The Big R-Book. Philippe J. S. De Brouwer
some measures of central tendency become more appropriate to use than others. In the following sections, we will look at the mean, mode, and median, and learn how to calculate them and under what conditions they are most appropriate to be used.
8.1.1 Mean
mean
Probably the most used measure of central tendency is the “mean.” In this section we will start from the arithmetic mean, but illustrate some other concepts that might be more suited in some situations too.
central tendency – mean
8.1.1.1 The Arithmetic Mean
mean – arithmetic
The most popular type of mean is the “arithmetic mean.” It is the average of a set of numerical values; and it is calculated by adding those values first together and then dividing by the number of values in the aforementioned set.
mean – arithmetic
Definition: Arithmetic mean
The unbiased estimator of the mean for K observations xk is:
mean
P()
probability
probability
f()
Not surprisingly, the arithmetic mean in R is calculated by the function mean().
probability density function
mean()
This is a dispatcher function1 and it will work in a meaningful way for a variety of objects, such as vectors, matrices, etc.
# The mean of a vector: x <- c(1,2,3,4,5,60) mean(x) ## [1] 12.5 # Missing values will block the override the result: x <- c(1,2,3,4,5,60,NA) mean(x) ## [1] NA # Missing values can be ignored with na.rm = TRUE: mean(x, na.rm = TRUE) ## [1] 12.5 # This works also for a matrix: M <- matrix(c(1,2,3,4,5,60), nrow=3) mean(M) ## [1] 12.5
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