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I am new to the field of Probability and Statistics and was wondering if there actually existed a number n as big enough sample size, that would be considered the cutoff for a close-enough approximation of population standard deviation from sample standard deviation.

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As you state it, the question is impossibly vague and does not have an answer. However, if you are willing to make an assumption about the distribution of the population, then it is possible to make some progress.

Suppose you have a sample of size $n$ from a normal population with unknown mean $\mu$ and unknown standard deviation $\sigma.$ Then you can get a 95% confidence interval for $\sigma$ using the chi-squared distribution with $n-1$ degrees of freedom.

To be specific, suppose $n = 50$ and the sample standard deviation is $S = 38.29.$ Then one can show that $$P\{31.555 < 49S^2/\sigma^2 < 70.222\} = 0.95.$$ After some algebra, for our data this becomes $$P\{49(38.29^2)/70.222 = 1023.042 < \sigma^2 < 49(38.29^2)/31.555 = 2276.662\}.$$ So that a 95% confidence interval for estimating $\sigma^2$ is about $(1023, 2277).$ Taking square roots of the endpoints we have $(31.98, 47.71)$ as a 95% confidence interval for $\sigma.$ Notice that the sample standard deviation $S = 38.29$ lies in this interval (but not exactly at the center of it).

With suitable software, such computations are easy to do. For example, in this particular situation a computation in R is shown below:

 n = 50; s = 38.29
 sqrt((n-1)*s^2/qchisq(c(.975,.025), n-1))
 ## 31.98494 47.71445

It would be up to you to decide if $n = 50$ observations from this normal sample have given a confidence interval with endpoints 'close enough' to each other to be useful, and whether a '95% level of confidence' in this particular kind of computation is 'sure enough' for your taste. Larger values of $n$ will give shorter confidence intervals. (A little exploration that I will not cover here allows one to find typical lengths of confidence intervals for various values of $n$ and various values of the population standard deviation $\sigma.$)

For populations with distributions other than normal, analogous confidence interval procedures are available.

Generally speaking, it requires more data to get close estimates of $\sigma$ than it does to get close estimates of $\mu.$ "Variances are very variable."

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