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136 lines
3.0 KiB
Markdown
Executable File
136 lines
3.0 KiB
Markdown
Executable File
# Statistics
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## Definitions
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### Range
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$$
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range = max(values) - min(values)
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$$
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```php
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ArrayUtils::range($values);
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```
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### Mean (arithmetic)
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$$
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\mu = mean = \bar{X} = \frac{1}{n}\sum_{i=1}^{n}a_i
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$$
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```php
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Average::arithmeticMean($values);
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```
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### Variance
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$$
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\sigma^{2} = Var(X) = \frac{1}{N} \sum_{i = 1}^{N}\left(x_{i} - \bar{X}\right)^{2}
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$$
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```php
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MeasureOfDispersion::empiricalVariance($x, $y);
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MeasureOfDispersion::sampleVariance($x, $y);
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```
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> The sample variance calculates with N - 1
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### Covariance
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$$
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cov(X,Y) = \frac{1}{N} \sum_{i = 1}^{N}\left(x_{i} - \bar{X}\right)\left(y_{i} - \bar{Y}\right)
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$$
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```php
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MeasureOfDispersion::empiricalCovariance($x, $y);
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MeasureOfDispersion::sampleCovariance($x, $y);
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```
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> The sample covariance calculates with N - 1
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## Variable types
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Variables are characteristics (i.e. height, weight)
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### Quantitive
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Quantitative variables represent a measer, they are numeric.
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#### Discrete
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Countable and finite possibilities/results/options.
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#### Continuous
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Not countable and infinite number of possibilities/results/options.
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### Qualitative
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Qualitative variables represent categories/levels, they are not numeric.
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#### Nominal
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Cannot be ordered, they have the same importance/level/value (i.e. hair color). A category can also by yes/no (= two levels).
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#### Ordinal
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Can be ordered, they have different importance/level/value (i.e. risk severity)
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### Transformation
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It is possible to transfrom variables:
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* Continuous to discrete: by removing steps in between (i.e. age only in years)
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* Quantitive to qualitive: by asigning numeric ranges a quality (i.e. school grades in pass and fail)
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## Tests
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### Chi-square
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#### Goodness of fit test
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This tests if observed values follows expected/known proportions (e.g. distributions.)
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* H0: The observation follows the expected frequency/distribution (i.e. normal distribution)
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* H1: The observation doesn't follow the expected frequency/distribution (i.e. normal distribution)
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> If H0 can be discarded or not depends on the significance (p-value).
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```php
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ChiSquaredDistribution::testHypothesis($observed, $expected, $significance = 0.05, $degreesOfFreedom = 0);
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```
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#### Test of independence
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This tests if there is a relationship between two categorical variables.
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* H0: The variables are independent (there is no relationship between them)
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* H1: The variables are dependent (there is a relationship between them)
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### t-test
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#### One sample
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This tests if the observed mean is different from a expected mean.
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#### Two samples
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This tests if the observed mean or median is different from two samples.
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### Correlation
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#### Pearson correlation
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Measures how two variables are related to each other (do they perform similarly, opposite or not related at all).
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$$
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\rho_{XY} = \frac{cov(X, Y)}{\sigma_X \sigma_Y}
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$$
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```php
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Correlation::bravaisPersonCorrelationCoefficientPopulation($x, $y);
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Correlation::bravaisPersonCorrelationCoefficientSample($x, $y);
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```
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> The sample correlation calculates with N - 1
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