Introduction to Sampling Distributions
- Define inferential statistics
- 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.
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 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.