Distribution of Mean

What is a sampling distribution?

The probability distribution of a given statistic based on a random sample of size n is called as a sampling distribution.

Sampling Distribution of the Mean:

This is a very important distribution. This is used to construct confidence intervals for the mean and for significance testing.

Definition:

The mean of the population from which the scores were sampled is defined as the mean of the sampling distribution of the mean. Consider a population with a mean μ and a standard deviation σ. The sampling distribution of the mean can be written as follows:

μ M = μ

The standard deviation and the variance of the sampling distribution can be written as follows:

M = σ / √n

σ M² = σ² / n

n = sample size

The standard deviation of the sampling distribution of the mean is named as the standard error of the mean. It is symbolically written as σ M.

The population variance divided by n, the sample size is called the variance of the sampling distribution of the mean.

It should be noted that the spread of the sampling distribution of the mean decreases as the sample size n increases.

Central Limit Theorem:

           Statement: Given a population with a finite mean σ and a finite non-zero variance σ², the sampling distribution of the mean tends to a normal distribution with a meanμand variance σ²/ n, as the sample size n increases.

The expressions for the mean and variance of the sampling distribution of the mean are novel or exemplary. What is novel is that regardless of the shape of the parent population, the sampling distribution of the mean tends to a normal distribution as the sample size n increases.

It can also be noted that larger the sample size n, the closer the sampling distribution of the mean would be to a normal distribution.