A low standard deviation indicates that the data points tend to be close to the mean
(also called the expected value) of the set.
A high standard deviation indicates that the data points are spread out over a wider range of values.
(https://en.wikipedia.org/wiki/Standard_deviation)
σ = √⟮σ²⟯ = √⟮variance⟯
SD+ = √⟮σ²⟯ = √⟮sample variance⟯
| public static standardDeviation ( array $numbers, boolean $SD＋ = false ) : numeric | ||
| $numbers | array | |
| $SD＋ | boolean | |
| return | numeric | |
/**
* Standard error of the mean (SEM)
* The standard deviation of the sample-mean's estimate of a population mean.
* https://en.wikipedia.org/wiki/Standard_error
*
* s
* SEₓ = --
* √n
*
* s = sample standard deviation
* n = size (number of observations) of the sample
*
* @param array $X list of numbers (random variable X)
*
* @return float
*/
public static function standardErrorOfTheMean(array $X) : float
{
$s = Descriptive::standardDeviation($X, Descriptive::SAMPLE);
$√n = sqrt(count($X));
return $s / $√n;
}
/**
* Sample correlation coefficient
* Pearson product-moment correlation coefficient (PPMCC or PCC or Pearson's r)
*
* A normalized measure of the linear correlation between two variables X and Y,
* giving a value between +1 and −1 inclusive, where 1 is total positive correlation,
* 0 is no correlation, and −1 is total negative correlation.
* It is widely used in the sciences as a measure of the degree of linear dependence
* between two variables.
* https://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient
*
* The correlation coefficient of two variables in a data sample is their covariance
* divided by the product of their individual standard deviations.
*
* Sxy
* rxy = ----------
* sx sy
*
* Sxy is the sample covariance
* σx is the sample standard deviation of X
* σy is the sample standard deviation of Y
*
* @param array $X values for random variable X
* @param array $Y values for random variable Y
*
* @return number
*/
public static function sampleCorrelationCoefficient(array $X, array $Y)
{
$Sxy = self::sampleCovariance($X, $Y);
$sx = Descriptive::standardDeviation($X, Descriptive::SAMPLE);
$sy = Descriptive::standardDeviation($Y, Descriptive::SAMPLE);
return $Sxy / ($sx * $sy);
}
