From d47c4e8ca983c2e8cc466c634c8b7133f55a4e38 Mon Sep 17 00:00:00 2001
From: Dennis Eichhorn <spl1nes.com@googlemail.com>
Date: Fri, 9 Sep 2022 21:01:11 +0200
Subject: [PATCH] continue statistics documentation writing

---
 math/statistics.md | 75 +++++++++++++++++++++++++++++++++++++++-------
 1 file changed, 65 insertions(+), 10 deletions(-)

diff --git a/math/statistics.md b/math/statistics.md
index 1b407a5..6fd0134 100644
--- a/math/statistics.md
+++ b/math/statistics.md
@@ -4,24 +4,50 @@
 
 ### Range
 
-```php
-ArrayUtils::range($values);
-```
-
 $$
 range = max(values) - min(values)
 $$
 
-### Mean (arithmetic)
-
 ```php
-Average::mean($values);
+ArrayUtils::range($values);
 ```
 
+### Mean (arithmetic)
+
 $$
-mean = \frac{1}{n}\sum_{i=1}^{n}a_i
+\mu = mean = \bar{X} = \frac{1}{n}\sum_{i=1}^{n}a_i
 $$
 
+```php
+Average::arithmeticMean($values);
+```
+
+### Variance
+
+$$
+\sigma^{2} = Var(X) = \frac{1}{N} \sum_{i = 1}^{N}\left(x_{i} - \bar{X}\right)^{2}
+$$
+
+```php
+MeasureOfDispersion::empiricalVariance($x, $y);
+MeasureOfDispersion::sampleVariance($x, $y);
+```
+
+> The sample variance calculates with N - 1
+
+### Covariance
+
+$$
+cov(X,Y) = \frac{1}{N} \sum_{i = 1}^{N}\left(x_{i} - \bar{X}\right)\left(y_{i} - \bar{Y}\right)
+$$
+
+```php
+MeasureOfDispersion::empiricalCovariance($x, $y);
+MeasureOfDispersion::sampleCovariance($x, $y);
+```
+
+> The sample covariance calculates with N - 1
+
 ## Variable types
 
 Variables are characteristics (i.e. height, weight)
@@ -63,18 +89,47 @@ It is possible to transfrom variables:
 
 #### Goodness of fit test
 
+This tests if observed values follows expected/known proportions (e.g. distributions.)
+
+* H0: The observation follows the expected frequency/distribution (i.e. normal distribution)
+* H1: The observation doesn't follow the expected frequency/distribution (i.e. normal distribution)
+
+> If H0 can be discarded or not depends on the significance (p-value).
+
+```php
+ChiSquaredDistribution::testHypothesis($observed, $expected, $significance = 0.05, $degreesOfFreedom = 0);
+```
+
 #### Test of independence
 
+This tests if there is a relationship between two categorical variables.
+
+* H0: The variables are independent (there is no relationship between them)
+* H1: The variables are dependent (there is a relationship between them)
+
 ### t-test
 
 #### One sample
 
+This tests if the observed mean is different from a expected mean.
+
 #### Two samples
 
-#### Paired samples
+This tests if the observed mean or median is different from two samples.
 
 ### Correlation
 
 #### Pearson correlation
 
-#### Spearman's rank correlation
\ No newline at end of file
+Measures how two variables are related to each other (do they perform similarly, opposite or not related at all).
+
+$$
+\rho_{XY} = \frac{cov(X, Y)}{\sigma_X \sigma_Y}
+$$
+
+```php
+Correlation::bravaisPersonCorrelationCoefficientPopulation($x, $y);
+Correlation::bravaisPersonCorrelationCoefficientSample($x, $y);
+```
+
+> The sample correlation calculates with N - 1