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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.
(optional) This expression can be derived very
easily from the variance
sum law. Let's begin by computing the
variance of the sampling distribution of the sum of three numbers
sampled from a population with variance σ2.
The variance of the sum would be σ2
+ σ2 + σ2.
For N numbers, the variance would be Nσ2.
Since the mean is 1/N times the sum, the variance of the sampling
distribution of the mean would be 1/N2
times the variance of the sum, which equals σ2/N.
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 non-zero 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. If you have used the "Central
Limit Theorem Demo," you have already seen this for yourself.
As a reminder, Figure 1 shows the results of the simulation for
N = 2 and N = 10. The parent population was a uniform
distribution. You can see that the distribution for N = 2 is far
from a normal distribution. Nonetheless, it does show that the
scores are denser in the middle than in the tails. For N = 10
the distribution is quite close to a normal distribution. Notice
that the means of the two distributions are the same, but that
the spread of the distribution for N = 10 is smaller.
Figure 2 shows how closely the sampling distribution
of the mean approximates a normal distribution even when the parent
population is very non-normal. If you look closely you can see
that the sampling distributions do have a slight positive skew.
The larger the sample size, the closer the sampling distribution
of the mean would be to a normal distribution.
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