```---------- ---------- ---------- ---------- ---------- ---------- ---------- ----------
# [  source("http://macosa.dima.unige.it/r.R")  ]
# To go deep see en.wikipedia.org/wiki/Skewness or mathworld.wolfram.com/Skewness.html.
# If we call M the mean of tha data X, the more used measure of skewness (asymmetry) is
#  M(((X–M(X))^3)/sigma^3: the mean of the cube of the deviation divided by the cube of
# the standard deviation (so that tha data dispersion is out of consideration).
# It is due to Pearson and Fisher (as well as other indices).
# If the data is symmetrical with respect to the mean, the index is null; if they have
# a tail to the right, the index is positive; if they have a tail to the left it is
# negative. It can vary from −∞ to ∞. Some example:
#
sk = function(da) sum((da-mean(da))^3/Sd(da)^3)/length(da)
x = c(1,rep(2,2),rep(3,4),rep(4,9),rep(5,16));  y = -x
Histogram(x, 0.5,5.5, 1); Histogram(y, -0.5,-5.5, -1)

sk(x); sk(y)
# -1.248292  1.248292
#
# Another index is Bowley's index: ((Q3-Q2)-(Q2-Q1))/(Q3-Q1)  where Qi is the i-th
# quartile. The geometric meaning is quite evident, but it disregards the tails (data
# before 1st quartile and after 3th quartile). In the case of previous data it does not
# make sense:
sk2 = function(da) ((Percentile(da,75)-median(da))-(median(da)-Percentile(da,25))) /
(Percentile(da,75)-Percentile(da,25))
sk2(x); sk2(y)
# 0       0
# I have a null index!
#
# Excel uses another index:
sk3 <- function(da) sk(da)*length(da)^2/((length(da)-1)*(length(da)-2))
sk3(x); sk3(y)
# -1.374464  1.374464
#
# Another index (that is due to Pearson and Fisher) is:
sk4 <- function(da) (mean(da)-median(da))/Sd(da)
sk4(x)
# -0.3231104
# This index is between -1 and 1.
#
# After all, the same asymmetric coefficient can be used in particular cases to compare
# data of the same type. But its meaningful use requires very special skills and
# attentions: it is best to refer to graphic representations or "own" indexes referring
# to percentiles!

```