Download Distribution of sample means

distribution of X and sample mean
Suppose we toss a die. There are six equally likely outcomes:
{1, 2, 3, 4, 5, 6}
{1, 2, 3, 4, 5, 6}
Now, consider all possible samples of size 2 that you can take
from this set of size 6.
{1, 2} {1, 3} {1, 4}
{1, 5} {1, 6}
{2, 3} {2, 4} {2, 5}
{2, 6} {3, 4}
{3, 5} {3, 6} {4, 5}
{4, 6} {5, 6}
Calculate the mean of each possible sample:
{1, 2} X = 1.5
{1, 3}
X = 2.0
{1, 4} X = 2.5
{1, 5}
X = 3.0
{1, 6} X = 3.5
{2, 3}
X = 2.5
{2, 4} X = 3.0
{2, 5}
X = 3.5
{2, 6} X = 4.0
{3, 4}
X = 3.5
{3, 5} X = 4.0
{3, 6}
X = 4.5
{4, 5} X = 4.5
{4, 6}
X = 5.0
{5, 6} X = 5.5
distribution of sample means
σX < σ
The probability distribution of X is a little closer to a bellshaped curve.
Uniform distribution
Central Limit Theorem
If all samples of a particular size n are selected from a large
population, the probability distribution of the sample mean
X is approximately a normal distribution with mean µ and
standard deviation given by:
σ
σx = √
n
The approximation improves if the sample size n is larger