Sample data.

Sample with weights. First element of sample is data, second is weight

O(n) Range. The difference between the largest and smallest elements of a sample.

O(n) Arithmetic mean. This uses Kahan-Babuška-Neumaier summation, so is more accurate than welfordMean unless the input values are very large.

O(n) Arithmetic mean. This uses Welford's algorithm to provide numerical stability, using a single pass over the sample data.

Compared to mean, this loses a surprising amount of precision unless the inputs are very large.

O(n) Arithmetic mean for weighted sample. It uses a single-pass algorithm analogous to the one used by welfordMean.

O(n) Harmonic mean. This algorithm performs a single pass over the sample.

O(n) Geometric mean of a sample containing no negative values.

Compute the kth central moment of a sample. The central moment is also known as the moment about the mean.

This function performs two passes over the sample, so is not subject to stream fusion.

For samples containing many values very close to the mean, this function is subject to inaccuracy due to catastrophic cancellation.

Compute the kth and jth central moments of a sample.

This function performs two passes over the sample, so is not subject to stream fusion.

For samples containing many values very close to the mean, this function is subject to inaccuracy due to catastrophic cancellation.

Compute the skewness of a sample. This is a measure of the asymmetry of its distribution.

A sample with negative skew is said to be left-skewed. Most of its mass is on the right of the distribution, with the tail on the left.

skewness $ U.to [1,100,101,102,103]
==> -1.497681449918257

A sample with positive skew is said to be right-skewed.

skewness $ U.to [1,2,3,4,100]
==> 1.4975367033335198

A sample's skewness is not defined if its variance is zero.

This function performs two passes over the sample, so is not subject to stream fusion.

For samples containing many values very close to the mean, this function is subject to inaccuracy due to catastrophic cancellation.

Compute the excess kurtosis of a sample. This is a measure of the "peakedness" of its distribution. A high kurtosis indicates that more of the sample's variance is due to infrequent severe deviations, rather than more frequent modest deviations.

A sample's excess kurtosis is not defined if its variance is zero.

This function performs two passes over the sample, so is not subject to stream fusion.

For samples containing many values very close to the mean, this function is subject to inaccuracy due to catastrophic cancellation.

Maximum likelihood estimate of a sample's variance. Also known as the population variance, where the denominator is n.

Unbiased estimate of a sample's variance. Also known as the sample variance, where the denominator is n-1.

Calculate mean and maximum likelihood estimate of variance. This function should be used if both mean and variance are required since it will calculate mean only once.

Calculate mean and unbiased estimate of variance. This function should be used if both mean and variance are required since it will calculate mean only once.

Standard deviation. This is simply the square root of the unbiased estimate of the variance.

Weighted variance. This is biased estimation.

Standard error of the mean. This is the standard deviation divided by the square root of the sample size.

Maximum likelihood estimate of a sample's variance.

Unbiased estimate of a sample's variance.

Standard deviation. This is simply the square root of the maximum likelihood estimate of the variance.

Covariance of sample of pairs. For empty sample it's set to zero

Correlation coefficient for sample of pairs. Also known as Pearson's correlation. For empty sample it's set to zero.

Pair two samples. It's like zip but requires that both samples have equal size.