The example of a function definition discussed in the recap session on April 19.

FunctionDefinition.R

# A step-by-step solution for the first function task of the "Basics" tutorial

# Goal: define a function that computes the sample variance of a vector

# Note: there are many strategies to develop functions; here we start by first
#  writing the code that solves our problem for one particular case outside the
#  function, and then generalize this code in a function.

# First step: think about the starting point for your function. In our case: 
#  we start with a vector containing some numbers, because this is from what
#  we compute the variance in the first place:

example_vector <- c(1,2,3,4)

# Second step: break the formula for the sample variance into parts, and 
#  translate each part into code. We start with the numerator of the formula:
vector_mean <- mean(example_vector) # The mean of the vector
vector_mean

vector_deviations <- example_vector - vector_mean # Deviation of each element from the mean
vector_deviations

vector_deviations_squared <- vector_deviations**2 # The squared deviations
vector_deviations_squared

numerator <- sum(vector_deviations_squared) # The sum of the squared deviations
numerator

# Now that we have computed the numerator, lets move to the denominator of the 
#  fraction:

nb_elements_vector <- length(example_vector) # This is the 'n' in the equation
denominator <- nb_elements_vector - 1

# To get the overall result, just divide the numerator by the denominator:
result <- numerator / denominator
result

# Third step: generalize our solution for the particular case into a general
#  function. To this end, think of a function name (here: 'var_manual') and
#  think about the number of arguments this function needs (here: one, i.e. 
#  the vector for which we want to compute the variance):

var_manual <- function(x){
  print(x)
}  
var_manual(example_vector)

# So far, the function only prints the argument we give to it. Now copy paste 
#  our code from above, and replace the name 'example_vector' with the name
#  of our argument (here: x). If we did not do so, the function would only work
#  if 'example_vector' was defined outside the function, and it would always
#  return the same result no matter what input we provide as an argument.

var_manual <- function(x){
  # Numerator:
  vector_mean <- mean(x) # The mean of the vector
  vector_deviations <- x - vector_mean # Deviation of each element from the mean
  vector_deviations_squared <- vector_deviations**2 # The squared deviations
  numerator <- sum(vector_deviations_squared) # The sum of the squared deviations

  # Denominator:
  nb_elements_vector <- length(x) # This is the 'n' in the equation
  denominator <- nb_elements_vector - 1
  
  # Result:
  result <- numerator / denominator
  result
} 

# Now we can use the function with arbitrary vectors as input. In the following
#  three examples, the same procedure from within the function was applied to 
#  three different vectors. The variable 'x' within the function can be thought
#  of as a placeholder that is replaced by the function input:

var_manual(x = c(1, 2, 3, 4, 5, 99))
var_manual(x = c(-4, 2, 4, 8, 10))
var_manual(x = c(50, 100, 82, 33))