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

SECTION 4.4 | Measuring Standard Deviation and Variance for a Sample 115
The following example is an opportunity for you to test your understanding by computing
sample variance and standard deviation yourself.
EXAMPLE 4.7
For the following sample of n 5 5 scores, compute the variance and standard deviation: 1,
5, 5. 1, and 8. You should obtain s 2 5 9 and s 5 3. Good luck. ■
Remember that the formulas for sample variance and standard deviation were constructed
so that the sample variability would provide a good estimate of population variability.
For this reason, the sample variance is often called estimated population variance,
and the sample standard deviation is called estimated population standard deviation. When
you have only a sample to work with, the variance and standard deviation for the sample
provide the best possible estimates of the population variability.
■ Sample Variability and Degrees of Freedom
Although the concept of a deviation score and the calculation SS are almost exactly the
same for samples and populations, the minor differences in notation are really very important.
Specifically, with a population, you find the deviation for each score by measuring
its distance from the population mean. With a sample, on the other hand, the value of m is
unknown and you must measure distances from the sample mean. Because the value of the
sample mean varies from one sample to another, you must first compute the sample mean
before you can begin to compute deviations. However, calculating the value of M places a
restriction on the variability of the scores in the sample. This restriction is demonstrated in
the following example.
EXAMPLE 4.8
Suppose we select a sample of n 5 3 scores and compute a mean of M 5 5. The first two
scores in the sample have no restrictions; they are independent of each other and they can
have any values. For this demonstration, we will assume that we obtained X 5 2 for the
first score and X 5 9 for the second. At this point, however, the third score in the sample
is restricted.
X A sample of n 5 3 scores with a mean of M 5 5
2
9
— d What is the third score?
For this example, the third score must be X 5 4. The reason that the third score is
restricted to X 5 4 is that the entire sample of n 5 3 scores has a mean of M 5 5. For 3
scores to have a mean of 5, the scores must have a total of SX 5 15. Because the first two
scores add up to 11 (9 1 2), the third score must be X 5 4.
■
In Example 4.8, the first two out of three scores were free to have any values, but the
final score was dependent on the values chosen for the first two. In general, with a sample
of n scores, the first n 2 1 scores are free to vary, but the final score is restricted. As a result,
the sample is said to have n 2 1 degrees of freedom.
DEFINITION
For a sample of n scores, the degrees of freedom, or df, for the sample variance are
defined as df 5 n 2 1. The degrees of freedom determine the number of scores in
the sample that are independent and free to vary.