### Class examples from Week 3: Central Tendency
### Ken Benoit Feb 2010


## mode example
x <- c(1, 2, 3, 1, 1, 5, 5, 4, 1, 4, 4, 3)
mode(x)       # this means something very different in R!
table(x)      # do a frequency distribution
# see which values are the maximum values
which(table(x)==max(table(x)))

## multiple modes example
x <- c(6, 6, 7, 2, 6, 1, 2, 3, 2, 4)
which(table(x)==max(table(x)))

## medians
x <- c(1, 5, 6, 7, 7, 8)
median(x)
x <- c(4, 8, 15, 16, 23, 42, 45, 2, 7, 33, 4, 46, 997)
sort(x)          # put the vector in order
length(x)        # length of vector is N
length(x)%%2     # modulus division: is there are remainder?
(length(x)+1)/2  # the position of the median by our formula
length(x)%%2==0  # no remainder means this number is ODD
median(x)        # use R to compute median
x <- c(x,998)    # add one more number to our test vector
x                # look at the vector
sort(x)          # look at the vector sorted
(length(x)+1)/2  # position of new median value
length(x)%%2    
(length(x)+1)/2  # the position of the median by our formula
length(x)%%2==0  # should be TRUE if even
median(x)

## means, means as deviations
x <- c(9,8,6,5,2)
xbar <- mean(x)
xdev <- x - xbar
cbind(x,xdev)
xbar

## mean sensitivity to extreme values
a <- c(5,6,6,7,8,9,10,10)
b <- c(5,6,6,7,8,9,10,95)
median(a)
median(b)
mean(a)
mean(b)

## geometric mean example
prod(10, 1, 1000, 1, 10)
prod(10, 1, 1000, 1, 10)^(1/5)
exp(1/5 * sum(10, 1, 100, 10))
exp(mean(log(c(10, 1, 1000, 1, 10))))

## harmonic mean example
x <- c(1,2,4,1)
1/mean(1/x)
length(x)/sum(1/x)

## applying mean and median to real data
# FIRST YOU NEED TO DOWNLOAD THE FILE dail2002.Rdata
# see the code from Week 2 on how to do this
load("dail2002.Rdata")
attach(dail2002)
summary(spend_total)
mean(spend_total)
plot(density(spend_total))
abline(v=mean(spend_total), col="red")
abline(v=median(spend_total), col="blue", lty="dashed")
detach(dail2002)

## weighted mean example -- uses a built-in dataset
data(state)                   # load the built-in state dataset
state.x77[1:5,1:2]            # inspecta snippet of the data
attach(data.frame(state.x77)) # coerce as data frame and attach
mean(Income)
weighted.mean(Income, Population) # automatically normalizes weight
median(Income)