In many cases public and nonprofi t managers can use actual numbers to measure phe- nomena: tons of garbage collected in a given town, number of arrests made by the police per week, response times in minutes of a local fi re department, number of chil- dren attending a daily church after-school program, miles driven in a week by Meals on Wheels volunteers, and so forth. Because this information consists of real numbers, it is possible to perform all types of arithmetic calculations with the data—addition, subtraction, multiplication, and division. As we will learn in Chapter 5 on measures of central tendency, when this is the case, we can readily compute mean or average scores, such as the mean or average number of tons of garbage collected per week, the average response time of the fi re department, and so on.
Unfortunately for public and nonprofi t administrators, available data are of- ten not measured in nearly as precise a fashion as are these variables. Th ere are several reasons for the lack of precision. In some cases, it is a refl ection of the state of the art of measurement. For instance, although it may be possible to say that a citizen is “very satisfi ed,” “satisfi ed,” “neutral,” “dissatisfi ed,” or “very dissatisfi ed”
with a new job training program contracted out to a nonprofi t agency, it usually is not possible to state that his or her level of satisfaction is exactly 2.3—or 5 or 9.856 or 1.003. Most measures of attitudes and opinions do not allow this level
of exactitude. In other instances, loss of precision results from errors in measure- ment or, perhaps, from lack of foresight. For example, you may be interested in the number of traffi c fatalities in the town of Berrysville over the past few years.
As a consequence of incomplete records or spotty reporting in the past, you may not be able to arrive at the exact number of fatalities in each of these years, but you may be quite confi dent in determining that there have been fewer fatalities this year than last year.
Finally, some variables inherently lack numerical precision: One could clas- sify the citizens of a community according to race (white, African American, His- panic, or other), gender (male, female), religion (Protestant, Catholic, Jewish, Buddhist, or other), and many other attributes. However, it would be futile to attempt to calculate the arithmetic average of race or religion, and it would be meaningless to say that a citizen is more female than male: A person is either one or the other.
In discussing these diff erent types of variables, social scientists usually refer to the concept of levels of measurement. Social scientists conventionally speak of three levels of measurement. Th e fi rst or highest (most precise) level is known as the interval level of measurement. Th e name derives from the fact that the measurement is based on a unit or interval that is accepted as a common stan- dard and that yields identical results in repeated applications. Weight is measured in pounds or grams, height in feet and inches, distance in miles, time in seconds or minutes, and so on. Th e variables discussed at the beginning of this section are all measured at the interval level: tons of garbage, number of arrests, response times in minutes. As a consequence of these standard units, it is possible to state not only that there were more arrests last week than this week but also that there were exactly 18 more arrests. (Some texts discuss a fourth level of measurement—
ratio—but for our purposes it is eff ectively the same as interval measurement.) Th e second level of measurement is called ordinal. At this level of measure- ment, it is possible to say that one unit or observation (or event or phenomenon) has more or less of a given characteristic than another, but it is not possible to say how much more or less. Generally, we lack an agreed-on standard or metric at this level of measurement. Almost all assessments of attitudes and opinions are at the ordinal level.
Consider the previous example, which focused on citizen satisfaction with a new job training program contracted out to a nonprofi t agency specializing in this function. At this writing, no one is quite sure how to measure satisfaction or how a unit of satisfaction may be defi ned. Nevertheless, an interviewer could be dispatched to the fi eld to ask participants, “How satisfi ed are you with the new job training program recently instituted in this community? Very satisfi ed, satis- fi ed, neutral, dissatisfi ed, or very dissatisfi ed?” To create an ordinal-level variable, we might attach numbers to the response categories for this survey question as a rank ordering. Th e numbered categories might look like those displayed in Table 2.1.
A participant who is “very satisfi ed” is assigned a score of “1.” A participant who is “satisfi ed” is assigned a score of “2,” and so on. Table 2.2 shows what the
satisfaction variable would look like if we entered data for a small sample of par- ticipants into a spreadsheet or statistical software package.
Of course, it would not be possible to ascertain from a citizen his or her exact numerical level of satisfaction (e.g., 4.37; 16.23). For example, if a “1” is assigned to the response of one participant and a “3” to another, the precise magnitude of diff erence between the participants cannot be determined (for more on why pre- cise distances across cases cannot be determined using nominal- and ordinal-level measures, see the section titled “Some Cautions” in Chapter 5). However, if one citizen answers that he is “very satisfi ed” with the program, it is safe to conclude that he is more satisfi ed than if he had stated that he was “satisfi ed” (or “neutral,”
“dissatisfi ed,” or “very dissatisfi ed”); similarly, a response of “very dissatisfi ed” in- dicates less satisfaction than one of “dissatisfi ed” (or “neutral,” “satisfi ed,” or “very satisfi ed”). How much more or less remains a mystery. As another example, a polling fi rm might ask a representative sample of citizens how good a job they believe the mayor is doing in running the city (very good, good, average, poor, or very poor) and to what extent they are interested in community aff airs (very interested, interested, neutral, uninterested, or very uninterested). Th ese variables are also ordinal-level and are subject to the same limitations as is the measure of satisfaction with the job training program.
Th e name ordinal measurement derives from the ordinal numbers: fi rst, sec- ond, third, and so on. Th ese numbers allow the ranking of a set of units or obser- vations (or events or phenomena) with respect to some characteristic, but they do not indicate the exact distances or diff erences between the objects. For example,
Table 2.1 An Ordinal Measure of the Concept “Satisfaction”
1 5 very satisfi ed 2 5 satisfi ed 3 5 neutral 4 5 dissatisfi ed 5 5 very dissatisfi ed
Table 2.2 An Example of an Ordinal Variable
Name Satisfaction
Jones 2
R. Smith 3
Franklin 1
Barnes 2
A. Smith 3
in an election, the order of fi nish of the candidates does not say anything about the number of votes each one received. Th e order of fi nish indicates only that the winner received more votes than did the runner-up, who in turn received more votes than the third-place fi nisher. By contrast, the exact vote totals of the can- didates are interval information that refl ects how many more or fewer votes each candidate received than the other candidates.
At the third level of measurement, one loses not only the ability to state exactly how much of a trait or characteristic an object or event possesses (in- terval measurement) but also the ability to state that it has more or less of the characteristic than has another object or event (ordinal measurement). In short, the n ominal level of measurement lacks any sense of relative size or magnitude:
It allows one to say only that things are the same or diff erent. Measurement not- withstanding, some of the most important variables in the social sciences are nominal. Th ese were mentioned before: race, gender, and religion. It is easy to ex- pand this list to public and nonprofi t management: occupation, type of housing, job classifi cation, sector of the economy, employment status. A nominal coding scheme for employee gender appears in Table 2.3.
If we entered data for a number of employees of a nonprofi t agency into a spreadsheet or statistical software package, the values for employee gender would look like those displayed in Table 2.4. In this example, employee Jones is female, whereas R. Smith is male.
Now that you have some idea of the three levels of measurement, write sev- eral examples of interval, ordinal, and nominal variables in the margins. After you have fi nished, fi ll in the level of measurement for each of the variables listed in Table 2.5.
Table 2.4 An Example of a Nominal Variable
Name Employee Gender
Jones 1
R. Smith 0
Franklin 1
Barnes 1
A. Smith 0
Table 2.3 A Nominal Measure of Employee Gender 1 5 female 0 5 male
Whether you were aware of it or not, if you are like most people, before you read this chapter, you probably assumed that measurement was easy and accurate: All you needed to do was to count whatever it is that interests you or rely on technology to do the same thing. From the number of dollars a person makes (income) to the number of people in your city (population) to the amount of degrees registered on a thermometer (temperature) to the number of miles recorded by your automobile odometer as you drive to and from work (commute mileage), measurement was straightforward and precise. In (management) practice, it simply is not that easy.
The Implications of Selecting a Particular Level of Measurement
When deciding what level of measurement to use, keep in mind that vari- ables originally coded at a higher level of measurement can be transformed into variables at lower levels of measurement. Th e opposite is generally not true. Variables
Table 2.5 Some Variables: What Is the Level of Measurement?
Variable Level of Measurement
1. Number of children
2. Opinion of the way the president is handling the economy (strongly approve; approve;
neutral; disapprove; strongly disapprove) 3. Age
4. State of residence
5. Mode of transportation to work
6. Perceived income (very low; below average;
average; above average; very high) 7. Income in dollars
8. Interest in statistics (low; medium; high) 9. Sector of economy in which you would like to
work (public; nonprofi t; private) 10. Hours of overtime per week
11. Your comprehension of this book (great;
adequate; forget it)
12. Number of memberships in clubs or associations
13. Dollars donated to nonprofi t organizations 14. Perceived success of animal rights association
in advocacy (very high; high; moderate;
low; very low)
15. Years of experience as a supervisor
16. Your evaluation of the level of “social capital”
of your community (very low; low;
moderate; high; very high)
originally coded at lower levels of measurement cannot be transformed into v ariables at higher levels of measurement. Th e reason is that higher levels of measurement include more information and more precise information than lower levels.
For example, let’s say that the director of a nonprofi t organization decides to collect data on the dollar amount of supplies and services spent on each client, to get a sense of how effi ciently services are provided. She can measure the cost per client at either the interval or the ordinal level. Th e director takes data for 10 recent clients and constructs both ordinal and interval variables, just to see what the diff erent data would look like. Th e results are displayed in Table 2.6.
If you were the director, how would you measure the cost variable?
Th ink about the advantages of measuring cost at the interval level. With the interval-level variable, we know exactly how much money was spent on each client. With the ordinal-level variable, we only know that a client falls into a particular cost range. If we construct the measure at the interval level, we can also determine the exact distance or numerical diff erence between individual cases. For example, it cost $19 more to serve the fi rst client than it did to serve the second client. In contrast, the ordinal-level measure would tell us only that the fi rst client was in the “moderate” cost range, whereas the second client was in the “low” cost range.
In terms of fl exibility, note that we can always recode or convert the interval- level version of the variable into the ordinal-level variable if we want or need to present the data diff erently at a later time. But what if we originally constructed the variable at the ordinal level and later found it necessary to report data on the actual dollar amount spent per client—for example, for an evaluation or grant proposal? We would be unable to obtain this information because ordinal-level data cannot be transformed into interval-level data. Although we would know that the cost per client was between $50 and $100 in four cases, we would be
Table 2.6 Different Levels of Measurement for the Same Concept Cost per Client in $ (interval) Cost per Client (ordinal)
57 2 where 1 5 low (less than $50)
38 1 2 5 moderate ($50 to $100)
79 2 3 5 high ($100 or more)
105 3
84 2
159 3
90 2
128 3
103 3
unable to determine the exact cost per client if our data were originally measured at the ordinal level.
As a public or nonprofi t manager, sometimes you will not be able to choose the level of measurement for the variables to be examined. Th is limitation often arises when working with data originally collected by another party, such as a government agency or private fi rm. In those cases where you are able to choose, keep in mind the many advantages of constructing variables at higher rather than lower levels of measurement.
In Parts V and VI of this book, you will see that levels of measurement have important implications for data analysis. If you plan on using a particu- lar statistical method to analyze data, you need to make sure to use an appro- priate level of measurement when constructing your variables. Otherwise you may spend a lot of time collecting data, only to fi nd that you are unable to an- alyze the data with the intended statistical technique. Contingency tables (the topic of Chapters 15–17) are used to analyze nominal- and ordinal-level data.
Regression analysis (covered in Chapters 18–23), one of the most commonly used techniques in statistical analysis, generally requires the use of interval- level data.
In public or nonprofi t administration, measurement can be a challenge.
In the fi rst place, as elaborated in the theory of measurement discussed ear- lier in the chapter, measurement often contains error. We do not have perfect measures of bureaucratic or organizational performance or employee morale or citizen satisfaction—or many of the other concepts that you may want to measure and use. Evaluating the quality and accuracy of measurement through validity and reliability assessment is desirable and frequently challenging. In the second place, the richness of public and nonprofi t administration calls for several diff erent types of measures. Th us, you will need to use subjective, objec- tive, and unobtrusive indicators. In addition, you will confront diff erent levels of measurement. Th e interval level corresponds most closely to our typical un- derstanding of measurement as counting things—for example, the number of hours of class attended in a week or the number of pages of reading required for an assignment is an interval measure. Th e other two levels of measurement, ordinal and nominal, do not allow such ready accounting. Nevertheless, you may have to assess work attitudes in your organization (ordinal) for an organi- zational development eff ort or identify fi ve areas of social need in your com- munity (nominal) for a grant proposal. Bringing the full menu of measurement concepts to bear on a problem or issue is a useful technique in public and non- profi t administration.