https://en.wikipedia.org/wiki/Skewness
1 ∑⟮xᵢ - μ⟯³
γ₁ = ----- × ---------
N - 1 σ³
μ is the mean
σ³ is the standard deviation cubed, or, the variance raised to the 3/2 power.
N is the sample size
/**
* Get a report of all the descriptive statistics over a list of numbers
* Includes mean, median, mode, range, midrange, variance, standard deviation, quartiles, etc.
*
* @param array $numbers
* @param bool $population: true means all possible observations of the system are present;
* false means a sample is used.
* @return array [ n, mean, median, mode, range, midrange, variance, sd, CV, mean_mad,
* median_mad, quartiles, skewness, kurtosis, sem, ci_95, ci_99 ]
*/
public static function describe(array $numbers, bool $population = false) : array
{
$n = count($numbers);
$μ = Average::mean($numbers);
$σ = self::standardDeviation($numbers, $population);
return ['n' => $n, 'min' => min($numbers), 'max' => max($numbers), 'mean' => $μ, 'median' => Average::median($numbers), 'mode' => Average::mode($numbers), 'range' => self::range($numbers), 'midrange' => self::midrange($numbers), 'variance' => $population ? self::populationVariance($numbers) : self::sampleVariance($numbers), 'sd' => $σ, 'cv' => $σ / $μ, 'mean_mad' => self::meanAbsoluteDeviation($numbers), 'median_mad' => self::medianAbsoluteDeviation($numbers), 'quartiles' => self::quartiles($numbers), 'midhinge' => self::midhinge($numbers), 'skewness' => $population ? RandomVariable::populationSkewness($numbers) : RandomVariable::skewness($numbers), 'ses' => RandomVariable::SES($n), 'kurtosis' => RandomVariable::kurtosis($numbers), 'sek' => RandomVariable::SEK($n), 'sem' => RandomVariable::standardErrorOfTheMean($numbers), 'ci_95' => RandomVariable::confidenceInterval($μ, $n, $σ, 95), 'ci_99' => RandomVariable::confidenceInterval($μ, $n, $σ, 99)];
}
