Example R programs and commands
		3. Range, mean deviation, variance, diversity

# All lines preceded by the "#" character are my comments.
# All other left-justified lines are my input.
# All other indented lines are the R program output.

# Input data
z <- c(1.1, 2.0, 1.5, 2.1, 1.9, 1.8, 3.5, 3.6, 3.7)

# Value range
range(z)
	[1] 1.1 3.7

# Deviation
z-mean(z)
 [1] -1.2555556 -0.3555556 -0.8555556 -0.2555556 -0.4555556 -0.5555556  1.1444444
 [8]  1.2444444  1.3444444

# Absolute deviation
abs(z-mean(z))
 [1] 1.2555556 0.3555556 0.8555556 0.2555556 0.4555556 0.5555556 1.1444444
 [8] 1.2444444 1.3444444

# Mean deviation
mean(abs(z-mean(z))) 
  [1] 0.8296296

# Variance
var(z)
   [1] 0.9602778

# Standard deviation
sd(z)
  [1] 0.9799376

# Coefficient of variance
sd(z)/mean(z)
   [1] 0.4160113

# Shannon diversity
f <- c(54, 13, 27, 2)
n <- sum(f)   # total number in the population
k <- length(f)  # number of categories
# Maximum Shannon diversity for this number of categories:
HSmax <- log(k);  HSmax
# Shannon diversity of these frequencies:
HS <- (n*log(n) - sum(f*log(f)))/n;  HS
# Shannon evenness:
HS/HSmax


# Brillouin diversity
f <- c(54, 13, 27, 2)   # frequency table
n <- sum(f)   # total number in the population
k <- length(f)  # number of categories
# Use integer quotient c and remainder d from n=ck+d:
c <- n %/% k; d <- n %% k;
# Maximum Brillouin diversity for this number of categories:
Hmax <- (lfactorial(n) - (k-d)*lfactorial(c) - d*lfactorial(c+1))/n;   Hmax
#  Brillouin diversity of these frequencies:
H <- (sum(log(1:n))-sum(lfactorial(f)))/n;   H
# Brillouin evenness: 
H/Hmax
# NOTE: The lfactorial() function in R computes the natural logarithm
# of the factorial of its argument, which is much more sensible for
# large values.  If you use common (base 10) logarithms, then you must
# multiply the numbers by log(10) ~ 2.3036 to get the natural (base e)
# logarithms used in Brillouin's index.