Scales of Measurement
There’s a difference in how weight and sex are measured and reported in Figure 1.2 that is fundamental to all statistical analysis—and to how you bring Excel’s tools to bear on the numbers. The difference concerns scales of measurement.
In Figures 1.2 and 1.3, the variable Sex is measured using a category scale, often called a nominal scale. Different values in a category variable merely represent different groups, and there’s nothing intrinsic to the categories that does anything but identify them. If you throw out the psychological and cultural connotations that we pile onto labels, there’s nothing about Male and Female that would lead you to put one on the left and the other on the right in Figure 1.3’s pivot chart, the way you’d put June to the left of July.
Another example: Suppose that you want to chart the annual sales of Ford, General Motors, and Toyota cars. There is no order that’s necessarily implied by the names themselves: They’re just categories. This is reflected in the way that Excel might chart that data (see Figure 1.4).
Figure 1.4 Excel’s Column charts always show categories on the horizontal axis and numeric values on the vertical axis.
Notice these two aspects of the car manufacturer categories in Figure 1.4:
- Adjacent categories are equidistant from one another. No additional information is supplied by the distance of GM from Toyota, or Toyota from Ford.
- The chart conveys no information through the order in which the manufacturers appear on the horizontal axis. There’s no implication that GM has less “car-ness” than Toyota, or Toyota less than Ford. You could arrange them in alphabetical order if you wanted, or in order of number of vehicles produced, but there’s nothing intrinsic to the scale of manufacturers’ names that suggests any rank order.
In contrast, the vertical axis in the chart shown in Figure 1.4 is what Excel terms a value axis. It represents numeric values.
Notice in Figure 1.4 that a position on the vertical, value axis conveys real quantitative information: the more vehicles produced, the taller the column. The vertical and the horizontal axes in Excel’s Column charts differ in several ways, but the most crucial is that the vertical axis represents numeric quantities, while the horizontal axis simply indicates the existence of categories.
In general, Excel charts put the names of groups, categories, products, or any other designation on a category axis and the numeric value of each category on the value axis. But the category axis isn’t always the horizontal axis (see Figure 1.5).
Figure 1.5 In contrast to Column charts, Excel’s Bar charts always show categories on the vertical axis and numeric values on the horizontal axis.
The Bar chart provides precisely the same information as does the Column chart. It just rotates this information by 90 degrees, putting the categories on the vertical axis and the numeric values on the horizontal axis.
I’m not belaboring the issue of measurement scales just to make a point about Excel charts. When you do statistical analysis, you choose a technique based in large part on the sort of question you’re asking. In turn, the way you ask your question depends in part on the scale of measurement you use for the variable you’re interested in.
For example, if you’re trying to investigate life expectancy in men and women, it’s pretty basic to ask questions such as, “What is the average life span of males? of females?” You’re examining two variables: sex and age. One of them is a category variable, and the other is a numeric variable. (As you’ll see in later chapters, if you are generalizing from a sample of men and women to a population, the fact that you’re working with a category variable and a numeric variable might steer you toward what’s called a t-test.)
In Figures 1.3 through 1.5, you see that numeric summaries—average and sum—are compared across different groups. That sort of comparison forms one of the major types of statistical analysis. If you design your samples properly, you can then ask and answer questions such as these:
- Are men and women paid differently for comparable work? Compare the average salaries of men and women who hold similar jobs.
- Is a new medication more effective than a placebo at treating a particular disease? Compare, say, average blood pressure for those taking an alpha blocker with that of those taking a sugar pill.
- Do Republicans and Democrats have different attitudes toward a given political issue? Ask a random sample of people their party affiliation, and then ask them to rate a given issue or candidate on a numeric scale.
Notice that each of these questions can be answered by comparing a numeric variable across different categories of interest.
Although there is only one type of category scale, there are three types of numeric scales: ordinal, interval, and ratio. You can use the value axis of any Excel chart to represent any type of numeric scale, and you often find yourself analyzing one numeric variable, regardless of type, in terms of another variable. Briefly, the numeric scale types are as follows:
- Ordinal scales are often rankings, and tell you who finished first, second, third, and so on. These rankings tell you who came out ahead, but not how far ahead, and often you don’t care about that. Suppose that in a qualifying race Jane ran 100 meters in 10.54 seconds, Mary in 10.83 seconds, and Ellen in 10.84 seconds. Because it’s a preliminary heat, you might care only about their order of finish, and not about how fast each woman ran. Therefore, you might convert the time measurements to order of finish (1, 2 and 3), and then discard the timings themselves. Ordinal scales are sometimes used in a branch of statistics called nonparametrics but are used infrequently in the parametric analyses discussed in this book.
- Interval scales indicate differences in measures such as temperature and elapsed time. If the high temperature Fahrenheit on July 1 is 100 degrees, 101 degrees on July 2, and 102 degrees on July 3, you know that each day is one degree hotter than the previous day. So, an interval scale conveys more information than an ordinal scale. You know, from the order of finish on an ordinal scale, that in the qualifying race Jane ran faster than Mary and Mary ran faster than Ellen, but the rankings by themselves don’t tell you how much faster. It takes elapsed time, an interval scale, to tell you that.
- Ratio scales are similar to interval scales, but they have a true zero point, one at which there is a complete absence of some quantity. The Celsius temperature scale has a zero point, but it doesn’t indicate a complete absence of heat, just that water freezes there. Therefore, 10 degrees Celsius is not twice as warm as 5 degrees Celsius, so Celsius is not a ratio scale. Degrees kelvin does have a true zero point, one at which there is no molecular motion and therefore no heat. Kelvin is a ratio scale, and 100 degrees kelvin is twice as warm as 50 degrees kelvin. Other familiar ratio scales are height and weight.
It’s worth noting that converting between interval (or ratio) and ordinal measurement is a one-way process. If you know how many seconds it takes three people to run 100 meters, you have measures on a ratio scale that you can convert to an ordinal scale—gold, silver, and bronze medals. You can’t go the other way, though: If you know who won each medal, you’re still in the dark as to whether the bronze medal was won with a time of 10 seconds or 10 minutes.
Telling an Interval Value from a Text Value
Excel has an astonishingly broad scope, and not only in statistical analysis. As much skill as has been built in to it, though, it can’t quite read your mind. It doesn’t know, for example, whether the 1, 2, and 3 you just entered into a worksheet’s cells represent the number of teaspoons of olive oil you use in three different recipes or 1st, 2nd, and 3rd place in a political primary. In the first case, you meant to indicate liquid measures on an interval scale. In the second case, you meant to enter the first three places in an ordinal scale. But they both look alike to Excel.
Text is a different matter. You might use the letters A, B and C to name three different groups, and in that case you’re using text values on a nominal, category scale. You can also use numbers: 1, 2 and 3 to represent the same three groups. But if you use a number as a nominal value, it’s a good idea to store it in the worksheet as a text value. For example, one way to store the number 2 as a text value in a worksheet cell is to precede it with an apostrophe: '2. (You’ll see the apostrophe in the formula box but not in the cell.)
On a chart, Excel has some complicated decision rules that it uses to determine whether a number is only a number. (Excel 2013 has some additional tools to help you participate in the decision-making process, as you’ll see later in this chapter). Some of those rules concern the type of chart you request. For example, if you request a Line chart, Excel treats numbers on the horizontal axis as though they were nominal, text values. But if instead you request an XY chart using the same data, Excel treats the numbers on the horizontal axis as values on an interval scale. You’ll see more about this in the next section.
So, as disquieting as it may sound, a number in Excel may be treated as a number in one context and not in another. Excel’s rules are pretty reasonable, though, and if you give them a little thought when you see their results, you’ll find that they make good sense.
If Excel’s rules don’t do the job for you in a particular instance, you can provide an assist. Figure 1.6 shows an example.
Figure 1.6 You don’t have data for all the months in the year.
Suppose that you run a business that operates only when public schools are in session, and you collect revenues during all months except June, July and August. Figure 1.6 shows that Excel interprets dates as categories—but only if they are entered as text, as they are in the figure. Notice these two aspects of the worksheet and chart in Figure 1.6:
- The dates are entered in the worksheet cells A2:A10 as text values. One way to tell is to look in the formula box, just to the right of the fx symbol, where you see the text value January.
- Because they are text values, Excel has no way of knowing that you mean them to represent dates, and so it treats them as simple categories—just like it does for GM, Ford, and Toyota. Excel charts the dates-as-text accordingly, with equal distances between them: May is as far from April as it is from September.
- You can see in the formula box that it’s an actual date, not just the name of a month, in cell A2, and the same is true for the values in cells A3:A10.
The Excel chart automatically responds to the type of values you have supplied in the worksheet. The program recognizes that the numbers entered represent monthly intervals and, although there is no data for June through August, the chart leaves places for where the data would appear if it were available. Because the horizontal axis now represents a numeric scale, not simple categories, it faithfully reflects the fact that in the calendar, May is four times as far from September as it is from April.
Figure 1.7 The horizontal axis accounts for the missing months.