Does more and more measurements of $X$ indefinitely improve $\sigma_X^2$? For a physical quantity $X$, does the standard deviation $\sigma^2_X\equiv\langle X^2\rangle-\langle X\rangle^2$ calculated from $N$ measurements smaller than that calculated from $N^\prime(<N)$ measurements? Is there any expectation that as the number of measurements increase, the SD decreases? I am talking about simple classical measurements; no quantum mechanics.
Additional remark There was an answer (now deleted) which proved that SD decreases with the increase in measurements.
Please also explain physically. Why do we expect (if it is obvious) the standard deviation to decrease with the increase in the number of measurements (even though for each successive measurement, we use identical instrument with a fixed least count)?
 A: Some info here might be helpful. 
https://sisu.ut.ee/measurement/31-normal-distribution
There is a definite dependence on the number of measurements for smaller measurements, but I'd expect it to die off.
Taking a measurement is like selecting an element from a set. As you sample elements from a set, you build a subset whose statistical properties more closely match those of the original population. The mean matches, the SD matches. 
The SD of a set consisting of one measurement is zero. The SD of two elements is $|x_1-x_2|/2$ with a mean of $(x_1+x_2)/2$. Measurements often follow a Normal Distribution. 
So the question is, how many measurements do you need to produce a subset of measurements that have a mean and SD you'd have if you took an infinite number of measurements? 
For that you need The Central Limit Theorem. With it's chief relevance to the question being The Law of Large Numbers.
The mean of a sub sample converges to the mean of the overall sample asthe number of measurements made goes to infinity. The SD is the difference between the mean of the squares of the measurements and the square of the mean of them measurements. In this case, the limit of the differences is the difference of the limits, so a series of measurements will have an SD converging to the population's SD over time. 
