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

SECTION 7.2 | The Distribution of Sample Means for any Population and any Sample Size 201
In commonsense terms, a sample mean is “expected” to be near its population mean.
When all of the possible sample means are obtained, the average value is identical to μ.
The fact that the average value of M is equal to μ was first introduced in Chapter 4
(page 120) in the context of biased versus unbiased statistics. The sample mean is an
example of an unbiased statistic, which means that on average the sample statistic produces
a value that is exactly equal to the corresponding population parameter. In this case, the
average value of all the sample means is exactly equal to μ.
DEFINITION
The mean of the distribution of sample means is equal to the mean of the population
of scores, μ, and is called the expected value of M.
■ The Standard Error of M
So far, we have considered the shape and the central tendency of the distribution of sample
means. To completely describe this distribution, we need one more characteristic, variability.
The value we will be working with is the standard deviation for the distribution
of sample means. This standard deviation is identified by the symbol σ M
and is called the
standard error of M.
When the standard deviation was first introduced in Chapter 4, we noted that this measure
of variability serves two general purposes. First, the standard deviation describes the
distribution by telling whether the individual scores are clustered close together or scattered
over a wide range. Second, the standard deviation measures how well any individual
score represents the population by providing a measure of how much distance is reasonable
to expect between a score and the population mean. The standard error serves the same two
purposes for the distribution of sample means.
1. The standard error describes the distribution of sample means. It provides a measure
of how much difference is expected from one sample to another. When the
standard error is small, all the sample means are close together and have similar
values. If the standard error is large, the sample means are scattered over a wide
range and there are big differences from one sample to another.
2. Standard error measures how well an individual sample mean represents the entire
distribution. Specifically, it provides a measure of how much distance is reasonable
to expect between a sample mean and the overall mean for the distribution of
sample means. However, because the overall mean is equal to μ, the standard error
also provides a measure of how much distance to expect between a sample mean
(M) and the population mean, μ.
Remember that a sample is not expected to provide a perfectly accurate reflection of
its population. Although a sample mean should be representative of the population mean,
there typically is some error between the sample and the population. The standard error
measures exactly how much difference is expected on average between a sample mean, M
and the population mean, μ.
DEFINITION
The standard deviation of the distribution of sample means, σ M
, is called the
standard error of M. The standard error provides a measure of how much distance
is expected on average between a sample mean (M) and the population mean (μ).