GRAPHS AND CHARTS
When looking at relationships between variables, the formula tells it all. Very often, however, a picture is indeed worth a thousand words in “giving us a feeling” for what the formula is telling us.
We will often be presented with a graph that we’ll study to gain some insight into the information the graph is presenting. Conversely, we will often need to be able to create a graph to show a formula that we are interested in. I’ll take this latter approach first.
Let’s start with a simple formula:
у = 27,000 – 2,000x.
This formula gives us a value for the variable у when we give it a value for the variable x. These variables might stand for the depreciation of a car’s value, the interest on a loan, the number of years that you will hold a loan, and so on.
Before I draw a graph, I have to know what values I will have to consider. Suppose that this formula tells us the value of a car over time: x will be the age of the car in years. The lowest value that x can possibly have is 0—that’s the year when the car is brand new. There’s no mathematical reason why x can’i be less than 0 (negative numbers), but it doesn’t make sense when I want x to stand for the age of the car. Let’s say I only want to look at the first 10 years of the car’s life. The largest value x can have is 10.
Next, I’ll create a table showing у for various values of x. If possible, I recommend that you create this or a similar table on a computer spreadsheet. Since I don’t know what spreadsheet program you’re using, I can’i give detailed directions for creating a graph from this table—but every spreadsheet program I’ve seen for the past 25 years has had graphing capability, so consult your manuals or do some searching online.
Table 1.4 Data for the First Graph Example
Figure 1.1 Graph of the data in Table 1.4.
As is seen in Table 1.4 and Figure 1.1, it’s conventional to show the units for a variable in parenthesis after the variable, for example, X (years) in Figure 1.1,
Table 1.4 shows the data generated using the above formula, for even number values of x between 0 and 10. Why even number values? Here I have a choice. I can get very detailed (x = 0, 0.2, 0.4, 0.6, … , 9.8, 10) or I can get very sparse (x = 0, 5, 10). There is no magic answer for what is the best thing to do. This is, unfortunately, as much an art as a science. You want enough detail so that the graph conveys all the details of the data, but you don’t want to bury yourself (or your computer) in reams of numbers.
Figure 1.1 shows the graph for the data in Table 1.4. The horizontal line along the bottom (the horizontal axis) has the label “X” beneath it and shows numbered points between 0 and 10. The vertical line along the right (the vertical axis) has the label “Y” to the left of it and shows numbered points between 0 and 30,000.
Each diamond corresponds to a row in the table. If you draw a vertical line up from the number 4 on the horizontal axis and draw a horizontal line to the right from the number 18,000 on the vertical axis, there is a small diamond at the point where these lines cross.
Figure 1.2 Continuous interpolation graph of data in Table 1.4.
Looking at the graph can quickly give you an idea of what’s happening: the car value is dropping rapidly with passing years.
What about, say, X = 5? There is no diamond on the graph. I could have included X = 5 and the corresponding Y value in the table. Alternatively, once I’m sure that there are no “surprises” in the curve, I can draw a continuous curve rather than a discrete point, as is shown in Figure 1.2.
In Figure 1.2, I’ve taken the same data that I used for Figure 1.1 but I connected all the diamonds (and I’m not showing the diamonds). This is a very convenient way to show things—when you ’ re very sure you know what’ s happening. Some formulas do funny things. Fortunately, we’ll only be looking at fairly “well-behaved” formulas in the upcoming chapters, so there’s no reason to dwell on mathematical curiosities. Connecting the dots between data points on a graph is called interpolation. Connecting these data points can sometimes lead to erroneous conclusions. Suppose the horizontal axis represents a number of apples and the vertical axis represents the cost of this number of apples in a store. The data might not be obvious because the store owner is giving discounts on large purchases. Connecting the points with a continuous line might give the impression that because you can estimate the cost of 1.5 apples, you can go into the store and buy 1.5 apples. In other words, connecting the points might give an incorrect impression that any value on the horizontal axis is possible.
When a graph describes how some variable such as the car value in the example above changes due to the change of another variable (the year in the example above), the graph is often entitled either “car values as a function of time,” “car values versus time,” or “car values with respect to time.” This is just a shorthand jargon to tell you to expect a graph with car values on the vertical axis and time on the horizontal axis.
Graphs are especially useful for comparing two or more sets of data, that is, data that came from tables with three or more columns. The first column almost invariably becomes the horizontal axis on a graph.
Table 1.5 Data for the Second Graph Example
Figure 1.3 Graph of the data in Table 1.5.
Consider Table 1.5 and Figure 1.3 . The table has three columns, labeled X, Y, and Z. Again considering car depreciation, Y and Z could represent the values of two different brands of cars, while X still represents the age of the car in years. In the figure, I’ve put the Y and Z labels inside the graph itself, each label nearest to the appropriate curve. The left (vertical) axis relates to both Y and Z.
Looking at Figure 1.3. it’s pretty clear that car brand Z. while costing more when new, depreciates faster than car brand Y. and after about 8 years, brand Z is actually worth less than brand Y.
Another popular style of graph is the “histogram.” The word histogram is derived from Greek roots and has to do with a drawing with information set upright. For practical purposes, it’s another way to show some graphical information.
Figure 1.4 shows the same data as Figure 1.1—that is, the data of Table 1.4 . As you can see, there is very little difference between these two figures. In the histogram version (Fig. 1.4), there is a solid bar reaching up from the horizontal axis to the place where Figure 1.4 had the diamond. Both graphs are read exactly the same way.
Figure 1.5 Histogram of two car depreciations.
Figure 1.5 shows the histogram version of Table 1.5 , where I am comparing two sets of data. Note that in this case, I am explaining which set of bars belongs to which variable by means of little squares, appropriately labeled. This type of histogram can be extended to three or more variables, but things start getting very crowded and hard to grasp.
The histograms I’ve just shown aren’t the only kinds of histograms; there are many variations on the theme. For the purposes of this book, however, these are enough.
The last type of graph that I want to show is the pie chart. Pie charts are typically used when you want to see how a total amount of some quantity is divided into pieces, or “slices of the pie.” A typical use is to show the major sections of an organization’s budget.
An example of a pie chart is shown in Figure 1.6. This is a hypothetical breakdown of a community’s total budget of $1,000,000. Each of the sections of the pie is labeled with the amount of money it represents and the legend on the right shows what this money is being spent on. The area of each pie slice is proportional to the fraction of the total that the slice represents as a fraction of the whole pie. In Figure 1.6, the salaries slice, representing $200,000, is twice as big as the capital slice, representing $100,000. As with the histogram, there are many variations of the pie chart, but Figure 1.6 is fairly representative. A pie chart can comfortably hold 8 or 10 categories. If you try to squeeze in many more than that, the chart starts getting too crowded to read.