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204 Part IV: Quality Assurance
Finally, note that the data in the above example contains two values, 250 and
300 thousand dollars, that seem to be the sales of top-performing sales personnel.
These two large values may be considered as extreme values.
In this case, the mean of these data is given by
X – = (7 + 8 + 10 + 12 + 12 + 15 + 15 + 16 + 17 + 18 +
19 + 20 + 22 + 25 + 250 + 300)/16 = 47.875
Since the mean of 47.875 is so much larger than the median of 16.5 it is obvious that
the mean of the data has been adversely affected by the extreme values. Since in
this case the mean does not adequately represent the measure of centrality of the
data set, the median would more accurately identify where the center of the data
is located.
Furthermore, if we replace the extreme values of 250 and 300, for example,
with 25 and 30 respectively, then the median does not change whereas the mean
becomes $16,937. Thus, the new data obtained by replacing 250 and 300 with 25
and 30 respectively did not contain any extreme values. Therefore, the new mean
value is more consistent with the true average sales.
Mode. The mode of a data set is the value that occurs most frequently. Mode is
the least used measure of centrality. When products are produced via mass
production, for example, clothes of certain sizes, rods of certain lengths, and so
on, the modal value is of great interest. Note that in any data set there may be no
mode, or conversely, there may be multiple modes. We denote the mode of a data
set by M 0 .
Part IV.A.1
EXAMPLE 18.5
Elizabeth took five courses in a given semester with 5, 4, 3, 3, and 2 credit hours. The
grade points she earned in these courses at the end of the semester were 3.7, 4.0, 3.3,
3.7, and 4.0 respectively. Find her GPA for the semester.
Solution:
Note that in this example the data points 3.7, 4.0, 3.3, 3.7, and 4.0 have different weights
attached to them, that is, credit hours for each course. Thus, to find Elizabeth’s GPA
we can not simply find the arithmetic mean. Rather, in this case we shall find the mean
called the weighted mean, which is defined as:
X
w
wX
1 1+ wX
2 2
+ … + wX
n
=
w + w + … w
1 2
n
n
wX
i i
=
w
i
(18.3)
where w 1 , w 2 , …, w n are the weights attached to X 1 , X 2 , . . . , X n respectively. In this
example, the GPA is given by:
( )+ ( )+ ( )+ ( )+ ( )
537 . 440 . 333 . 337 . 240 .
X w
=
5+ 4+ 3+
3+
2
= 3. 735.