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 203
Note that the formula satisfies all of the requirements for the standard error. Specifically,
a. As sample size (n) increases, the size of the standard error decreases. (Larger
samples are more accurate.)
b. When the sample consists of a single score (n = 1), the standard error is the same
as the standard deviation (σ M
= σ).
In Equation 7.1 and in most of the preceding discussion, we have defined standard error
in terms of the population standard deviation. However, the population standard deviation
(σ) and the population variance (σ 2 ) are directly related, and it is easy to substitute variance
into the equation for standard error. Using the simple equality σ = Ïs 2 , the equation for
standard error can be rewritten as follows:
standard error = s M
5 Î s Ïn 5 s2
n 5 s2
n
(7.2)
Throughout the rest of this chapter (and in Chapter 8), we will continue to define standard
error in terms of the standard deviation (Equation 7.1). However, in later chapters
(starting in Chapter 9) the formula based on variance (Equation 7.2) will become more
useful.
Figure 7.3 illustrates the general relationship between standard error and sample
size. (The calculations for the data points in Figure 7.3 are presented in Table 7.2.)
Again, the basic concept is that the larger a sample is, the more accurately it represents
its population. Also note that the standard error decreases in relation to the square root
of the sample size. As a result, researchers can substantially reduce error by increasing
sample size up to around n = 30. However, increasing sample size beyond n = 30
does not produce much additional improvement in how well the sample represents the
population.
The following example is an opportunity for you to test your understanding of the standard
error by computing it for yourself.
Standard Error
(based on s 5 10)
Standard distance
between a sample
mean and
the population
mean
10
9
8
7
6
5
4
3
2
1
0
1
4 9 16 25 36 49 64 100
Number of scores in the sample (n)
FIGURE 7.3
The relationship between standard error and sample size. As the sample size is increased, there is less error between the
sample mean and the population mean.