Sampling Distribution of the Mean
Prerequisites
Introduction
to Sampling Distributions, Variance
Sum Law I
Learning Objectives
 State the mean and variance of the sampling distribution of the mean
 Compute the standard error of the mean
 State the central limit theorem
The sampling distribution of the mean was defined
in the section introducing sampling distributions. This section
reviews some important properties of the sampling distribution
of the mean that were introduced in the demonstrations in this
chapter.
Mean
The mean of the sampling distribution of the mean
is the mean of the population from which the scores were sampled.
Therefore, if a population has a mean, μ,
then the sampling distribution of the mean is also μ.
The symbol μM
is used to refer to the mean of the sampling distribution of the
mean. Therefore, the formula for the mean of the sampling distribution
of the mean can be written as:
μM
= μ
Variance
The variance of the sampling distribution of the
mean is computed as follows:
That is, the variance of the sampling distribution
of the mean is the population variance divided by N, the sample
size (the number of scores used to compute a mean). Thus, the
larger the sample size, the smaller the variance of the sampling
distribution of the mean.
The standard error of the mean is the standard deviation
of the sampling distribution of the mean. It is therefore the
square root of the variance of the sampling distribution of the
mean and can be written as:
The standard error is represented by a σ
because it is a standard deviation. The subscript(M) indicates
that the standard error in question is the standard error of the
mean.
Central Limit Theorem
The central limit theorem states that:
Given a population with a finite mean μ and
a finite nonzero variance σ2,
the sampling distribution of the mean approaches a normal
distribution with a mean of μ and a variance of σ2/N
as N, the sample size increases.
The expressions for the mean and variance of the
sampling distribution of the mean are not new or remarkable. What
is remarkable is that regardless of the shape of the parent population,
the sampling distribution of the mean approaches a normal distribution
as N increases.
