Truncated Normal -- Reproduce a randomly generated data set help.
Problem:
Given a bounded Gaussian Distribution -- looking reproduce similar results i.e. same mean and standard deviation randomly.
Definition:
Data set exhibits properties of a Gaussian distribution.
Data set is bounded.
Each value of the data set is an integer.
Let mu and sigma represent the mean and standard deviation.
Let a and b represent the lower and upper bounds.
Comments:
Tried to employ both rtnorm or rttruncnorm functions in R programming.
When I get a value from either of these functions, I'm simply rounding the result to the nearest integer. When I elect to unbound either function ie. from -Inf to Inf, I find that I can reproduce an experimental data set that is quite similar to the original data except for having data values that are obviously falling outside of the original boundaries [a,b], but yes do achieve the goal mu and sigma.
When I bound my result using these functions, the effective standard deviation is smaller as compared to the original standard deviation; I'm cutting away the head and tail of the distribution leaving an effective sigma that spans a smaller range. 
While I appreciate that I can multiply and shift between truncated and non-truncated distributions using (X-mu)/sigma where X is a random value, the real goal is to reproduce an experimental truncated Gaussian distribution corresponding to the original truncated Gaussian data set.
For example, I tried simply increasing the goal standard deviation in rtnorm function for a bounded data set [a,b] to generate an effective standard deviation that is equivalent to the original standard deviation; however, this approach failed. This resulted in the standard deviation increasing somewhat but the effective standard deviation hit a ceiling falling short of the goal sigma thus never realizing the original standard deviation, mean and boundaries.
Remember, I am going on the premise that since I have an actual, real-world bounded Gaussian distribution with a given mean and standard deviation, I'm thinking that I can absolutely reproduce a similar randomly generated result. 
Am I mistaken? What might I do differently?
I appreciate any insight that any of you might lend. Your help would
really be welcomed and valued.
 A: As the variance of your simulated untruncated normal distribution increases the density gets flatter across the [a,b] interval.  At its limit it becomes almost a uniform distribution hence with a maximum standard deviation of $\sqrt{\frac{1}{12}(b-a)^2}$.  If the standard deviation of the population you are looking at is more than that, it must mean you have a bimodal distribution of some sort.  
Even in that case (which means you don't have a truncated normal distribution) the maximum theoretical variance happens when $\mu$ is bang in the middle of $a$ and $b$ and the data is all at the extremes - half at the lower bound and half at the upper.  The maximum possible standard deviation from a bounded distribution of any sort is $\sqrt{.5(a-\mu)^2+.5(b-\mu)^2}$ which reduces to $b-\mu$.
I would suggest you look at histograms and density line plots of your original data and question your starting assumption that comes from a truncated rounded Gaussian distribution.  I would also compare its actual standard deviation with the two theoretical limits above.  If it is less than the standard deviation of a uniform distribution you should be able to simulate a truncated normal distribution with that standard deviation, and you probably have a coding error.  If it is more than that amount, you need to find an alternative model.  
