The mean is highly influenced by the outliers. To mitigate this to some extend the parameter trim allows to remove the tails. It will sort all values and then remove the x% smallest and x% largest observations.
v <- c(1,2,3,4,5,6000) mean(v) ## [1] 1002.5 mean(v, trim = 0.2) ## [1] 3.5
8.1.1.2 Generalised Means
mean – generalized
More generally, a mean can be defined as follows:
Definition: f-mean
Popular choices for f() are:
f(x) = x : arithmetic mean,
f(x) = xm: power mean,
f(x) = lnx : geometric mean,
arithmetic mean
mean – harmonic
harmonic mean
mean – power
power mean
mean – geometric
geometric mean
The Power Mean
One particular generalized mean is the power mean or Hölder mean. It is defined for a set of K positive numbers xk by
holder mean
mean – holder
by choosing particular values for m one can get the quadratic, arithmetic, geometric and harmonic means.
mean – quadratic
m → ∞: maximum of xk
m = 2: quadratic mean
m = 1: arithmetic mean
m → 0: geometric mean
m = 1: harmonic mean
m → −∞: minimum of xk
Example: Whichmeanmakes most sense?
What is the average return when you know that the share price had the following returns: −50%, +50%,−50%, +50%. Try the arithmetic mean and the mean of the log-returns.
returns <- c(0.5,-0.5,0.5,-0.5) # Arithmetic mean: aritmean <- mean(returns) # The ln-mean: log_returns <- returns for(k in 1:length(returns)) { log_returns[k] <- log( returns[k] + 1) } logmean <- mean(log_returns) exp(logmean) - 1 ## [1] -0.1339746 # What is the value of the investment after these returns: V_0 <- 1 V_T <- V_0 for(k in 1:length(returns)) { V_T <- V_T * (returns[k] + 1) } V_T ## [1] 0.5625 # Compare this to our predictions: ## mean of log-returns V_0 * (exp(logmean) - 1) ## [1] -0.1339746 ## mean of returns V_0 * (aritmean + 1) ## [1] 1
8.1.2 The Median
median
While the mean (and the average in particular) is widely used, it is actually quite vulnerable to outliers. It would therefore, make sense to have a measure that is less influenced by the outliers and rather answers the question: what would a typical observation look like. The median is such measure.
central tendency – median
The median is the middle-value so that 50% of the observations are lower and 50% are higher.
x <- c(1:5,5e10,NA) x ## [1] 1e+00 2e+00 3e+00 4e+00 5e+00 5e+10 NA median(x) # no meaningful result with NAs ## [1] NA median(x,na.rm = TRUE) # ignore the NA ## [1] 3.5 # Note how the median is not impacted by the outlier, # but the outlier dominates the mean: mean(x, na.rm = TRUE) ## [1] 8333333336
8.1.3 The Mode
mode
central tendency – mode
The mode is the value that has highest probability to occur. For a series of observations, this should be the one that occurs most often. Note that the mode is also defined for variables that have no order-relation (even labels such as “green,” “yellow,” etc. have amode, but not a mean or median—without further abstraction or a numerical representation).
In R, the function mode() or storage.mode() returns a character string describing how a variable is stored. In fact, R does not have a standard function to calculate mode, so let us create our own:
mode()
storage.mode()
# my_mode # Finds the first mode (only one) # Arguments: # v -- numeric vector or factor # Returns: # the first mode my_mode <- function(v) { uniqv <- unique(v) tabv <- tabulate(match(v, uniqv)) uniqv[which.max(tabv)] } # now test this function x <- c(1,2,3,3,4,5,60,NA) my_mode(x) ## [1] 3 x1 <-
