Page 86 - Becoming Metric Wise

76 Becoming Metric-Wise

their harmonic mean is the Dice coefficient. For a proof and a definition
of these terms we refer the reader to Egghe and Rousseau (2006b).

4.3.3 The Median
The median of a sequence of observations is determined as follows: Rank all
observations from smallest to largest. If the number of observations is an odd
number, then the one in the middle is the median. If the number of observa-
tions is even, then the median is the average of the two middle observations.
For example:
The median of 10, 12, 13, 20, 30 is 13, while the median of 10, 12,
13, 20, 30, 50 is 16.5.
The average can be heavily influenced by a few outliers. Hence the
median provides a more robust alternative. For example, the average of
(1,3,5,10,21) is 8 (influenced by 21 which is an outlier for this set of
numbers); its median is 5.
If values are symmetric with respect to the average, then the median
coincides with the average.
With regard to the median, we include some food for thought or
challenges for the reader. These are included to illustrate that even simple
notions may have unexpected outcomes. This serves as a warning that the
more complicated research indicators discussed further on do not always
lead to intuitively obvious results.
Is it possible that for a finite sequence of numbers average and median
coincide, but that the sequence is not symmetric with respect to the aver-
age? The answer is yes: try to find an example yourself.
A sequence of numbers is given and its median is determined. Now add
a number to this sequence (not to each number of the sequence) which is
strictly larger than the median. Is it possible that the median of this new
sequence stays the same?
Yes. Consider the sequence (1,0,3,2,2). Its median is 2. Now we add
the number 10, leading to the sequence (1,0,3,2,2,10). Its median is still 2.
Another question. A sequence of numbers is given and its median is
determined. Now add to this sequence a number which is strictly larger
than its mean. Is it possible that the median of this new sequence is smal-
ler than the median of the original sequence?
Yes. Consider the sequence (10,10,10,0,0). Its median is 10 and its
mean is 6. We add the number 8, leading to (10,10,10,0,0,8). The new
median is 9.