Now, observe again the 10 digits we use to represent numbers Notice that not only is each digit different from every other

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1 VARIABLES- NOMINAL, ORDINAL and INTERVAL/SCALE LEVELS OF MEASUREMENT Variables: traits or characteristics that vary from one individual, group, or society to another individual, group, or society. Examples: 1) An individual s age, sex, religion. 2) A family s socio-economic status (SES). 3) A group s size (number of members), cohesiveness (feelings of belonging together), duration (length of time it has existed). 3) A society s crime rate, population density, demographic composition. Variables are what we study in data analysis. While some variables may be qualitative in nature, in this course we will concentrate on quantitative datathat is, variables which are represented or expressed using numbers. We begin with the recognition that not all quantitative representations or expressions of variables have the same numerical properties.

2 Observe the 10 digits we use to represent numbers Notice that each digit is different from every other digit. In measuring certain variables, we use these digits to represent different individuals or different categories of people. Consider this example from a questionnaire distributed to a sample of respondents. Sex: 1) Male 2) Female It should be clear from this example that the numerals 1 and 2 only represent different categories of gender. We could just as easily have used letters (M,F; A; B; etc.) to represent these categories. In this instance, the numerals 1 and 2 are simply alternative ways of specifying the NAMES of the categories- just as your Social Security number is an alternative to identifying your name. (The same is true of your 855 number here at WPU.) Other examples are questions (on a questionnaire) that provide either/or answers, such as: Do you support stricter gun control legislation in the United States? 1) Yes 2) No Numbers that are unique to individuals- no one else has your Social Security or 855 numbersare also examples of this level of measurement. Social science researchers refer to variables such as these (and many, many others) as NOMINAL variables because they name a category of respondents.

3 Now, observe again the 10 digits we use to represent numbers Notice that not only is each digit different from every other digit, there is also a rank ordering of the digits. That is, 7 is greater than 6 which in turn is greater than 5 which is then greater than 4, etc. In this instance, the numbers are used not only to designate different individuals or categories, but also to place them in an ordered arrangement, typically from highest- first place- to lowest- last place. One way of understanding this use of numbers is to think of a horse race like the Kentucky Derby with a field of 12 horses entered (actually, any type of race will serve in this analogy). The winner finishes in first (1) place, the runner-up finishes in second (2) place, next comes third (3) place, fourth (4) place, fifth (5) place, and so on down to the twelfth (12) or final finisher. The use of numbers here represents the order in which the horses finish the race; for this use of numbers we use the term ORDINAL measurement. Here is an example from a (fairly) typical questionnaire used by students in sociology at William Paterson University. Year in college: 1. Freshman 2. Sophomore 3. Junior 4. Senior 5. Other

4 Students responding to this item would be coded as follows. 1 indicates the student is a freshman (first year of college); 2 indicates the student is a sophomore (second year of college); 3 indicates the student is a junior (third year of college); 4 indicates the student is a senior (fourth year of college); 5 indicates the student belongs to some alternative year (perhaps a second degree student. Other examples of ordinal measures are: a) Rating scales, such as asking respondents to indicate how favorably they rate Pres. Obama after his first term in office; b) Rankings, such as U.S. News and World Reports rankings of American colleges and universities; c) Birth order, such as asking respondents if they are the oldest child (first born), a middle child, or youngest child in a family.

5 Now, let s make one final observation of the10 digits we use to represent numbers Now, note that not only are all the digits different and in order, there are also equal intervals of difference between them. That is, 9 is greater than 8 by the same amount (1) that 8 is greater than 7 (1) which in turn is greater than 6 by the same amount (1), etc. Each digit is 1 interval distant from the digit(s) adjacent to it. It is the property of equal intervals between numbers than makes it possible for us to do mathematical calculations- such as computing an average- on the numbers. Here are some examples from sociology. 1. Age 2. Income 3. Number of brothers and sisters Notice that, in all of these instances, a respondent can answer by writing a number in the space provided. All 3 variables are measured in equal units: year in age- a 20 year old individual is as much older than someone 19 as a 65 year old is older than a 64 year old individual (each year is the same length); dollar of income- every dollar is worth 100 cents (each penny has the same value); siblings in number of brothers and sisters- an individual with 1 brother and 1 sister has 1 more sibling than someone with 1 sister (each brother or sister counts as 1 sibling).

6 Quantitative variables such as these (and many, many others), are referred to by social science researchers as Interval or SCALE variables. (Scale is the term used in SPSS to refer to these types of variables.) If a variable is the result of a calculation, such as a ratio, rate, or percentage, it is a variable measured at the interval level. Here are some commonly used examples from social science. 1) The sex ratio of a population: a demographic concept that measures the proportion of males to females in a given population. It is usually expressed as the number of males per 100 females. The ratio is stated in the form of 105:100, where in this example there would be 105 males for every 100 females in a population. Note that this ratio is based on the notion that each male counts as 1 male and each female counts as 1 female; the numbers of males and females can be added to get the total numbers of males and females in a society. 2) The crime rate of an area, such as a (US) city or state: this is calculated by the FBI as the number of crimes reported to police in a specific area (city or state) divided by the total population of that area; the result of this division is then multuplied by 100,000 to provide a standard measure of criminal activity across cities and/or states. Every crime counts equally- that is why it is possible to add the total number of all crimes; similarly, in the population, every individual counts equally. Each term in this computation is based on equal units (intervals) of measurement.

7 3) A final way to determine if a variable is measured at the interval level is to ask a simple question: Presented with a series of numbers, does it make sense to add these numbers up and divide the sum by the number of numbers to find the arithmetic average of the numbers? If it doesn t make (common) sense to find the average of a set of numbers, then the variable is NOT measured at the interval level. Think: Does it make sense to compute the average Social Security number of a group of people? Does it make sense to compute the average cell phone number of a group? If computing an average violates this common sense principle, then the variable in question is NOT measured at an interval level. [Most likely, it s a nominal level variable.] There are also some guidelines you can use to determine if a variable is measured at the NOMINAL level. Recall that the root of the term nominal is the Latin word- nomen, meaning name. A variable measured at the nominal level uses numerals as an alternative way of designating the name of some person (855 number), some category (1. male 2. female), some object ( names a telephone device in the northern New Jersey area), or some other item of interest in research is the ZIP code for Wayne, NJ. These last 2 examples provide a clue to Determining if a variable is measured at the nominal level.

8 1) Notice that each cell phone number, as well as each ZIP code, is unique- no other phone device has that number; no other geographic area has that ZIP code. In general, if a number is unique in that it names one and only one person, category, or other characteristic/trait, that variable is measured at the nominal level. 2) Some numbers have letters in them- your driver s license number begins with the first letter of your last name. This should be an obvious clue that any number which includes a letter in it is not a number, but a designator or namer. Hence, the variable is measured at the nominal level. [It should be equally obvious that you cannot calculate an average of driver s license numbers, or any other number with a letter in it; at the least, the presence of a letter in a number should be a tipoff that this is NOT an interval variable.