Graph a probability distribution for the mean of a discrete variable

Describe a sampling distribution in terms of "all possible outcomes"

Describe a sampling distribution in terms of repeated sampling

Describe the role of sampling distributions in inferential statistics

Define the standard error of the mean

Suppose you randomly sampled 10 people from the
population of women in Houston, Texas, between the ages of 21 and
35 years and computed the mean height of your sample. You would
not expect your sample mean to be equal to the mean of all women
in Houston. It might be somewhat lower or it might be somewhat
higher, but it would not equal the population mean exactly. Similarly,
if you took a second sample of 10 people from the same population,
you would not expect the mean of this second sample to equal the
mean of the first sample.

Recall that inferential statistics concern generalizing from a sample to a population. A critical part of inferential statistics involves
determining how far sample statistics are likely to vary from
each other and from the population parameter. (In this example, the sample statistics are the sample means
and the population parameter is the population mean.) As the later
portions of this chapter show, these determinations are based
on sampling distributions.

Discrete Distributions

We will illustrate the concept of sampling distributions
with a simple example. Figure 1 shows three pool balls, each with
a number on it. Two of the balls are selected randomly (with replacement)
and the average of their numbers is computed.

Figure 1. The pool balls.

All possible outcomes are shown below in Table
1.

Table 1. All possible outcomes when two
balls are sampled with replacement.

Outcome

Ball 1

Ball 2

Mean

1

1

1

1.0

2

1

2

1.5

3

1

3

2.0

4

2

1

1.5

5

2

2

2.0

6

2

3

2.5

7

3

1

2.0

8

3

2

2.5

9

3

3

3.0

Notice that all the means are either 1.0, 1.5,
2.0, 2.5, or 3.0. The frequencies of these means are shown in
Table 2. The relative frequencies are equal to the frequencies
divided by nine because there are nine possible outcomes.

Table 2. Frequencies of means for N = 2.

Mean

Frequency

Relative Frequency

1.0

1

0.111

1.5

2

0.222

2.0

3

0.333

2.5

2

0.222

3.0

1

0.111

Figure 2 shows a relative frequency distribution of the means based on Table 2.
This distribution is also a probability distribution since the Y-axis is the probability
of obtaining a given mean from a sample of two balls in addition
to being the relative frequency.

Figure 2. Distribution of means for N
= 2.

The distribution shown in Figure 2 is called the sampling distribution of the mean. Specifically, it is the sampling distribution of the mean for a sample size of 2 (N = 2). For this simple example, the distribution of pool balls and the sampling distribution are both discrete distributions. The pool balls have only the values 1, 2, and 3, and a sample mean can have one of only five values shown in Table 2.

There is an alternative way of conceptualizing a
sampling distribution that will be useful for more complex distributions.
Imagine that two balls are sampled (with replacement) and the
mean of the two balls is computed and recorded. Then this process
is repeated for a second sample, a third sample, and eventually
thousands of samples. After thousands of samples are taken and
the mean computed for each, a relative frequency distribution
is drawn. The more samples, the closer the relative frequency
distribution will come to the sampling distribution shown in Figure
2. As the number of samples approaches infinity, the relative frequency
distribution will approach the sampling distribution. This means
that you can conceive of a sampling distribution as being a relative frequency
distribution based on a very large number of samples. To be strictly correct, the relative frequency distribution approaches the sampling distribution as the number of samples approaches infinity.

It is important to keep in mind that every statistic,
not just the mean, has a sampling distribution. For example, Table
3 shows all possible outcomes for the range of two numbers (larger
number minus the smaller number). Table 4 shows the frequencies
for each of the possible ranges and Figure 3 shows the sampling
distribution of the range.

Table 3. All possible outcomes when two balls are sampled with replacement.

Outcome

Ball 1

Ball 2

Range

1

1

1

0

2

1

2

1

3

1

3

2

4

2

1

1

5

2

2

0

6

2

3

1

7

3

1

2

8

3

2

1

9

3

3

0

Table 4. Frequencies of ranges for N = 2.

Range

Frequency

Relative Frequency

0

3

0.333

1

4

0.444

2

2

0.222

Figure 3. Distribution of ranges for N
= 2.

It is also important to keep in mind that there is a sampling
distribution for various sample sizes. For simplicity, we have
been using N = 2. The sampling distribution of the range for N
= 3 is shown in Figure 4.

Figure 4. Distribution of ranges for N
= 3.

Continuous Distributions

In the previous section, the population consisted
of three pool balls. Now we will consider sampling distributions
when the population distribution is continuous. What if we had
a thousand pool balls with numbers ranging from 0.001 to 1.000
in equal steps? (Although this distribution is not really continuous,
it is close enough to be considered continuous for practical purposes.)
As before, we are interested in the distribution of means we would
get if we sampled two balls and computed the mean of these two balls.
In the previous example, we started by computing the mean for
each of the nine possible outcomes. This would get a bit tedious
for this example since there are 1,000,000 possible outcomes (1,000
for the first ball x 1,000 for the second). Therefore, it is more
convenient to use our second conceptualization of sampling distributions
which conceives of sampling distributions in terms of relative
frequency distributions. Specifically, the relative frequency
distribution that would occur if samples of two balls were repeatedly
taken and the mean of each sample computed.

When we have a truly continuous distribution, it
is not only impractical but actually impossible to enumerate all
possible outcomes. Moreover, in continuous distributions, the
probability of obtaining any single value is zero. Therefore,
as discussed in the section "Introduction
to Distributions," these values are called probability
densities rather than probabilities.

Sampling Distributions and Inferential Statistics

As we stated in the beginning of this chapter,
sampling distributions are important for inferential statistics.
In the examples given so far, a population was specified and the
sampling distribution of the mean and the range were determined.
In practice, the process proceeds the other way: you collect sample
data and from these data you estimate parameters of the sampling
distribution. This knowledge of the sampling distribution can
be very useful. For example, knowing the degree to which means
from different samples would differ from each other and from the
population mean would give you a sense of how close your particular
sample mean is likely to be to the population mean. Fortunately,
this information is directly available from a sampling distribution.
The most common measure of how much sample means differ from each
other is the standard deviation of the sampling distribution of
the mean. This standard deviation is called the standard
error of the mean. If all the sample means were very close
to the population mean, then the standard error of the mean would
be small. On the other hand, if the sample means varied considerably,
then the standard error of the mean would be large.

To be specific, assume your sample mean were 125
and you estimated that the standard error of the mean were 5 (using
a method shown in a later section). If you had a normal distribution,
then it would be likely that your sample mean would be within
10 units of the population mean since most of a normal distribution
is within two standard deviations of the mean.

Keep in mind that all statistics have sampling distributions,
not just the mean. In later sections we will be discussing the
sampling distribution of the variance, the sampling distribution
of the difference between means, and the sampling distribution
of Pearson's correlation, among others.