Page 111 - Modern Analytical Chemistry

1400-CH04 9/8/99 3:55 PM Page 94

94 Modern Analytical Chemistry

point. The test statistic, Q exp , is calculated using equation 4.23 if the suspected out-
lier is the smallest value (X 1 )
X - X 1
2
Q exp = 4.23
X n - X 1
or using equation 4.24 if the suspected outlier is the largest value (X n )
X n - X n-1
Q exp = 4.24
X n - X 1
where n is the number of members in the data set, including the suspected outlier.
It is important to note that equations 4.23 and 4.24 are valid only for the detection
of a single outlier. Other forms of Dixon’s Q-test allow its extension to the detection
10
of multiple outliers. The value of Q exp is compared with a critical value, Q(a, n), at
a significance level of a. The Q-test is usually applied as the more conservative two-
tailed test, even though the outlier is the smallest or largest value in the data set.
Values for Q(a, n) can be found in Appendix 1D. If Q exp is greater than Q(a, n),
then the null hypothesis is rejected and the outlier may be rejected. When Q exp is
less than or equal to Q(a, n) the suspected outlier must be retained.

4
EXAMPLE .22
The following masses, in grams, were recorded in an experiment to determine
the average mass of a U.S. penny.
3.067 3.049 3.039 2.514 3.048 3.079 3.094 3.109 3.102
Determine if the value of 2.514 g is an outlier at a= 0.05.

SOLUTION
To begin with, place the masses in order from smallest to largest
2.514 3.039 3.048 3.049 3.067 3.079 3.094 3.102 3.109

and calculate Q exp
X - X 1 . 3 039 -2 .514
2
Q exp = = = . 0 882
X - X 1 . 3 109 -2 .514
9
The critical value for Q(0.05, 9) is 0.493. Since Q exp > Q(0.05, 9) the value is
assumed to be an outlier, and can be rejected.

The Q-test should be applied with caution since there is a probability, equal to
a, that an outlier identified by the Q-test actually is not an outlier. In addition, the
Q-test should be avoided when rejecting an outlier leads to a precision that is un-
reasonably better than the expected precision determined by a propagation of un-
certainty. Given these two concerns it is not surprising that some statisticians cau-
tion against the removal of outliers. 11 On the other hand, testing for outliers can
provide useful information if we try to understand the source of the suspected out-
lier. For example, the outlier identified in Example 4.22 represents a significant
change in the mass of a penny (an approximately 17% decrease in mass), due to a
change in the composition of the U.S. penny. In 1982, the composition of a U.S.
penny was changed from a brass alloy consisting of 95% w/w Cu and 5% w/w Zn, to
a zinc core covered with copper. 12 The pennies in Example 4.22 were therefore
drawn from different populations.