Partitioning of Data - Spatial Mathematics: Theory and Practice through Mapping - page 131

6.2.5 Standard deviations
When the data set is ranged using the standard deviations method, the soft-
ware finds the mean value and then places range breaks above and below
the mean at intervals of (usually) either 1/4, 1/3, 1/2, or 1 standard deviation
until all the data values are contained within these ranges. The software
should aggregate any values that are beyond three standard deviations from
the mean into two classes, greater than three standard deviations above the
mean (“ > 3 Std Dev.”) and less than three standard deviations below the mean
(“ <- 3 Std. Dev.”). Figure 6.5 shows the world's land areas grouped according
to this ranging method. In an effort to reinforce interactive thinking between
maps and math, readers familiar with calculus might reasonably ask what, if
any, role the Mean Value Theorem or the Intermediate Value Theorem might
play in the classification method described here.
Merits and limitations of the different ranging methods to partition raw data
are summarized ( Table 6.1 ) . What is critical is to understand that the choice
of partition can produce vastly different analysis and interpretation. Thus, it
is important in research to be clear why one ranging method was chosen over
another. Indeed, it may often be prudent to display data mapped using several
different ranging methods.
Which of the maps in Figures 6.1 through 6.5 is “best”? In looking at the
merits and limitations of each of this set of commonly used ranging methods,
we see that the pattern of the underlying data may help us to answer this
question. The answers that come from the data should make good intuitive
sense when looking at the maps, themselves, based on our general knowl-
edge. Figure 6.6 shows the underlying attribute table of the map opened
in Microsoft Excel and graphed as a simple line chart in Excel. (Or, one can
employ the onboard classification histogram embedded in ArcGIS software
as is done in one of the activities near the end of this chapter.) Clearly, the
red line of the chart is not linear. Thus, one would expect that the quantile
method of ranging of data would not be appropriate. That idea fits with the
appearance of the map in Figure 6.2 . That map conveys very little informa-
tion. Thus, we would reject quantiles as a mapping classification method for
this set of data.
Similarly, the Geometrical Interval method, while it makes more distinction
among data entries than does the quantile method, it still does not show clear
distinctions among the four classes. The entries appear bunched into two of
the four classes. Partition by Equal Interval makes reasonable distinctions
for the larger countries, but mid-sized and small countries appear grouped
together in a single class. Partition by Standard Deviation is skewed, as is
Equal Interval, to have more distinctions made among larger countries; how-
ever, the degree of skew is not as severe in the Standard Deviation partition
as in the Equal Interval partition. Thus, of this set of maps, we would likely
choose either Natural Breaks or Standard Deviation, depending on the appli-
cation, as a method for partitioning this data set.