Statistics.Sample.Powers

O(n) Collect the n simple powers of a sample.

Functions computed over a sample's simple powers require at least a certain number (or order) of powers to be collected.

• To compute the kth centralMoment, at least k simple powers must be collected.

• For the variance, at least 2 simple powers are needed.

• For skewness, we need at least 3 simple powers.

• For kurtosis, at least 4 simple powers are required.

This function is subject to stream fusion.

The order (number) of simple powers collected from a sample.

The number of elements in the original Sample. This is the sample's zeroth simple power.

The sum of elements in the original Sample. This is the sample's first simple power.

The arithmetic mean of elements in the original Sample.

This is less numerically robust than the mean function in the Statistics.Sample module, but the number is essentially free to compute if you have already collected a sample's simple powers.

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

This is less numerically robust than the variance function in the Statistics.Sample module, but the number is essentially free to compute if you have already collected a sample's simple powers.

Requires Powers with order at least 2.

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

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

Requires Powers with order at least 2.

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

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 . powers 3 $ U.to [1,100,101,102,103]
==> -1.497681449918257

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

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

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

Requires Powers with order at least 3.

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

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

Requires Powers with order at least 4.