If the parent distribution is normal, the sampling distribution of the mean is normal with the same mean and a standard deviation that is reduced by a factor of the square root of n (the sample size) relative to the parent distribution. Remarkably, this conclusion holds approximately even if the parent distribution is not normal. This latter result is called the Central Limit Theorem.
Example #1 illustrates the central limit theorem when the underlying distribution is normal or chi-square.
Example #2 illustrates the central limit theorem when the underlying distribution is uniform, bowtie, right wedge, left wedge, and triangular.
Exercise #1 requires you to discuss the central limit theorem when the underlying distribution is uniform, bowtie, right wedge, left wedge, and triangular.
Exercise #2 requires you to discuss how the sampling distribution of the mean depends on n and the shape of the underlying parent distribution.
Exercise #3 requires you to discuss how the variability of the sampling distribution of the mean depends on n and the underlying symmetric parent distributions.