::
+   s_{\overline{x}}=\frac{s}{\sqrt{n}}
+   
 Degree of Freedom
 --------------------