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80 Modern Analytical Chemistry
degrees of freedom
The number of independent values on population and the sample, respectively. In the case of a population, the degrees of
which a result is based (n).
Degrees of Freedom Unlike the population’s variance, the variance of a sample in-
cludes the term n – 1 in the denominator, where n is the size of the sample
n 2
å ( X i - X)
2
=
s = i 1 4.12
n - 1
Defining the sample’s variance with a denominator of n, as in the case of the popu-
2
lation’s variance leads to a biased estimation of s . The denominators of the vari-
ance equations 4.8 and 4.12 are commonly called the degrees of freedom for the
freedom is always equal to the total number of members, n, in the population. For
–
the sample’s variance, however, substituting X for m removes a degree of freedom
from the calculation. That is, if there are n members in the sample, the value of the
–
th
n member can always be deduced from the remaining n – 1 members and X. For
example, if we have a sample with five members, and we know that four of the
members are 1, 2, 3, and 4, and that the mean is 3, then the fifth member of the
sample must be
–
(X ´n)– X 1 – X 2 – X 3 – X 4 =(3 ´5)–1–2–3–4=5
4 5 Confidence Intervals for Samples
D.
Earlier we introduced the confidence interval as a way to report the most probable
value for a population’s mean, m, when the population’s standard deviation, s, is
2
2
known. Since s is an unbiased estimator of s , it should be possible to construct
confidence intervals for samples by replacing s in equations 4.10 and 4.11 with s.
2
Two complications arise, however. The first is that we cannot define s for a single
member of a population. Consequently, equation 4.10 cannot be extended to situa-
2
2
tions in which s is used as an estimator of s . In other words, when sis unknown,
we cannot construct a confidence interval for mby sampling only a single member
of the population.
The second complication is that the values of z shown in Table 4.11 are derived
2
2
2
for a normal distribution curve that is a function of s , not s . Although s is an un-
2
2
biased estimator of s , the value of s for any randomly selected sample may differ
2
2
significantly from s . To account for the uncertainty in estimating s , the term z in
equation 4.11 is replaced with the variable t, where t is defined such that t ³z at all
confidence levels. Thus, equation 4.11 becomes
ts
m=X ± 4.13
n
Values for t at the 95% confidence level are shown in Table 4.14. Note that t be-
comes smaller as the number of the samples (or degrees of freedom) increase, ap-
proaching z as n approaches infinity. Additional values of t for other confidence lev-
els can be found in Appendix 1B.
4 5
EXAMPLE .1
What is the 95% confidence interval for the data in Table 4.1?
SOLUTION
The mean and standard deviation for this sample are, respectively, 3.117 g and
0.051 g. Since the sample consists of seven measurements, there are six degrees