Distribution of the Sample Means

A sample mean is a numerical set which is an average value of a particular portion of a number in a certain group. A sample mean is expressed as x.suppose if we take a sample of size n and there is a n independent variables x1,x2…xn and each value is respect to one randomly selected observation, then the distribution of the population  of these variables has the mean value mhu and the standard deviation sigma. Then the sample value is,  
                             X=1/n(x₁+x₂+…..x_n)

Mean and variance of the distribution of the sample means:

Using the mean and variance property of the random variable ,the mean and variance of the sample mean is given below.
µ_x= µ
sigmax=sigma/sqrt(n)`
therefore the mean value of the sample mean is similar to the mean of the population distribution and the variance is smaller than the variance of the population distribution.

For example:
In population distribution  where the mean value µ=20 and the standard deviation σ=2 that means ( N(20,2).then we have the simple random sample of 100 students ,then what is the  mean and variance of  the sample mean distribution?
                                                µ_x= µ
                                                µ_x=20
                                                σ_x=σ/`sqrt(n)`
                                                σ_x =2/`sqrt(100)`
                                                σ_x =2/10
                                                 σ_x =0.2

shape property of the distribution of the sample means:

It may be in the following shapes like normal, skewed, bimodal.

Explanation for distribution of the sample means:
If the population distribution is normal,then the distribution of  the sample mean is normal. If in the population distribution   the mean and variance value is respectively  (µ, σ), then the mean and variance for the distribution of the sample mean is(µ, σ/`sqrt(n)` ).
 For linearcombination the sample mean x=(1/n)(x₁+x₂+..x _n)
By using the above all concept the following theorem is illustrated.

Central Limit Theorem- distribution of the sample mean:
Theorem definition:

If we take a  population with a  mean μ and a variance σ², then the  sampling distribution of the mean approach a normal distribution with a mean of μ and a variance of σ²/N as N means   increases value of the sample size.