#   look at a population
source("../pop_sd.R")
pop_main <- c(33, 27, 16, 18, 14, 19, 26,  7, 28, 30,
              13, 20, 17, 21, 21, 27, 16, 21, 39,  7)
#  what does it look like
hist( pop_main)

#  remember that a histogram, especially for small
#  sets of data, can change its shape if we change
#  the break ppoints
hist( pop_main, breaks=seq(3,45,7) )
#  or
hist( pop_main, breaks=seq(5,41,4),
      main="doctored break points")

# However the mean and standard deviation do not
# depend on such break points
mean( pop_main )
pop_sd( pop_main )

#  Now, observe the change in the mean and standard
#  deviation if we add the same value to each item
#  in the population to get a new population
pop_add_15 <- pop_main + 15
pop_add_15
mean( pop_add_15 )
pop_sd( pop_add_15 )

# The mean went up by 15 but the standard deviation
# did not change.
##########
# Now, observe the change in the mean and standard 
# deviation if we multiply each value in the population 
# by a given value.
pop_mult_3 <- pop_main * 3
pop_mult_3
mean( pop_mult_3 )
pop_sd( pop_mult_3 )

# The effect of such multiplication was to change
# both the mean and the standard deviation.
###########
# Now look at the pattern of changing the mean to 0
# and then changing the standard deviation to 1
pop_mean_0 <- pop_main - mean( pop_main )
pop_mean_0
mean( pop_mean_0 )
pop_mean_0_sd <- pop_sd( pop_mean_0 )
pop_mean_0_sd
pop_m0_sd1 <- pop_mean_0 / pop_mean_0_sd
pop_m0_sd1
mean( pop_m0_sd1 )
pop_sd( pop_m0_sd1 )

# just to remind us that the values displayed
# are rounded
options(digits=22)
pop_m0_sd1
options( digits=7 )

#  go back and look at a histogram of this
#  new population
hist( pop_m0_sd1 )

#  so, now, let us generate a new population
#  but this time let us have mean=14.5 and
#  standard deviation=3.7
#########
#  First set the standard deviation
pop_sd3p7 <- pop_m0_sd1 * 3.7
mean( pop_sd3p7  )
pop_sd( pop_sd3p7 )

# Then change the mean to be 14.5
pop_m14p5_sd3p7 <- pop_sd3p7 + 14.5
mean( pop_m14p5_sd3p7 )
pop_sd( pop_m14p5_sd3p7 )


# look at the historgram of the new population 
hist( pop_m14p5_sd3p7)
hist( pop_m14p5_sd3p7, 
      breaks=seq( 7.1, 23.75, 1.85),
      main="doctored again")

##########
#  We will move onto a much larger population.
#  Here is a population of IQ scores.
source("../gnrnd5.R")
gnrnd5(43073899904, 15000100)

# we can look at the first and last 15 values
head( L1, 15)
tail( L1, 15)

# here is a summary of the population
summary( L1 )
# and a histogram
hist(L1, main="IQ values")


#  Then look at the mean and standard deviation
pop_mean <- mean(L1)
pop_mean
pop_stndev <- pop_sd( L1 )
pop_stndev
# Now redo the histogram  using the info from
#  the mean and standard deviation to set break 
#  points
hist( L1, main="IQ values, using special breaks",
      breaks=seq(40,160, 7.5), right=FALSE)

# and a box and whisker plot
boxplot( L1, horizontal = TRUE,
         main="IQ values")
#  Mark off the box plot with standard deviation lines
std_markers <-pop_mean+(-3:3)*pop_stndev
std_markers
abline( v=std_markers,
        lty="dashed", col="red")

#  Now find the 4th, 16th, 50th, 84th, and 96th percentiles
#  then compare those to the mean and standard deviation
#  markers
q_marks <-quantile(L1, c(0.04, 0.16, 0.50, 0.84, 0.96))
q_marks
abline( v=q_marks, lty="dotted", col="blue")

#  now convert L1 to L2 where L2 has mean 0 and
#  standard deviation 1
L2 <- (L1-pop_mean)/pop_stndev
mean( L2 )
pop_sd( L2 )
hist( L2, right=FALSE )
boxplot( L2, horizontal=TRUE )