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8.2. THE MEASURE OF ROUGHNESS 139
Interestingly, we all know the device that measures the opposite: if numbers
are not spread out, they are all close to average. In mathematics we tend to
use mean and average interchangeably, and, of course,
sum of the numbers
average = mean =
number of numbers .
Since each of the sets in our example contain 4 numbers that sum to 12, each
has a mean or average of 3. This gives us an idea of how to measure roughness:
a set of numbers is rougher when it contains many numbers that are far from
the mean. So we might try to define roughness by “sum how far the numbers
are from their mean.” How far implies distance which implies subtraction. But
it also implies that 5 and 1 are both a distance of 2 from 3, no matter that
one is larger than 3 and one is smaller. So we cannot use simple subtraction to
measure distance (5 − 3 = +2 but 1 − 3 = −2) but would have to use absolute
value ((|5−3| = +2 and |1−3| = +2). Unfortunately, as a mathematical device,
absolute value is somewhat of a pain to work with.
What is traditionally chosen instead is to square the result (as the square
of a number is positive). This gives us our second attempt at a definition
for roughness: “the sum of the squares of the distances from the mean.” For
example, the middle set of numbers then would have roughness
(4 − 3) 2 + (0 − 3) 2 + (4 − 3) 2 + (4 − 3) 2 = 1 1 + 3 2 + 1 2 + 1 2 = 1 + 9 + 1 + 1 = 11,
while the final set would have roughness 18. (Check this!)
While this agrees with our intuition that the final set is rougher than the
middle one, the definition is not quite right yet. The sets {103, 103, 103, 103},
{104, 100, 104, 104} and {101, 104, 101, 106} created by adding 100 to our sets
have 103 as their means and also would have roughness 0, 11 and 18. But it
ought to be that an error of 2 is more forgivable when the target, the mean, is
103 than when it is 3. So we must adjust for the sum. Since we are squaring
the numbers, it makes sense to take the total sum into account by dividing by
its square.
We thus define what Sinkov calls the Measure of Roughness or M.R. 4
to be
M.R. =
sum of the squares of the distances to the mean
.
square of the sum of the numbers
The first set has roughness 0, since all of its numbers, and so its mean, are 3’s.
For the second we compute:
M.R. of {4, 0, 4, 4} = (4 − 3)2 + (0 − 3) 2 + (4 − 3) 2 + (4 − 3) 2
12 2
= 1 + 9 + 1 + 1
144
= .076.
Check that you understand this by computing M.R. for the third set. 5
4 For those with some statistics background, M.R. is very closely related to variance.
5 .125.
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