Sampling distribution and normal distribution formulas

The normal distribution is when you have enough samples so that the non-normal distribution converges upon normality with a mean and a standard error.

The formula for sampling distribution = variance (StDev squared) / number of samples

The formula for normal distribution = a mean with the standard deviation / number of samples square rooted.

Why does the square from the variance (sample distribution), get transferred to the square root of the number of samples (normal distribution)