Biostatistics

40 CHAPTER 2 DESCRIPTIVE STATISTICS
Properties of the Mean The arithmetic mean possesses certain properties, some
desirable and some not so desirable. These properties include the following:
1. Uniqueness. For a given set of data there is one and only one arithmetic mean.
2. Simplicity. The arithmetic mean is easily understood and easy to compute.
3. Since each and every value in a set of data enters into the computation of the mean, it
is affected by each value. Extreme values, therefore, have an influence on the mean
and, in some cases, can so distort it that it becomes undesirable as a measure of
central tendency.
As an example of how extreme values may affect the mean, consider the following
situation. Suppose the five physicians who practice in an area are surveyed to determine
their charges for a certain procedure. Assume that they report these charges: $75, $75, $80,
$80, and $280. The mean charge for the five physicians is found to be $118, a value that is
not very representative of the set of data as a whole. The single atypical value had the effect
of inflating the mean.
Median The median of a finite set of values is that value which divides the set into
two equal parts such that the number of values equal to or greater than the median is
equal to the number of values equal to or less than the median. If the number of values is
odd, the median will be the middle value when all values have been arranged in order of
magnitude. When the number of values is even, there is no single middle value. Instead
there are two middle values. In this case the median is taken to be the mean of these two
middle values, when all values have been arranged in the order of their magnitudes. In
other words, the median observation of a data set is the ðn þ 1Þ=2th one when the
observation have been ordered. If, for example, we have 11 observations, the median is
the ð11 þ 1Þ=2 ¼ 6th ordered observation. If we have 12 observations the median is the
ð12 þ 1Þ=2 ¼ 6:5th ordered observation and is a value halfway between the 6th and 7th
ordered observations.
EXAMPLE 2.4.3
Let us illustrate by finding the median of the data in Table 2.2.1.
Solution:
The values are already ordered so we need only to find the two middle values.
The middle value is the ðn þ 1Þ=2 ¼ ð189 þ 1Þ=2 ¼ 190=2 ¼ 95th one.
Counting from the smallest up to the 95th value we see that it is 54.
Thus the median age of the 189 subjects is 54 years.
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EXAMPLE 2.4.4
We wish to find the median age of the subjects represented in the sample described in
Example 2.4.2.
Solution: Arraying the 10 ages in order of magnitude from smallest to largest gives 38,
43, 50, 57, 57, 59, 61, 64, 65, 66. Since we have an even number of ages, there