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

202 CHAPTER 7 | Probability and Samples: The Distribution of Sample Means
Once again, the symbol for the standard error is σ M
. The σ indicates that this value is
a standard deviation, and the subscript M indicates that it is the standard deviation for the
distribution of sample means. Similarly, it is common to use the symbol μ M
to represent the
mean of the distribution of sample means. However, μ M
is always equal to μ and our primary
interest in inferential statistics is to compare sample means (M) with their population
means (μ). Therefore, we simply use the symbol μ to refer to the mean of the distribution
of sample means.
The standard error is an extremely valuable measure because it specifies precisely
how well a sample mean estimates its population mean—that is, how much error you
should expect, on the average, between M and μ. Remember that one basic reason
for taking samples is to use the sample data to answer questions about the population.
However, you do not expect a sample to provide a perfectly accurate picture of
the population. There always is some discrepancy or error between a sample statistic
and the corresponding population parameter. Now we are able to calculate exactly
how much error to expect. For any sample size (n), we can compute the standard
error, which measures the average distance between a sample mean and the population
mean.
The magnitude of the standard error is determined by two factors: (1) the size of the
sample and (2) the standard deviation of the population from which the sample is selected.
We will examine each of these factors.
The Sample Size Earlier we predicted, based on commonsense, that the size of a
sample should influence how accurately the sample represents its population. Specifically,
a large sample should be more accurate than a small sample. In general, as the sample size
increases, the error between the sample mean and the population mean should decrease.
This rule is also known as the law of large numbers.
DEFINITION
The law of large numbers states that the larger the sample size (n), the more probable
it is that the sample mean will be close to the population mean.
The Population Standard Deviation As we noted earlier, there is an inverse relationship
between the sample size and the standard error: bigger samples have smaller
error, and smaller samples have bigger error. At the extreme, the smallest possible sample
(and the largest standard error) occurs when the sample consists of n = 1 score. At this
extreme, each sample is a single score and the distribution of sample means is identical to
the original distribution of scores. In this case, the standard deviation for the distribution
of sample means, which is the standard error, is identical to the standard deviation for the
distribution of scores. In other words, when n = 1, the standard error = σ M
is identical to
the standard deviation = σ.
When n = 1, σ M
= σ (standard error = standard deviation).
This formula is contained
in the central
limit theorem.
You can think of the standard deviation as the “starting point” for standard error. When
n = 1, the standard error and the standard deviation are the same: σ M
= σ. As sample size
increases beyond n = 1, the sample becomes a more accurate representative of the population,
and the standard error decreases. The formula for standard error expresses this relationship
between standard deviation and sample size (n).
standard error = s M
5 s Ïn
(7.1)