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Summary: We consider how to use Scheme and GIMP to
visualize statistical data.
Contents:
We have explored GIMP primarily as a platform for making visually
interesting images (calling them art would probably be a stretch).
However, it is also possible to use the GIMP (or any graphics platform)
to make informative images. In particular, many statisticians
have found that complex data sets are easier to understand when
the data are represented visually in addition to textually.
For example, in a data set with one independent variable (say, the
size of an input for an algorithm) and one dependent variable (say, the
number of calls to cons that the algorithm makes on
one input of that size), we might plot a graph to better understand
that relationship. If the points form a straight, but not horizontal, line,
we might conclude that the relationship is linear. If the points form a
more complex shape, we need to do more analysis.
One particularly challenging aspect of data visualization is what to
do with multivariate data, data in which we have more than
two variables, and in which it is not necessarily clear which variables
are dependent and which variables are independent. In such situations,
one needs to explore how best to represent the various variables for
each observation.
For example, consider the wealth of information about countries of
the world available at
http://devdata.worldbank.org/data-query/. There are
a few hundred countries represented, fifty four possible attributes
(amount of agricultural land, population, life expectancy, per-capita
income, etc.), and six years of data (2000 to 2005, inclusive).
For these data, someone might be interested in exploring relationships that
involve population, life expectancy, per-capita income, and year.
How can we represent these data to best explore these relationships?
Before we delve into the particular techniques one might use to visualize
data, let us consider first how statisticians tend to characterize data.
The broadest characterization that statisticians make is between categorical
data and numerical data.
Categorical data are data that have been broken up into
categories, and in which the categories have no numerical relationship,
other than set-theoretic relationship. For example, we would call
gender (male or female) a categorical variable and (except for some
outliers that we'll ignore for now), the relationship between the two
is that anyone who is not a woman is a man, and vice versa. For the
World Bank data described in the introduction, one obvious categorical
variable is the continent to which the country belongs.
Quantitative data are data that are represented numerically, and
that often therefore have a numeric relationship. For example, for
numeric grades, we might say that a grade of 100 is twice as large as
a grade of 50. For the World Bank data described in the introduction,
most variables are quantitative.
While it is not generally possible to represent categorical data
quantitatively (except, of course, in terms of the number of data
points that fall in each category), it is straightforward to
represent quantitative data categorically. For example, we might
separate grades into those below 60 (failing) and those above 60 (passing).
Sometimes, we will categorize less broadly (e.g., A, B, C, D, and F).
At times, we may enumerate categories (e.g., Male is category 0, Female is
category 1, No Answer is category 2, and Other is category 3). However, such
enumeration should never be taken to mean that there is any real
quantitative aspect to the data. For example, we would never average
genders (nor continents nor ...).
In general, we think about representing each data point with a single
symbol. We map different attributes of the point to different attributes
of the symbol.
For categorical attributes, we can associate a color, shape, or pattern
with each category. For a student, we might use one shape (e.g., a
circle) for humanities majors, another (e.g., a square) for science
majors, and a third (e.g., a triangle) for social studies majors.
We might then use color to represent the dormitory in which the student
resides (e.g., red for North Campus, yellow for South Campus, blue for
East Campus, and purple for off campus). Experienced designers pay attention
to those who are color blind or color deficient in choosing colors, but
we leave that topic for another course.
For quantitative attributes, we might map the number to the size of the
symbol, the shade of the color (lower numbers lighter, higher numbers
darker), the weight of the border of the shape, the x position of the
center, or the y position of the center.
In the corresponding laboratory, you will have a chance to explore
some of these variations.
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