This month's column focuses on the nature of quantitative information. Graphs display quantitative information: numbers that measure performance, predict the future and identify opportunities. The nature of quantitative information varies in some fundamental ways that tie directly to some of the choices you must make when graphing that information.
Quantitative and Categorical Data
Quantitative information consists not only of numbers, but also of data that identifies what the numbers mean. If I walked up to you, looked you in the eye, and said, "The answer is 24,901," you would probably be confused, understandably suspicious that I had a few screws loose. By itself, a number means nothing. However, if I were to tell you that the circumference of the earth at the equator is 24,901 miles, that would mean something. To be complete and meaningful, quantitative information consists of both quantitative data - the numbers - and categorical data - the labels that tell us what the numbers measure. The graph in Figure 1 highlights this distinction by displaying the categorical data in black and the quantitative data in various other colors. (Note: The gray axes, excluding the red tick marks, are neither quantitative nor categorical data, and in fact are not data at all, but simply visual objects that support the graph by defining the plot area.)
Figure 1: Illustration of the difference between quantitative data (red) and categorical data (black).
The graph in Figure 1 displays two scales: a quantitative scale along the vertical axis and a categorical scale along the horizontal axis. They differ in what they identify: quantitative values on the one hand and categorical items on the other. Most two-dimensional graphs consist of one quantitative scale and one categorical scale, although a familiar exception is the scatterplot, which has quantitative scales along both axes (see Figure 2). In a line graph, the categorical scale always appears on the horizontal axis. In a bar graph, the categorical scale can appear on either axis, with bars running horizontally or vertically. Data points - simple symbols such as dots, squares, triangles and so forth - are rarely used by themselves to encode values other than in scatterplots. Unlike bars and lines, data points can encode a quantitative value simultaneously along two scales.
Figure 2: A scatterplot is the only commonly used 2-D graph that lacks a categorical scale along one of its two axes.
Three Types of Categorical Scales
When used in graphs, categorical scales come in three fundamental types: nominal, ordinal and interval.
Nominal scales consist of discrete items that belong to a common category, but really don't relate to one another in any particular way. They differ in name only (that is, nominally). The items in a nominal scale, in and of themselves, have no particular order and don't represent quantitative values. Typical examples include regions (e.g., The Americas, Asia and Europe) and departments (e.g., sales, marketing and finance).
Ordinal scales consist of items that have an intrinsic order, but like a nominal scale, the items in and of themselves do not represent quantitative values. Typical examples involve rankings, such as "A, B and C," "small, medium and large," and "poor, below average, average, above average and excellent."
Figure 3: Examples of nominal, ordinal and interval scales.
Interval scales also consist of items that have an intrinsic order; but in this case, they represent quantitative values as well. An interval scale starts out as a quantitative scale that is then converted into a categorical scale by subdividing the range of values into a sequential series of smaller ranges of equal size and by giving each range a label. Consider the quantitative range that appears along the vertical scale in Figure 2. This range, from 55 to 80, could be converted into a categorical scale consisting of the following smaller ranges:
- > 55 and 3/4 60,
- > 60 and 3/4 65,
- > 65 and 3/4 70,
- > 70 and 3/4 75, and
- > 75 and 3/4 80.
Here's a quick (and somewhat sneaky) test to see how well you've grasped these concepts. Can you identify the type of categorical scale that appears in Figure 4?
Figure 4: Example of a categorical scale that is commonly used in graphs. Can you determine which kind it is?
Months of the year obviously have an intrinsic order, which leaves the question: "Do the items correspond to quantitative values?" In fact, they do. Units of time such as years, quarters, months, weeks, days and hours are measures of quantity, and the individual items in any given unit of measure (e.g., years) represent equal intervals. Actually, months aren't exactly equal and even years vary in size occasionally due to leap years, but they are close enough in size to constitute an interval scale.