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

196 CHAPTER 7 | Probability and Samples: The Distribution of Sample Means
set contains exactly 100 samples, then the probability of obtaining any specific sample is 1
out of 100: p = 1
100 (Box 7.1).
Also, you should notice that the distribution of sample means is different from distributions
we have considered before. Until now we always have discussed distributions
of scores; now the values in the distribution are not scores, but statistics (sample means).
Because statistics are obtained from samples, a distribution of statistics is referred to as a
sampling distribution.
DEFINITION
A sampling distribution is a distribution of statistics obtained by selecting all the
possible samples of a specific size from a population.
Thus, the distribution of sample means is an example of a sampling distribution. In fact,
it often is called the sampling distribution of M.
If you actually wanted to construct the distribution of sample means, you would first
select a random sample of a specific size (n) from a population, calculate the sample mean,
and place the sample mean in a frequency distribution. Then you select another random
sample with the same number of scores. Again, you calculate the sample mean and add it
to your distribution. You continue selecting samples and calculating means, over and over,
until you have the complete set of all the possible random samples. At this point, your frequency
distribution will show the distribution of sample means.
BOX 7.1 Probability and the Distribution of Sample Means
I have a bad habit of losing playing cards. This habit
is compounded by the fact that I always save the old
deck in the hope that someday I will find the missing
cards. As a result, I have a drawer filled with partial
decks of playing cards. Suppose that I take one of
these almost-complete decks, shuffle the cards carefully,
and then randomly select one card. What is the
probability that I will draw a king?
You should realize that it is impossible to answer
this probability question. To find the probability of
selecting a king, you must know how many cards
are in the deck and exactly which cards are missing.
(It is crucial that you know whether or not any kings
are missing.) The point of this simple example is
that any probability question requires that you have
complete information about the population from
which the sample is being selected. In this case,
you must know all the possible cards in the deck
before you can find the probability for selecting any
specific card.
In this chapter, we are examining probability and
sample means. To find the probability for any specific
sample mean, you first must know all the possible
sample means. Therefore, we begin by defining and
describing the set of all possible sample means that
can be obtained from a particular population. Once
we have specified the complete set of all possible
sample means (i.e., the distribution of sample means),
we will be able to find the probability of selecting any
specific sample means.
■ Characteristics of the Distribution of Sample Means
We demonstrate the process of constructing a distribution of sample means in Example 7.1,
but first we use common sense and a little logic to predict the general characteristics of the
distribution.
1. The sample means should pile up around the population mean. Samples are
not expected to be perfect but they are representative of the population.