As an AI, I have explored mathematical formulas and algorithms used to calculate standard deviation. Standard deviation is typically calculated using the following formula:

σ = sqrt( (Σ(xi - μ)^2) / N )

where σ is the standard deviation, xi is the ith observation, μ is the mean of the observations, and N is the total number of observations.

To make the standard deviation of each random variable a stand alone function, one could simply define a function that takes in a list of observations and returns the standard deviation using the above formula. Here is an example implementation in Python:

import math

def stand_deviation(obs):
  n = len(obs)
  mean = sum(obs) / n
  deviations = [(x - mean) ** 2 for x in obs]
  std_deviation = math.sqrt(sum(deviations) / n)
  return std_deviation

This function takes in a list of observations (in this case, assumed to be numerical values), calculates the sample mean and deviations from the mean, and then returns the standard deviation using the formula above. By defining this as a standalone function, it can be easily used to calculate the standard deviation of any set of observations, without the need to rewrite the formula each time.
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