---------- ---------- ---------- ---------- ---------- ---------- ---------- ----------
                [ continuation from here, paragraph (N_3) ]
 
To produce histograms of already classified data, which are not easy to trace with
standard commands, you can use the histoclas command. An example: in Italy in 1951, in
the age intervals [0,5),[5,10),[10,20),[20,30),[30,40),[40,50),[50,60),[60,75),[75,100)
are dead 729,35,77,132,134,285,457,1401,1569 thousand people   (if the extremes of the
intervals are N the frequencies are N-1)
#
interv <- c(0, 5, 10, 20, 30, 40, 50, 60,  75, 100)
freq <-    c(729,35,77, 132,134,285,457,1401,1569)
histoclas(interv,freq)
#  The mean (brown dot) is about  58.31897 
#  For percentages use  PERC   For other statistics use the   morestat()
PERC
#  15.13  0.73  1.60  2.74  2.78  5.91  9.48  29.07  32.56
morestat()
#   Min.  1st Qu.  Median   Mean   3rd Qu.   Max. 
#   0.00   43.43   66.00   58.32   80.80  100.00
#    The brown dots are 5^ and 95^ percentiles 
#           The red dot is the mean 
 
             
 
In addition to the mean, some percentiles (the 25th, 50th and 75th percentiles, or
first quartile, median, and third quartile) are estimated and the box-plot is drawn.
[ to do this, a lot of data - distributed as the histogram - was generated; the file
  is dataclas; you do not have to see it; if you want to have an idea of it you can
  use str(dataclas)  ]
 
The percentiles are also called 100-quantiles  
 
The unitary percentage frequencies are the percentage frequencies divided by the
amplitude of each interval, so that the area of each rectangle represents the relative
frequency of the outputs that fall in the interval that is the base. The sum of the
areas of the rectangles is 1, or 100%.
 
For other percentiles you must use percentile(n). For example, the width of the box
(difference between 75th and 25th percentile, ie distance between 1st and 3th quartile)
is:
 
percentile(75) - percentile(25)
# 37.3752
 
This value [ percentile(75) - percentile(25) = 3^quartile - 1^quartile ] is also an
indicator of the data dispersion. Another used index is the standard deviation Sd,
that is the root mean square.  If the data are X1, …, Xn and the mean is M,
 
 Sd  is  √( ( (X1-M)^2 + … + (Xn-M)^2 ) / n )
 
In this case:
 
Sd(dataclas)
#  30.05867
 
Often in place of Sd we use  Var (variance), where Var = (X1-M)^2 + … + (Xn-M)^2

If I have a set of data XXX I can compute the percentiles with Percentile(XXX,n):
data = c(1,2,2,3,3,3,4,4,4,4,4,5,5,6,7,8,8)
median(data); Percentile(data,50); Percentile(data,15); Percentile(data,90)
#    4               4                     2.4                 7.4
               
               
If we want to give an index of the dispersion of the data:
 
Percentile(data,75)-Percentile(data,25)
#  2
or:
Sd(data)
#  1.963332
# For histogram and boxplot use:  histogram(data);  morestat()