3 ms·

SD is defined by dividing by the square root of the number of samples. It is never divided by the mean - as this metric has no relevance. The SD as it relates

SD is defined by dividing by the square root of the number of samples. It is never divided by the mean - as this metric has no relevance. The SD as it relates to the central limit theorem is the confidence in the estimation of the average E(X) of a distribution. It means that the SD reduces i.e. confidence increases as the number of samples increase. You don't need the central limit theorem for this. It follows from the definition of standard deviation where it has the square root of the number of samples in the denominator.

whatshisface 4y ago
The variation in the random variable relative to the value of the random variable goes away for large sums.
- random314 4y ago
  The variation in the random variable relative to the value of the random variable is fixed. It doesn't change, no matter how many times you sample the random variable. It's called the per-sample standard deviation. If 2 people A and B have a salary difference of 100K$ , then their salaries will not start to cluster together if you add C,D,E,F and more into the mix because of Central Limit theorem. Their salary difference will stay exactly at 100K as before. The variation/standard deviation in the sum of random variables increases as the square root of number of samples, not on the basis of how large the sum is. This follows from the definition of standard deviation. If you divide this "sum of random variables" which is itself a new kind of random variable (that is different from the per sample random variables you are starting out with) by number of samples then as alpha*sqrt(n)/n = alpha/sqrt(n) the variation in the summation random variable reduces(if divided by n) by a factor of 1/sqrt(n) Once again, all of this simply follows from the definition of standard deviation. Central limit theorem doesn't come into the picture.
  - whatshisface 4y ago
    https://en.wikipedia.org/wiki/Law_of_large_numbers https://en.wikipedia.org/wiki/Law_of_large_numbers
    - random314 4y ago
      I am not really sure what you are on about. So I will stop responding here.

whatshisface 4y ago

The variation in the random variable relative to the value of the random variable goes away for large sums.

random314 4y ago
The variation in the random variable relative to the value of the random variable is fixed. It doesn't change, no matter how many times you sample the random variable. It's called the per-sample standard deviation. If 2 people A and B have a salary difference of 100K$ , then their salaries will not start to cluster together if you add C,D,E,F and more into the mix because of Central Limit theorem. Their salary difference will stay exactly at 100K as before. The variation/standard deviation in the sum of random variables increases as the square root of number of samples, not on the basis of how large the sum is. This follows from the definition of standard deviation. If you divide this "sum of random variables" which is itself a new kind of random variable (that is different from the per sample random variables you are starting out with) by number of samples then as alpha*sqrt(n)/n = alpha/sqrt(n) the variation in the summation random variable reduces(if divided by n) by a factor of 1/sqrt(n) Once again, all of this simply follows from the definition of standard deviation. Central limit theorem doesn't come into the picture.
- whatshisface 4y ago
  https://en.wikipedia.org/wiki/Law_of_large_numbers https://en.wikipedia.org/wiki/Law_of_large_numbers
  - random314 4y ago
    I am not really sure what you are on about. So I will stop responding here.