“N/A”, “undesired”, “great”, “great”) my_mode(x1) ## [1] “great” # text from https://www.r-project.org/about.html t <- “R is available as Free Software under the terms of the Free Software Foundation’s GNU General Public License in source code form. It compiles and runs on a wide variety of UNIX platforms and similar systems (including FreeBSD and Linux), Windows and MacOS.” v <- unlist(strsplit(t,split=” “)) my_mode(v) ## [1] “and”
unique()
Linux
FreeBSD
tabulate()
uniqv()
While this function works fine on the examples provided, it only returns the first mode encountered. In general, however, the mode is not necessarily unique and it might make sense to return them all. This can be done by modifying the code as follows:
# my_mode # Finds the mode(s) of a vector v # Arguments: # v -- numeric vector or factor # return.all -- boolean -- set to true to return all modes # Returns: # the modal elements my_mode <- function(v, return.all = FALSE) { uniqv <- unique(v) tabv <- tabulate(match(v, uniqv)) if (return.all) { uniqv[tabv == max(tabv)] } else { uniqv[which.max(tabv)] } } # example: x <- c(1,2,2,3,3,4,5) my_mode(x) ## [1] 2 my_mode(x, return.all = TRUE) ## [1] 2 3
We were confident that it was fine to over-ride the definition of the function my_mode. Indeed, if the function was already used in some older code, then one would expect to see only one mode appear. That behaviour is still the same, because we chose the default value for the optional parameter return.all to be FALSE. If the default choice would be TRUE, then older code would produce wrong results and if we would not use a default value, then older code would fail to run.
8.2. Measures of Variation or Spread
measures of spread
Variation or spread measures how different observations are compared to the mean or other central measure. If variation is small, one can expect observations to be closer to each other.
Definition: Variance
variance
8.2.1 Standard Deviation
Definition: Standard deviation
spread – standard deviation
standard deviation
The estimator for standard deviation is:
t <- rnorm(100, mean=, sd=20) var(t) ## [1] 248.2647 sd(t) ## [1] 15.75642 sqrt(var(t)) ## [1] 15.75642 sqrt(sum((t - mean(t))∧2)/(length(t) - 1)) ## [1] 15.75642
sd()
8.2.2 Median absolute deviation
Definition: mad
mad
median absolute deviation
mad(t) ## [1] 14.54922 mad(t,constant=1) ## [1] 9.813314
mad()
The default “constant=1.4826” (approximately
for Xi distributed as N(μ, σ2) and large n.
8.3. Measures of Covariation
covariation
When there is more than one variable, it is useful to understand what the interdependencies of variables are. For example when measuring the size of peoples hands and their length, one can expect that people with larger hands on average are taller than people with smaller hands. The hand size and length are positively correlated.
The basic measure for linear interdependence is covariance, defined as
8.3.1 8.3.1 The Pearson Correlation
An important metric for linear relationship is the Pearson correlation coefficient ρ.
correlation – Pearson
Definition: Pearson Correlation Coefficient
cor(mtcars$hp,mtcars$wt) ## [1] 0.6587479
cor()
Of course, we also have functions that provide the covariance matrix and functions that convert the one in the other.
d <- data.frame(mpg = mtcars$mpg, wt = mtcars$wt, hp = mtcars$hp) # Note that we can feed a whole data-frame in the functions. var(d) ## mpg wt hp ## mpg 36.324103 -5.116685 -320.73206 ## wt -5.116685 0. 957379 44.19266 ## hp -320.732056 44.192661 4700.86694 cov(d) ## mpg wt hp ## mpg 36.324103 -5.116685 -320.73206 ##
