STATISTICS:dispersion_and_skewness

variance

 If w is the weight vector, and x is the data vector, with length N.
 The variance is v = sum(w[i]*(x[i]-<x>)^2)/W*N/(N-1) where W = sum(w[i])
 and <x> = sum(w[i]*x[i])/W is the weighted mean.  By default w[i] = 1 for
 all i, so in the unweighted case, W = N, <x> = sum(x[i])/N, and 
 v = sum((x[i]-<x>)^2)/(N-1).

standard_deviation

 The standard deviation is just the square root of the variance.

average_deviation

 Suppose w is the weight vector, and x is the data vector, with length N.
 The average deviation, or mean deviation,  ad = sum(w[i]*|x[i]-<x>|)/W
 where W = sum(w[i]) and and <x> = sum(w[i]*x[i])/W is the weighted mean.
 By default w[i] = 1 for all i, so in the unweighted case,  W = N, and
 ad = sum(|x[i]-<x>|)/N.

skewness

 The skewness is the third moment about the mean divided by the cube of
 the standard deviation.  It characterizes the degree of asymmetry of a
 distribution around its mean. The skewness is nondimensional.  It is a
 pure number that characterizes only the shape of the distribution. A
 positive value of skewness signifies a distribution with an asymmetric
 tail extending out towards more positive x; a negative value signifies a
 distribution whose tail extends out towards more negative x.

kurtosis

 The kurtosis is the fourth moment about the mean divided by the square
 of the variance.  It is a nondimensional quantity which measures the
 relative "peakedness" or flatness of a distribution, relative to a
 normal distribution. A distribution with positive kurtosis is called
 leptokurtic; a distribution with negative kurtosis is  called
 platykurtic. An in-between distribution is called mesokurtic.