Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

200 CHAPTER 7 | Probability and Samples: The Distribution of Sample Means
the number of possible samples increases dramatically and it is virtually impossible to actually
obtain every possible random sample. Fortunately, it is possible to determine exactly
what the distribution of sample means looks like without taking hundreds or thousands
of samples. Specifically, a mathematical proposition known as the central limit theorem
provides a precise description of the distribution that would be obtained if you selected
every possible sample, calculated every sample mean, and constructed the distribution of
the sample mean. This important and useful theorem serves as a cornerstone for much of
inferential statistics. Following is the essence of the theorem.
Central Limit Theorem For any population with mean μ and standard deviation
σ, the distribution of sample means for sample size n will have a mean of μ and a
standard deviation of syÏn and will approach a normal distribution as n approaches
infinity.
The value of this theorem comes from two simple facts. First, it describes the distribution
of sample means for any population, no matter what shape, mean, or standard deviation.
Second, the distribution of sample means “approaches” a normal distribution very
rapidly. By the time the sample size reaches n = 30, the distribution is almost perfectly
normal.
Note that the central limit theorem describes the distribution of sample means by identifying
the three basic characteristics that describe any distribution: shape, central tendency,
and variability. We will examine each of these.
■ The Shape of the Distribution of Sample Means
It has been observed that the distribution of sample means tends to be a normal distribution.
In fact, this distribution is almost perfectly normal if either of the following two conditions
is satisfied:
1. The population from which the samples are selected is a normal distribution.
2. The number of scores (n) in each sample is relatively large, around 30 or more.
(As n gets larger, the distribution of sample means will closely approximate a normal
distribution. When n > 30, the distribution is almost normal regardless of the shape of the
original population.)
As we noted earlier, the fact that the distribution of sample means tends to be normal
is not surprising. Whenever you take a sample from a population, you expect the
sample mean to be near to the population mean. When you take lots of different samples,
you expect the sample means to “pile up” around μ, resulting in a normal-shaped
distribution. You can see this tendency emerging (although it is not yet normal) in
Figure 7.2.
■ The Mean of the Distribution of Sample Means:
The Expected Value of M
In Example 7.1, the distribution of sample means is centered at the mean of the population
from which the samples were obtained. In fact, the average value of all the sample
means is exactly equal to the value of the population mean. This fact should be intuitively
reasonable; the sample means are expected to be close to the population mean, and they do
tend to pile up around μ. The formal statement of this phenomenon is that the mean of the
distribution of sample means always is identical to the population mean. This mean value
is called the expected value of M.