Levels of Measurement

Levels of Measurement
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Data can be classified according to levels of measurement. The level of measurement of the data dictates the calculations that can be done to summarize and present the data. It will also determine the statistical tests that should be performed. For example, there are six colors of candies in a bag of M&M’s. Suppose we assign brown a value of 1, yellow 2, blue 3, orange 4, green 5, and red 6. From a bag of candies, we add the assigned color values and divide by the number of candies and report that the mean color is 3.56. Does this mean that the average color is blue or orange? Of course not! As a second example, in a high school track meet there are eight competitors in the 400 meter run. We report the order of finish and that the mean finish is 4.5. What does the mean finish tell us? Nothing! In both of these instances, we have not properly used the level of measurement.

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There are actually four levels of measurement: nominal, ordinal, interval, and ratio. The lowest, or the most primitive, measurement is the nominal level. The highest, or the level that gives us the most information about the observation, is the ratio level of measurement.

Nominal-Level Data

Statistics in Action

Where did statistics get its start? In 1662 John Graunt published an article called “Natural and Political Observations Made upon Bills of Mortality.” The author’s “observations” were the result of a study and analysis of a weekly church publication called “Bill of Mortality,” which listed births, christenings, and deaths and their causes. Graunt realized that the Bills of Mortality represented only a fraction of all births and deaths in London. However, he used the data to reach broad conclusions about the impact of disease, such as the plague, on the general population. His logic is an example of statistical inference. His analysis and interpretation of the data are thought to mark the start of statistics.

For the nominal level of measurement observations of a qualitative variable can only be classified and counted. There is no particular order to the labels. The classification of the six colors of M&M’s milk chocolate candies is an example of the nominal level of measurement. We simply classify the candies by color. There is no natural order. That is, we could report the brown candies first, the orange first, or any of the colors first. Gender is another example of the nominal level of measurement. Suppose we count the number of students entering a football game with a student ID and report how many are men and how many are women. We could report either the men or the women first. For the nominal level the only measurement involved consists of counts. Table 1–1 shows a breakdown of the sources of the world oil supply. The variable of interest is the country or region. This is a nominal-level variable because we record the information by source of the oil supply and there is no natural order. Do not be distracted by the fact that we summarize the variable by reporting the number of barrels produced per day.

Table 1-1	Source of World Oil Supply for 2004.

Table 1–1 shows the essential feature of the nominal scale of measurement: There is no particular order to the categories.

In order to process data on oil production, gender, employment by industry, and so forth, the categories are often numerically coded 1, 2, 3, and so on, with 1 representing OPEC, 2 representing OECD, for example. This facilitates counting by the computer. However, because we have assigned numbers to the various categories, this does not give us license to manipulate the numbers. For example, 1 + 2 does not equal 3, that is, OPEC + OEDC does not equal former U.S.S.R. To summarize, the nominal-level data have the following properties:

Data categories are represented by labels or names.
Even when the labels are numerically coded, the data categories have no logical order.

Ordinal-Level Data

The next higher level of data is the ordinal level. Table 1–2 lists the student ratings of Professor James Brunner in an Introduction to Finance course. Each student in the class answered the question “Overall, how did you rate the instructor in this class?” The variable rating illustrates the use of the ordinal scale of measurement. One classification is “higher” or “better” than the next one. That is, “Superior” is better than “Good,” “Good” is better than “Average,” and so on. However, we are not able to distinguish the magnitude of the differences between groups. Is the difference between “Superior” and “Good” the same as the difference between “Poor” and “Inferior”? We cannot tell. If we substitute a 5 for “Superior” and a 4 for “Good,” we can conclude that the rating of “Superior” is better than the rating of “Good,” but we cannot add a ranking of “Superior” and a ranking of “Good,” with the result being meaningful. Further we cannot conclude that a rating of “Good” (rating is 4) is necessarily twice as high as a “Poor” (rating is 2). We can only conclude that a rating of “Good” is better than a rating of “Poor.” We cannot conclude how much better the rating is.

Table 1-2	Rating of a Finance Professor

Another example of ordinal-level data is the Homeland Security Advisory System. The Department of Homeland Security publishes this information regarding the risk of terrorist activity to federal, state, and local authorities and to the American people. The five risk levels from lowest to highest including a description and color codes are shown to the left.

This is an example of the ordinal scale because we know the order or the ranks of the risk levels—that is, orange is higher than yellow—but the amount of the difference in risk is not necessarily the same. To put it another way, the difference in the risk level between yellow and orange is not necessarily the same as between green and blue. You can check the current risk level and read more about the various levels by going to: www.whitehouse.gov/homeland.

In summary the properties of the ordinal level of data are:

Data classifications are represented by sets of labels or names (high, medium, low) that have relative values.
Because of the relative values, the data classified can be ranked or ordered.

Interval-Level Data

The interval level of measurement is the next highest level. It includes all the characteristics of the ordinal level, but, in addition, the difference between values is a constant size. An example of the interval level of measurement is temperature. Suppose the high temperatures on three consecutive winter days in Boston are 28, 31, and 20 degrees Fahrenheit. These temperatures can be easily ranked, but we can also determine the difference between temperatures. This is possible because 1 degree Fahrenheit represents a constant unit of measurement. Equal differences between two temperatures are the same, regardless of their position on the scale. That is, the difference between 10 degrees Fahrenheit and 15 degrees is 5, the difference between 50 and 55 degrees is also 5 degrees. It is also important to note that 0 is just a point on the scale. It does not represent the absence of the condition. Zero degrees Fahrenheit does not represent the absence of heat, just that it is cold! In fact 0 degrees Fahrenheit is about–18 degrees on the Celsius scale.

Another example of the interval scale of measurement is women’s dress sizes. Listed below is information on several dimensions of a standard U.S. women’s dress.

Why is the “size” scale an interval measurement? Observe as the size changes by 2 units (say from size 10 to size 12 or from size 24 to size 26) each of the measurements increases by 2 inches. To put it another way the intervals are the same.

There is no natural zero point for dress size. A “size 0” dress does not have “zero” material. Instead it would have a 24-inch bust, 16-inch waist and 27-inch hips. Moreover, the ratios are not reasonable. If you divide a size 28 by a size 14, you do not get the same answer as dividing a size 20 by 10. Neither ratio is equal to two as the “size” number would suggest. In short, if the distances between the numbers make sense, but the ratios do not, then you have an interval scale of measurement.

The properties of the interval-level data are:

Data classifications are ordered according to the amount of the characteristic they possess.
Equal differences in the characteristic are represented by equal differences in the measurements.

Ratio-Level Data

Practically all quantitative data is recorded on the ratio level of measurement. The ratio level is the “highest” level of measurement. It has all the characteristics of the interval level, but in addition, the 0 point is meaningful and the ratio between two numbers is meaningful. Examples of the ratio scale of measurement include wages, units of production, weight, changes in stock prices, distance between branch offices, and height. Money is a good illustration. If you have zero dollars, then you have no money. Weight is another example. If the dial on the scale of a correctly calibrated device is at 0, then there is a complete absence of weight. The ratio of two numbers is also meaningful. If Jim earns $40,000 per year selling insurance and Rob earns $80,000 per year selling cars, then Rob earns twice as much as Jim.

Table 1–3 illustrates the use of the ratio scale of measurement. It shows the incomes of four father-and-son combinations.

Table 1-3	Father–Son Income Combinations

Observe that the senior Lahey earns twice as much as his son. In the Rho family the son makes twice as much as the father.

In summary, the properties of the ratio-level data are:

Data classifications are ordered according to the amount of the characteristics they possess.
Equal differences in the characteristic are represented by equal differences in the numbers assigned to the classifications.
The zero point is the absence of the characteristic and the ratio between two numbers is meaningful.

Chart 1–3 summarizes the major characteristics of the various levels of measurement.

CHART 1-3	Summary of the Characteristics for Levels of Measurement

Self-Review 1–2

Click here for Solution

What is the level of measurement reflected by the following data?

The age of each person in a sample of 50 adults who listen to one of the 1,230 talk radio stations in the United States is:
In a survey of 200 luxury-car owners, 100 were from California, 50 from New York, 30 from Illinois, and 20 from Ohio.

Exercises

The answers to the odd-numbered exercises are at the end of the book.

1. What is the level of measurement for each of the following variables?

Student IQ ratings.
Distance students travel to class.
Student scores on the first statistics test.
A classification of students by state of birth.
A ranking of students as freshman, sophomore, junior, and senior.
Number of hours students study per week.

2. What is the level of measurement for these items related to the newspaper business?

The number of papers sold each Sunday during 2006.
The departments, such as editorial, advertising, sports, etc.
A summary of the number of papers sold by county.
The number of years with the paper for each employee.

3. Look in the latest edition of USA Today or your local newspaper and find examples of each level of measurement. Write a brief memo summarizing your findings.

4. For each of the following, determine whether the group is a sample or a population.

The participants in a study of a new cholesterol drug.
The drivers who received a speeding ticket in Kansas City last month.
Those on welfare in Cook County (Chicago), Illinois.
The 30 stocks reported as a part of the Dow Jones Industrial Average.

Chapter Exercises 10

Exercises 1, 2, 3, 4

exercises.com 15

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Home > Chapter 1 > Levels of Measurement

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	Course-wide Content
		Data Sets Tables ALEKS Interactive Business S... Visual Statistics Business Statistics Ce... Updates and Errata Appendix A Appendix B MH Enhanced Cartridge

	Instructor Resources
		Solutions Manual Solutions to Study Gui...
	News, Articles & Links
		Web Links
	Quizzes
		Multiple Choice Quiz
	More Resources
		Chapter Outline Glossary PowerPoint Presentations Assignments iPod® Content
	eBook
		Chapter Opener Introduction Why Study Statistics? What is Meant by Stati... Types of Statistics Types of Variables Levels of Measurement Ethics and Statistics Computer Applications Chapter Summary Chapter Exercises exercises.com Data Set Exercises Chapter 1 Answers to S...